http://www.csie.dyu.edu.tw/~spring/Exercises/search1_utf8.phpfile=index_utf8.xml題庫科目相關來源日期題目類型題目內容解答備註formal languagescontext-free languagescontext-free grammar賴映潔Tue May 12 8:36:21 2009設計grammar5.1.8 (g) Find context-free grammars for L = {anbmck:k ≠ n + m},n≥ 0, m ≥ 0, k ≥ 0.formal languages_60213659.odp習題5.1.8(g)formal languagescontext-free languagescontext-free grammarspringFri Apr 10 10:26:13 2009設計grammar5.1.7 (c) find context-free grammars for the following languages
(c) L = anbm : n ≠ 2m} (n ≥ 0, m ≥ 0)formal languages_57455451.odp習題 5.1.7(c)algorithmsbinary search treeF9506037Sat Mar 14 9:44:53 2009證明12.3-5 Is the operation of deletion \"commutative\" in
the sense that deleting x and the y from a binary
search tree leaves the same tree as deleting y
and then x ? Argue why it is or give a
counterexample.algorithms_55120171.odp習題12.3-5algorithmsbinary search treeF9506049Sat Mar 14 9:28:53 2009證明12.1-5 Argue that since sorting n elements takes Ω(n lg n) time in the worst case in the comparison model, any comparison-based algorithm for constructing a binary search tree from an arbitrary list of n elements takes Ω(n lg n) time in the worst case.algorithms_55119211.odp習題12.1-5algorithmsHash Tablesopen addressingF9506237Sat Mar 14 9:24:40 2009計算11.4-4 Consider an open-address hash table with uniform hashing. Give upper bounds on the expected number of probes in an unsuccessful search and on the expected number of probes in a successful search when the load factor is 3/4 and when it is 7/8.algorithms_55118958.odp習題11.4-4algorithmsbinary search treered-black treesF9506219Sat Mar 14 9:20:51 2009問答13.3-1 In line 16 of RB-INSERT, we set the color of the newly inserted node z to red. Notice that if we had chosen to set z\'s color to black, then property 4 of a red-black tree would not be violated. Why didn\'t we choose to set z\'s color to black?algorithms_55118729.odp習題13.3-1algorithmsbinary search treered-black treesF9506011Sat Mar 14 9:18:12 2009證明13.1-3 Let us define a relaxed red-black tree as a binary
search tree that satisfies red-black properties 1, 3, 4,
and 5. In other words, the root may be either red or
black. Consider a relaxed red-black tree T whose root
is red. If we color the root of T black but make no
other changes to T, is the resulting tree a red-black
tree?algorithms_55118570.odp習題13.1-3algorithmsstackqueueF9506055Sat Mar 14 9:13:35 2009設計演算法10.1-7 Show how to implement a stack using two queues. Analyze the running time of the stack operations.algorithms_55118293.odp習題10.1-7algorithmslinked list李育德 賴宗佑 林哲賢 陳炯桓Sat Mar 14 9:01:40 2009基本性質Problem 10-1 Comparisons among lists
For each of the four types of lists in the following table, what is the asymptotic worst-case running time for each dynamic-set operation listed?algorithms_55117577.odpProblem 10-1algorithmsred-black treesbinary search tree曾一平Sat Mar 14 8:55:55 2009基本性質13.2-3 Let a, b, and c be arbitrary nodes in subtrees α,
β, and γ, respectively, in the left tree of Figure 13.2.
How do the depths of a, b, and c change when a left
rotation is performed on nodes x in the figure?algorithms_55117233.odp習題13.2-3algorithmsHash TablesF9506246Sat Mar 14 8:52:37 2009執行演算法11.2-2 Demonstrate the insertion of the
keys 5, 28, 19, 15, 20, 33, 12, 17, 10 into
a hash table with collisions resolved by
chaining. Let the table have 9 slots, and
let the hash function be h(k) = k mod 9algorithms_55117035.odp習題11.2-2algorithmscomplexitybig-OspringSat Mar 7 11:28:23 2009證明3.1-4 Is 2n+1 = O(2n)? Is 22n = O(2n)?
(請證明您的答案)algorithms_54521581.odp習題3.1-4formal languagescontext-free languagescontext-free grammarGreibach normal formf9306023Fri Feb 27 16:07:20 2009設計grammarA6-2 Find a Greibach normal-form grammar equivalent tothe following CFG:
S ->AB, A -> BS | b, B -> SA |aformal languages_53847118.odp習題A6-2formal languagescontext-free languagescontext-free grammar胡偉諄Fri Feb 27 16:04:50 2009設計grammarA5-3 Construct a context-free grammar that generates L = {xcn | x ∈ {a, b}*, na(x) + nb(x) ≥ n}.formal languages_53846968.odp習題A5-3formal languagesset洪喜音Fri Feb 27 15:49:21 2009證明A1-1. Let L1 and L2 be two languages and |L1| = m, |L2| = n. Is it true that |L1L2| = mn for all L1, L2?formal languages_53846039.odp習題A1-1formal languagescontext-free languagescontext-free grammartransforming grammars陳羿亘Fri Feb 27 12:39:36 2009設計grammar6.1.24 Use the result of the preceding exercise to rewrite
the grammar
A → Aa | aBc | λ
B → Bb | bc
So that it no longer has productions of the form A →Ax or B→Bxformal languages_53834654.odp習題6.1.24formal languagescontext-free languagescontext-free grammarChomsky normal formF9406228Fri Feb 27 12:35:04 2009設計grammar6.2.3 Transform the grammar S -> aSaA | A , A->abA|b into Chomsky normal form.formal languages_53834382.odp習題6.2.3formal languagescontext-free languagescontext-free grammarChomsky normal form朱致宏Fri Feb 27 12:27:15 2009設計grammar6.2.2 Convert the grammar S → aSb | ab into Chomsky
normal form.formal languages_53833913.odp習題6.2.2formal languagescontext-free languagescontext-free grammartransforming grammars賴映潔Fri Feb 27 12:25:16 2009證明6.1.16 Let G be a grammar without λ-production, but possibly with some unit-productions. Show that the construction of Theorem 6.4 does not then introduce any λ-productions.formal languages_53833794.odp習題6.1.16formal languagescontext-free languagespushdown automataF9406235Fri Feb 27 12:19:18 2009設計automaton7.1.4 (i) Construct an npda that accepts the language L = {w : na(w) + nb(w) = nc(w)} on = {a, b, c}.formal languages_53833436.odp習題7.1.4(i)formal languagescontext-free languagescontext-free grammarF9206101Fri Feb 27 12:16:28 2009設計grammar5.1.18 Show that the language L = { w1cw2 : w1,w2 ∈{a,b}+,
w1 ≠ w2R} , with Σ = { a,b,c }, is context-free.formal languages_53833266.odp習題5.1.18formal languagescontext-free languagescontext-free grammarambiguous朱致宏Fri Feb 27 12:08:54 2009證明5.2.13 Show that the following grammar is ambiguous.
S → aSbS | bSaS | λformal languages_53832812.odp習題5.2.13formal languagescontext-free languagescontext-free grammarf9306023Fri Feb 27 12:06:22 2009設計grammar5.1.15 Show that the following language is context-free. L ={uvwvR : u, v, w ∈ {a, b}+, |u| = |w| = 2}.formal languages_53832660.odp習題5.1.15formal languagescontext-free languagescontext-free grammarGreibach normal form林祐誼Fri Feb 27 11:18:46 2009設計grammar6.2.13 Convert the grammar
S → Abb | a,
A → aaA | B,
B → bAb
Into Greibach normal form.formal languages_53829803.odp習題6.2.13formal languagescontext-free languagescontext-free grammarf9406228Fri Feb 27 11:15:52 2009設計grammar5.1.8(c) Find context-free grammars for L = {anbmck: k = n + m}, n ≥ 0, m ≥ 0, k ≥ 0.formal languages_53829629.odp習題5.1.8(c)formal languagescontext-free languagescontext-free grammartransforming grammars胡偉諄Fri Feb 27 11:14:00 2009設計grammar6.1.6 Eliminate useless productions from
S → a | aA | B | C,
A → aB | λ,
B → Aa,
C → cCD,
D → ddd.formal languages_53829518.odp習題6.1.6formal languagescontext-free languagescontext-free grammarF9406234Fri Feb 27 11:11:50 2009設計grammar5.1.13 (c) Let L = {anbn: n ≥ 0}. Show that the complement of L and L* are context-free.formal languages_53829388.odp習題5.1.13(c)formal languagescontext-free languagescontext-free grammarGreibach normal formF9206101Fri Feb 27 11:04:44 2009設計grammar6.2.10 Convert the grammar S → aSb | bSa | a | b into
Greibach normal form.formal languages_53828962.odp習題6.2.10formal languagescontext-free languagescontext-free grammartransforming grammars洪喜音Fri Feb 27 11:02:16 2009描述語言6.1.5 Eliminate all useless productions from the grammar
S → aS | Ab,
A →bA,
B →AA.
What language does this grammar generate?formal languages_53828814.odp習題6.1.5formal languagesregular languagespumping lemmaF9306038Fri Feb 27 10:59:24 2009證明4.3.7 Show that the language L =
{anbn: n ≥ 0} ∪ {anbn+1: n ≥ 0} ∪
{anbn+2: n ≥ 0}is not regular.formal languages_53828641.pdf習題4.3.7formal languagesregular languagespumping lemma曾愷銘Fri Feb 27 10:56:52 2009證明4.3.4 f) Prove that L= {ww: w Î{a, b}*} is not regular.formal languages_53828490.odp習題4.3.4(f)formal languagesregular languagesF9406207Fri Feb 27 10:52:55 2009證明4.3.6 a) Determine whether or not L = {anbn: n ≥ 1} ∪ {anbm : n ≥ 1, m ≥ 1} is regular.formal languages_53828253.odp習題4.3.6(a)formal languagesregular languagespumping lemma林祐誼Fri Feb 27 10:50:33 2009證明4.3.15g) L={anbl : | n-l | = 2}是Regular Language嗎?
請證明。formal languages_53828111.odp習題4.3.15(g)formal languagesregular languagesnfaf9306023Fri Feb 27 10:47:26 2009設計automaton4.1.2 b) Find the nfa that accept L(ab*a*) ∩ L(a*b*a)formal languages_53827924.odp習題4.1.2(b)formal languagesregular languagespumping lemma林柏存Fri Feb 27 10:43:52 2009證明4.3.4 g) Prove that L= {w w w w r : w ∈ {a, b}*} is not regular.formal languages_53827710.odp習題4.3.4(g)formal languagescontext-free languagescontext-free grammar蔡學仁Fri Feb 27 10:40:14 2009設計grammar5.1.11 Find a context-free grammar for Σ = {a, b} for the languageL = {anwwRbn: w ∈ Σ*, n ≥ 1}formal languages_53827492.odp習題5.1.11formal languagescontext-free languagescontext-free grammar陳羿亘Fri Feb 27 10:26:13 2009設計grammar5.1.8 (b) Find context-free grammars for L = {anbmck: n = m or m ≠ k}, n ≥ 0, m ≥ 0, k ≥ 0.formal languages_53826651.odp習題5.1.8(b)formal languagescontext-free languagess-grammar洪喜音Fri Feb 27 8:59:11 2009設計grammar5.2.1 Find an s-grammar for L(aaa*b+b).formal languages_53821429.odp習題5.2.1formal languagesregular languages胡偉諄Fri Feb 27 8:54:15 2009設計演算法4.2.7 Exhibit an algorithm that, given any three regular languages,L,L1,L2,determines whether or not L=L1L2.formal languages_53821133.odp習題4.2.7algorithmssortheap sortF9406009Thu Feb 26 17:51:18 2009問答6.2-4 What is the effect of calling Max-Heapify(A, i) for i> heap-size[A]/2?algorithms_53766956.odp習題6.2-4algorithmssortf9406241Thu Feb 26 17:48:02 2009基本性質8.3-2 Which of the following sorting algorithms are stable: insertion sort, merge sort, heapsort, and quicksort? Give a simple scheme that makes any sorting algorithm stable. How much additional time and space does your scheme entail?algorithms_53766760.odt習題8.3-2algorithmssortbucket-sortf9506803Thu Feb 26 17:42:52 2009執行演算法8.4-1 Using Figure 8.4 as a model, illustrate the operation of Bucket-Sort on the array A = 〈.79, .13, .16, .64, .39, .20, .89, .53, .71, .42〉.algorithms_53766450.odp習題8.4-1algorithmssortlower bound林煥傑Thu Feb 26 17:38:59 2009證明9.1-1 Show that the second smallest of n elements can be found with n + ⌊lg n – 2⌋ comparisons in the worst case. (Hint: Also find the smallest element.)algorithms_53766217.odp習題9.1-1algorithmssortheap sort桂子龍Thu Feb 26 17:34:36 2009證明6.1-2 Show that an n-element heap has height ⌊lg n⌋algorithms_53765954.odp習題6.1-2algorithmssortheap sortF9506213Thu Feb 26 17:30:33 2009證明6.1-7 Show that, with the array representation for storing an n-element heap, the leaves are the nodes indexed by ⌊n/2⌋ + 1, ⌊n/2⌋ + 2, ..., n.algorithms_53765711.odp習題6.1-7algorithmssortheap sortF9506237Thu Feb 26 17:23:43 2009問答6.2-3 What is the effect of calling Max-Heapify(A, i) when the element A[i] is larger than its children?algorithms_53765301.odp習題6.2-3algorithmssortcounting sortF9506029Thu Feb 26 17:00:52 2009證明8.2-3 Suppose that the for loop header in line 9 of the Counting-Sort procedure is rewritten as
9 for j ← 1 to length[A]
Show that the algorithm still works properly. Is the modified algorithm stable?algorithms_53763930.pdf習題9.2-3algorithmssortselectionF9506011Thu Feb 26 16:57:38 2009設計演算法9.3-1 In the algorithm Select, the input elements are divided into groups of 5. Will the algorithm work in linear time if they are divided into groups of 7? Argue that Select does not run in linear time if groups of 3 are used.algorithms_53763736.odp習題9.3-1algorithmssortF9506229Thu Feb 26 16:53:20 2009證明8.1-4 You are given a sequence of n elements to sort. The input sequence consists of n/k subsequences, each containing k elements. The elements in a given subsequence are all smaller than the elements in the succeeding subsequence and larger than the elements in the preceding subsequence. Thus, all that is needed to sort the whole sequence of length n is to sort the k elements in each of the n/k subsequences. Show an Ω(n lg k) lower bound on the number of comparisons needed to solve this variant of the sorting problem. (Hint: It is not rigorous to simply combine the lower bounds for the individual subsequences.)algorithms_53763478.pdf習題8.1-4algorithmssortheap sortF9506035Thu Feb 26 16:49:19 2009time complexity6.2-6 Show that the worst-case running time of Max-
Heapify on a heap of size n is Ω(lg n).algorithms_53763237.pdf習題6.2-6algorithmssortheap sort張文龍Thu Feb 26 16:46:15 2009基本性質6.1-4 Where in a max-heap might the smallest element
reside, assuming that all elements are distinct?algorithms_53763053.odp習題6.1-4algorithmssortheap sort王立恩Thu Feb 26 16:41:24 2009修改演算法6.2-5 The code for Max-Heapify is quite efficient in terms of constant factors, except possibly for the recursive call in line 10, which might cause some compilers to produce inefficient code. Write an efficient Max-Heapify that uses an iterative contro construct (a loop) instead of recursion.algorithms_53762762.odt習題6.2-5algorithmssortingheapsortf9506021Thu Feb 26 6:27:57 2009分析演算法6.4-3 What is the running time of heapsort on an array A of length n that is already sorted in increasing order? What about decreasing order?algorithms_53725955.odp習題 6.4-3algorithmssortingbucket-sorttime complexityf9506015Thu Feb 26 6:21:19 2009修改演算法8.4-2 What is the worst-case running time for the bucket-sort algorithm? What simple change to the algorithm preserves its linear expected running time and makes its worst-case running time O(nlogn)?algorithms_53725557.odp習題 8.4-2algorithmsdata structureheap曾一平Thu Feb 26 6:16:22 2009執行演算法6.5-2 Illustrate the operation of MAX-HEAP-INSERT(A, 10) on the heap A={15, 13, 9, 5, 12, 8, 7, 4, 0, 6, 2, 1}. Use the heap of Fig6,5 as a model for the HEAP-INCREASE-KEY call.algorithms_53725260.odp習題 6.5-2formal languagescontext-free languagespushdown automatanpdaspringWed May 28 9:56:03 2008設計 automaton7.1.4 (k) Construct an npda that accepts the language L = {w : na(w) ＜ nb(w)} on Σ = {a, b, c}.formal languages_30064841.odp習題7.1.4 k)formal languagescontext-free languagespushdown automatanpdaspringWed May 28 9:46:54 2008設計 automaton7.1.4 (d) Construct an npda that accepts the language L = {anbn+mcm : n ≥ 0, m ≥ 0}.formal languages_30064293.odp習題7.1.4 d)formal languagescontext-free languagespushdown automataspringWed May 21 9:58:56 2008設計 automaton7.1.4 (e) Construct an npda that accepts the language L = {a3bncn:n ≥ 0}.formal languages_29460213.odp習題7.1.4 e)formal languagescontext-free languagespushdown automata李京釜Wed May 21 9:52:25 2008設計 automaton7.1.4(b) Construct an npda that accepts the language L = {wcwR : w ∈ {a, b}* }formal languages_29459823.odp習題7.1.4 b)formal languagescontext-free languagespushdown automataspringWed May 21 9:46:04 2008設計 automaton7.1.4 (g) Construct an npda that accepts the language L = {w :na(w) = nb(w) + 1} on Σ = {a, b, c}.formal languages_29459443.odp習題7.1.4 g)formal languagescontext-free languagespushdown automataregular expressionspringWed May 21 9:40:25 2008設計 automaton7.1.3 (c) Construct an npda that accepts the regular languageL(aaa*b) ∪ L(aab*aba*)formal languages_29459103.odp習題7.1.3 c)formal languagescontext-free languagescontext-free grammars陳柏緯Wed May 14 14:44:38 2008設計 grammars5.1.8 (d) Find context-free grammars for
L = {anbmck: k = n + 2m,n ≥ 0, m ≥ 0, k ≥
0}.formal languages_28872556.odp習題 5.1.8 d)formal languagesregular languagesregular expression黃弘碩Wed May 14 10:11:03 2008設計 regular expression3.1.6 d) Give regular expressions for complement of the
language L2 = {anbm : n 〈 3, m ≤ 4}.formal languages_28856141.odp習題 3.1.6 d)formal languagesregular languagesregular grammars吳威毅Wed May 14 10:02:56 2008設計 grammars3.3.13 e) Find regular grammars for L = {w: |na(w) - nb(w)| is odd} on {a, b}.formal languages_28855654.odp習題 3.3.13 e)formal languagesregular languagesregular expression柯衍慶Wed May 14 9:57:31 2008設計 regular expression3.1.28 Find a regular expression for all bit strings, with leading bit 1, interpreted as a binary integer, with values not between 10 and 30.formal languages_28855329.odp習題 3.1.28formal languagesregular languagesregular expression廖本宇Wed May 14 9:51:13 2008設計 regular expression3.2.13 d) Find a regular expression for L = {w: 2na(w)+3nb(w) is even} on {a, b}.formal languages_28854951.odp習題 3.2.13 d)formal languagesregular languagesregular expression朱致宏Wed May 14 9:45:15 2008設計 regular expression3.1.17 a) Write regular expressions for all strings ending in 01 on Σ= {0, 1}.formal languages_28854593.odp習題 3.1.17 a)formal languagesregular languagesregular grammars林祐誼Wed May 14 9:38:31 2008設計 grammars3.3.12 Find a regular grammar that generates the language L={w ∈ {a,b}*:na(w)+3nb(w) is even}.formal languages_28854188.odp習題 3.3.12formal languagesregular languagesregular expressionDFA簡裕哲Wed May 14 9:31:55 2008設計 automaton3.2.4 d) Find a dfa that accept L(((aa*)*b)*).formal languages_28853793.odp習題 3.2.4 d)formal languagescontext-free languagesGreibach normal formspringWed May 7 10:11:30 2008設計 grammars6.2.11 Convert the following grammar into Greibach normal form.
S → aSb | ab.formal languages_28251368.odp習題6.2.11formal languagescontext-free languagesλ-productionsunit-productionsspringWed May 7 10:09:10 2008證明Let G = (V, T, S, P) be any context-free grammar without any λ-productions or unit-productions. Let k be the maximum number of symbols on the right of any production in P. Show that there is an equivalent grammar in Chomsky normal form with no more than (k-1) |P| + |T| production rules.formal languages_28251228.odp習題6.2.6formal languagescontext-free languagesCYK algorithm洪惠美Wed May 7 10:03:29 2008執行演算法Use the CYK algorithm to determine whether the strings aabb, aabba, and abbbb are in the language generated by the grammar in Example 6.11.formal languages_28250887.odp習題6.3.1formal languagescontext free languagesλ-productionsspringWed May 7 6:21:40 2008設計 grammar6.1.7 Eliminate λ-productions from
S → AaB | aaB,
A → λ,
B → bbA | λ.formal languages_28237578.odp習題 6.1.7formal languagescontext-free languagess-grammarsambiguousspringFri Apr 25 17:16:00 2008證明5.2.4 Show that every s-grammar is unambiguous.formal languages_27240038.odp習題5.2.4formal languagescontext-free languagescontext-free grammarsspringFri Apr 25 17:11:32 2008設計 grammars5.1.7 (f) Find context-free grammars for L = {w ∈ {a, b}*: na(v) ≥ nb(v), where v is any prefix of w}, n ≥ 0, m ≥ 0.formal languages_27239770.odp習題5.1.7 f)formal languagesregular languagespumping lemma洪惠美Fri Apr 25 16:45:18 2008證明4.3.5 b) Determine whether or not L = {an : n is not prime number} on Σ = {a} is regular.formal languages_27238196.odp習題4.3.15 b)formal languagesregular languagespumping lemmaspringFri Apr 25 16:34:10 2008證明4.3.15 e) L = {anbl : n ≤ l ≤ 2n} 是 regular language 嗎？請證明。formal languages_27237528.odp習題4.3.15 e)formal languagesregular languagespumping lemma吳忠融Fri Apr 25 16:28:42 2008證明4.3.15 c) L = {anbl : n/l is an integer} 是 regular language 嗎？請證明。formal languages_27237200.odp習題4.3.15 c)formal languagesregular languagesregular grammarsregular expressionspringFri Apr 25 13:01:10 2008設計 grammars3.3.2 Find a regular grammar that generates the language
L(aa*(ab+a)*)formal languages_27224748.odp習題3.3.2formal languagesregular languagesregular grammarsDFA何振剛Fri Apr 25 12:57:06 2008設計 automaton3.3.1 Construct a dfa that accepts the language generated by the grammar
S → abA,
A → baB,
B → aA | bb.formal languages_27224503.odp習題3.3.1formal languagesregular languagesregular expressionDFAspringFri Apr 25 12:28:44 2008設計 automaton3.2.4 b) Find a dfa that accept L(ab(a+ab)*(a+aa)).formal languages_27222802.odp習題3.2.4 b)formal languagesregular languagesregular expression洪惠美Fri Apr 25 11:43:01 2008設計 regular expression3.2.13 c) Find a regular expression for L = {w: (na(w) - nb(w)) mod 3 ≠ 0} on {a, b}.formal languages_27220059.odp習題3.2.13 c)algorithmssortingQuicksortspringFri Apr 25 11:20:55 2008翻譯請寫出下列英文的中文大意
The running time of quicksort depends on whether the partition is balanced or unbalanced, and this in turn depends on which elements are used for partitioning. If the partitioning is balanced, the algorithm runs asymptotically as fast as merge sort. If the partitioning is unbalanced, however, it can run asymptotically as slowly as insertion sort.algorithms_27218733.odp2008平常測驗一algorithmssortingcounting sortspringFri Apr 25 11:19:28 2008執行演算法Perform the operation of Counting-Sort on the array A = 6, 5, 2, 3, 2, 4, 4, 2, 1, 7, 3.要寫下陣列 C 的三個主要內容與最後陣列 B 的內容。algorithms_27218646.odp2008平常測驗一algorithmssortingstablein placeaverage casespringFri Apr 25 11:18:10 2008計算Fill the following table.algorithms_27218568.odp2008平常測驗一algorithmssortingQuicksortPartitionspringFri Apr 25 11:15:53 2008執行演算法Perform the operation of Partition on the array A = 〈13, 19, 9, 5, 12, 8, 7, 4, 11, 2, 6,10〉.algorithms_27218431.odp2008平常測驗一algorithmssortingheap sortheapspringFri Apr 25 11:14:22 2008執行演算法請用 heap sort 的方式來將 〈13, 2, 25, 7, 17, 20〉 作遞減排序
a)請用 min-heapify 將 heap 建出來。寫出調整每一個元素後的情形。
b)寫出建完 heap 後的排序過程，寫出搬走每個元素、調整完後的結果。algorithms_27218340.odp2008平常測驗一algorithmsfunctionMaster TheoremspringFri Apr 25 11:12:02 2008計算Give asymptotic upper and lower bounds for T(n) in each of the following recurrences.
a. T(n) = T(n-1) + 1/n.
b. T(n) = T(n-1) + lg n.
c. T(n) = 7T(n/2) + n2.
d. T(n) = 16T(n/4) + n2.
e. T(n) = 7T(n/3) + n2.
f. T(n) = T(n-1) + n.algorithms_27218200.odp2008平常測驗一algorithmsfunctionspringFri Apr 25 11:08:38 2008計算Rank the following functions by order of growth; that is, find an arrangement g1, g2, ..., g15 of the functions satisfying g1 = (g2), g2 = (g3), ..., g14 = (g15).
n2, n!, (lg n)!, n3, lg2n, lg(n!), 22n, 2nn, n2n, nlg lgn, (lg n)lg n, 4lg n, 2n, n lg n, 22n+1algorithms_27217996.odp2008平常測驗一algorithmssortingselection sortinsertion sortspringFri Apr 25 11:06:10 2008執行演算法請分別使用 selection sorting 、insertion sort的方式對 〈35, 32, 24, 33, 18, 11, 15〉作遞增排序；寫下每一個元素被排好後的情形。algorithms_27217848.odp2008平常測驗一algorithmstime complexityheapThetaBig OspringFri Apr 25 11:04:39 2008是非題是非題
f(n) + g(n) = (max(f(n), g(n))).
f(n) = O(g(n) g(n) = O(f(n)).
An array that is sorted in non-decreasing order is a min-heap.
The sequence 〈23, 17, 14, 6, 13, 10, 11, 5, 4, 12〉 is a max-heap.
The time complexity of the best case of heap sort is O(n).algorithms_27217757.odp2008平常測驗一algorithmssortingcounting sortspringFri Apr 25 10:57:12 2008翻譯請寫出下列英文的中文大意
The basic idea of counting sort is to determine, for each input element x, the number of elements less than x. This information can be used to place element x directly into its position in the output array. For example, if there are 17 elements less than x, then x belongs in output position 18.algorithms_27217310.odp2008平常測驗一algorithmssortingradix sortspringFri Apr 25 10:55:37 2008執行演算法Sort the following number in descending order with Radix Sort: 3852, 4621, 1357, 2489, 8543, 5781, 8799, 1076, 4893, 3651.algorithms_27217215.odp2008平常測驗一algorithmssortingstablein placeworst casespringFri Apr 25 10:54:00 2008基本性質Fill the following table.algorithms_27217118.odp2008平常測驗一algorithmssortingQuicksortPartitionspringFri Apr 25 10:51:35 2008執行演算法Perform the operation of Partition on the array A = 〈10, 19, 9, 5, 12, 8, 7, 4, 11, 2, 6,13〉.algorithms_27216973.odp2008平常測驗一algorithmssortingheapheap sortspringFri Apr 25 10:44:20 2008執行演算法請用 heap sort 的方式來將 〈13, 2, 3, 25, 7, 17, 8〉 作遞增排序
a)請用 max-heapify 將 heap 建出來。寫出調整每一個元素後的情形。
b)寫出建完 heap 後的排序過程，寫出搬走每個元素、調整完後的結果。algorithms_27216538.odp2008平常測驗一algorithmsfunctionMaster TheoremspringFri Apr 25 10:40:57 2008計算Give asymptotic upper and lower bounds for T(n) in each of the following recurrences.
a. T(n) = 2T(n/4) + n1/2
b. T(n) = T(n-1) + lg n
c. T(n) = 7T(n/2) + n2
d. T(n) = T(n1/2) + 1
e. T(n) = 4T(n/2) + n3
f. T(n) = T(n-1) + nalgorithms_27216335.odp2008平常測驗一algorithmsfunctiontime complexityspringFri Apr 25 10:39:20 2008計算Rank the following functions by order of growth; that is, find an arrangement g1, g2, ..., g15 of the functions satisfying g1 = O(g2), g2 = O(g3), ..., g14 = O(g15).
lg(lg*n), n2n, n!, (lg n)!, (3/2)n, lg2n, 22n, lg*n, 2nn, nlg lgn,(lg n)lg n, (n+1)!, lg*(lg n), 2n, 22n+1algorithms_27216238.odp2008平常測驗一algorithmssortingheapspringFri Apr 25 10:36:59 2008設計演算法6.5-8 Give an O(n lg k)-time algorithm to merge k sorted lists into one sorted list, where n is the total number of elements in all the input lists. (Hint: Use a min-heap for k-way merging.)algorithms_27216097.odp習題6.5-8algorithmssortingselection sortinsertion sortspringFri Apr 25 10:33:34 2008執行演算法請分別使用 selection sorting 、insertion sort的方式對 〈35, 32, 24, 33, 18, 11, 15〉作遞減排序；寫下每一個元素被排好後的情形。algorithms_27215892.odp2008平常測驗一algorithmstime complexityBig OheapsortingspringFri Apr 25 10:31:26 2008是非題是非題
f(n) + g(n) = (min(f(n), g(n))).
f(n) = O(g(n)) ==> 2f(n) = O(2g(n)).
The height of a heap with 33 elements is 4.
The sequence 〈3, 7, 4, 6, 13, 10, 11, 15, 14, 16〉 is a min-heap.
Any comparison sort algorithm requires Ω(n lg n) comparisons in the worst case.algorithms_27215764.odp2008平常測驗一algorithmsfunctionMaster TheoremspringFri Apr 25 8:41:13 2008計算4.3-1 Use the master method to give tight asymptotic bounds for the following recurrences.
a. T(n) = 4T(n/2) + n.
b. T(n) = 4T(n/2) + n2.
c. T(n) = 4T(n/2) + n3.algorithms_27209151.odp習題4.3-1algorithmsfunctionBig OThetaBig OmegaspringFri Apr 25 8:39:17 2008證明Problem 3-4 Asymptotic notation properties
Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures.
a. f(n) = O(g(n)) implies g(n) = O(f(n)).
b. f(n) + g(n) = Θ(min(f(n), g(n))).
c. f(n) = O(g(n)) implies lg(f(n)) = O(lg(g(n))), where lg(g(n)) ≥ 1 and f(n) ≥ 1 for all sufficiently large n.
d. f(n) = O(g(n)) implies 2f(n) = O(2g(n)).
e. f(n) = O((f(n))2).
f. f(n) = O(g(n)) implies g(n) = Ω(f(n))
g. f(n) = Θ(f(n/2))
h. f(n) + o(f(n)) = Θ(f(n)).algorithms_27209035.odp習題 Problem 3-4algorithmsfunctionspringFri Apr 25 8:35:05 2008計算Problem 3-3 Ordering by asymptotic growth ratesalgorithms_27208783.odp習題 Problem 3-3algorithmstime complexityspringFri Apr 25 8:31:19 2008證明3.1-7 Prove that o(g(n)) ∩ ω(g(n)) = Ø.algorithms_27208557.odp習題3.1-7algorithmssortingtime complexityspringFri Apr 25 8:29:07 2008設計演算法2.3-7 Describe a (n lg n)-time algorithm that, given a set S of n integers and another integer x, determines whether or not there exist two elements in S whose sum is exactly x.algorithms_27208424.odp習題2.3-7algorithmssortingselection sortspringFri Apr 25 8:27:14 2008設計演算法2.2-2 Consider sorting n numbers sotred in array A by first finding the smallest element of A and exchanging it with the element in A[1]. Then find the second smallest element of A, and exchange it with A[2]. Continue in this manner for the first n-1 elements of A. Write pseudocode for this algorithm, which is known as selection sort. What loop invariant does this algorithm maintain? Why does it need to run for only the first n – 1 elements, rather than for all n elements? Give the best-case and worst case running times of selection sort in -notation.algorithms_27208312.odp習題2.2-2algorithmstime complexitymerge sortinsertion sortspringFri Apr 25 8:24:18 2008計算1.2-2 Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in 64 n log2n steps. For which values of n does insertion sort beat merge sort?algorithms_27208136.odp習題1.2-2formal languagesregular languagesregular expressionspringFri Apr 25 8:18:21 2008簡答3.1.8 Give a simple verbal description of the language L((aa)*b(aa)* + a(aa)*ba(aa)*).formal languages_27207779.odp習題3.1.8formal languagesregular languagesregular expression李京釜Fri Apr 25 6:34:57 2008設計 regular expression3.1.6 b) Give regular expressions for the language L2 ={anbm : n ＜ 4, m ≤ 3}.formal languages_27201575.odp習題 3.1.6 b)formal languagesregular languagesregular expressionspringFri Apr 25 6:31:50 2008設計 regular expression3.1.5 Find a regular expression for the set {anbm : (n + m) is even}.formal languages_27201388.odp習題 3.1.5formal languagesregular languagesregular expressionspringFri Apr 25 6:26:38 2008設計 regular expression3.1.4 Find a regular expression for the set {anbm : n ≥ 3, m is even}.formal languages_27201076.odp習題 3.1.4formal languagesregular languagesregular expressionspringFri Apr 25 6:12:59 2008是非題3.1. 20 Determine whether or not the following claims are true fro all regular expressions r1 and r2. The symbol ≡ stands for equivalence of regular expressions in the sense that both expressions denote the same language.
a). (r1*)* ≡ r1*
b). r1*(r1 + r2)* ≡ (r1 + r2)*
c). (r1 + r2)* ≡ (r1* r2*)*
d). (r1* r2*)* ≡ r1* r2*formal languages_27200257.odp習題 3.1.20formal languagesregular languagesregular expressionspringFri Apr 25 6:02:13 2008設計 regular expression3.1.17 b) Write regular expressions for all strings not ending in 01 on Σ= {0, 1}.formal languages_27199611.odp習題 3.1.17 b)formal languagesregular languagesregular expression吳 蕙 如Thu Apr 24 14:49:00 2008設計 regular expression3.1.16 b) Give regular expression for the language with all strings containing no more than three a\'s on Σ = {a, b, c}.formal languages_27144818.odp習題3.1.16 b)formal languagesregular languagesregular expression許鴻仁Thu Apr 24 14:44:31 2008設計 regular expression3.1.16 d) Give regular expression for the language with all strings that contain no run of a\'s of length greater than two on Σ = {a, b, c}.formal languages_27144549.odp習題3.1.16 d)formal languagesregular languagesregular expression劉驥雷Thu Apr 24 14:41:15 2008設計 regular expression3.1.15 Find a regular expression for L = {w ∈ {0, 1}* : w has exactly one pair of consecutive zeros}formal languages_27144353.odp習題3.1.15formal languagesregular languagesregular expressionspringThu Apr 24 14:36:02 2008設計 regular expression3.1.14 Find a regular expression for L = {vwv : v, w ∈ {a,b}*, |v| ≤ 3}formal languages_27144039.odp習題3.1.14formal languagesregular languagesspringThu Apr 24 14:07:00 2008翻譯請寫出下列英文的中文意義。
We have defined regular languages, studied some ways in which they can be represented, and have seen a few examples of their usefulness. We now raise the question of how general regular languages are. Could it be that every formal language is regular? Perhaps any set can be accepted by some, albeit very complex, finite automaton. As we will see shortly, the answer to this conjecture is definitely no. But to understand why this is so, we must inquire more deeply into the nature of regular languages and see what properties the whole family has.formal languages_27142298.odp2008期中考formal languagesregular languagesDFAspringThu Apr 24 14:04:35 2008設計 automatonConstruct a dfa that accepts strings on {0, 1} if and only if the value of the string, interpreted as a binary representation of an integer, is 2 modula 6.formal languages_27142153.odp2008期中考formal languagesregular languagesright quotientspringThu Apr 24 14:03:05 2008設計 automatonLet L1 = L(a*baa*) and L2 = L(aba*). Find L1/L2.formal languages_27142063.odp習題4.1.10formal languagesregular languagesNFAspringThu Apr 24 14:00:51 2008設計 automatonFind an nfa that accepts the language L(ab*aa + bba*ab).formal languages_27141929.odp2008期中考formal languagesregular languagesspringThu Apr 24 13:58:21 2008證明Show that if L is regular, so is LR.formal languages_27141778.odp習題2.3.12formal languagesregular languagesDFAspringThu Apr 24 13:55:51 2008設計 automatonConstruct a DFA to accept all strings on {0,1} that all strings of length three or more in which the second symbol from the right end is different from the leftmost symbol.formal languages_27141629.odp2008期中考formal languagesregular languagesregular expressionregular grammarDFAspringThu Apr 24 13:50:16 2008設計Find the dfa, regular grammar and regular expression for the language L = {w: (na(w) - nb(w)) mod 3 > 0} on ∑ = {a, b}.formal languages_27141294.odp2008期中考formal languagesregular languagesregular expressionregular grammarDFAspringThu Apr 24 13:44:54 2008設計Find the dfa, regular expression and regular grammar for the language L = {w : |w| mod 3 ≠ |w| mod 2} on Σ = {a}.formal languages_27140972.odp2008期中考formal languagesregular languagesregular expressionspringThu Apr 24 6:08:41 2008設計 regular expression3.1.11 Find a regular expression for L = {abnw : n ≥ 3, w∈ {a, b}+}.formal languages_27113599.odp習題3.1.11formal languagesregular languagesregular grammarsright-linear grammars涂毓城Thu Apr 24 5:45:57 2008設計 regular grammars3.3.6 Construct a right-linear grammar for the language L((aab*ab)*).formal languages_27112235.odp習題3.3.6formal languagesregular languagesregular expression謝岦陞Wed Apr 23 17:18:44 2008設計 regular expression3.1.18 b) Find regular expressions for L = {w :na(w) mod 3 = 0} on {a, b}.formal languages_27067402.odp習題3.1.18 b)formal languagesregular languagespumping lemma陳柏緯Wed Apr 23 17:04:18 2008證明4.3.9 Is the language L = {w ∈ {a, b, c}* : |w| = 3na(w)} regular ?formal languages_27066536.odp習題4.3.9formal languagesregular languagesDFA張家銘Wed Apr 23 16:57:44 2008設計 automaton3.2.7 Find the minimal dfa that accepts
L(a*bb) ∪ L(ab*ba).formal languages_27066142.odp習題3.2.7formal languagesregular languagesDFA楊良吉Wed Apr 23 16:26:07 2008設計 automaton2.1.8 c) Find dfa\'s for the language {w: there are at most two runs of a\'s of length three} on {a, b}.formal languages_27064244.odp習題2.1.8 c)formal languagesregular languagesDFA朱致宏Wed Apr 23 15:36:03 2008設計 automaton2.1.7 f) Find a dfa for the language L on Σ = {a, b}. L = {w: (na(w) + 2nb(w)) mod 3 ＜ 2}.formal languages_27061240.odp習題2.1.7 f)formal languagesregular languagesDFA吳威毅Wed Apr 23 14:45:49 2008設計 automatonA2.10 Construct a DFA that accepts the language of binary strings in which every block of four symbols contains at least three 0’s.formal languages_27058227.odp習題A2.10formal languagesregular languagesregular expression胡偉諄Wed Apr 23 14:33:19 2008設計 regular expression3.1.18 c) Find regular expressions for L = {w : na(w) mod 5 > 0} on {a, b}.formal languages_27057477.odp習題3.1.18 c)formal languagesregular languagesDFA陳羿亘Wed Apr 23 12:47:05 2008設計 automaton3.2.5 b) Find a dfa that accept L(ab*a*) ∩ L((ab)*ba).formal languages_27051103.odp習題3.2.5 b)formal languagesregular languagesNFA張家銘Wed Apr 23 12:32:18 2008設計 automatonA2.2 Construct an NFA that accepts the set of binary strings that contain both substrings010 and 101.
[Hint: This is equivalent to the set of binary strings that contain either a substring 0101 or a substring 1010 or a substring 010 followed by 101 or a substring101 followed by 010.]formal languages_27050215.odp習題A2.2formal languagesregular languagesDFA林祐誼Wed Apr 23 9:58:01 2008證明2.1.11 Show that the language L = {vwv: v, w ∈ {a, b}*, |v| = 2} is regular.formal languages_27040959.odp習題2.1.11formal languagesregular languagesregular expression賴映潔Wed Apr 23 9:49:28 2008設計 regular expression3.1.16 (a) Give regular expression for the language with all strings containing exactly one a on Σ = {a, b, c}.formal languages_27040446.odp習題3.1.16 a)formal languagesregular languagesDFA黃弘碩Wed Apr 23 9:45:21 2008設計 automaton2.1.9e) Construct a DFA to accept all strings on {0,1} that all strings of length five or more in which the fourth symbol from the right end is different from the leftmost symbol.formal languages_27040198.odp習題2.1.9 e)formal languagesregular languagesregular grammarsleft-linear grammar蔡學仁Wed Apr 23 9:40:15 2008設計 grammars3.3.10 Find a left-linear grammar for the language L((aab*ab)*).formal languages_27039893.odp習題3.3.10formal languagesregular languagesDFA廖本宇Wed Apr 23 9:21:43 2008設計 automaton2.1.5 c) Give a dfa for the language L = {w1abw2: w1∈ {a, b}*, w2 ∈ {a, b}*}.formal languages_27038780.odp習題2.1.5 c)formal languagessets朱致宏Wed Apr 23 9:14:58 2008證明1.1.9 Show that S1 ∪ S2 – (S1 ∩ S2) = S2.formal languages_27038376.odp習題1.1.9formal languagesregular languagesDFA胡偉諄Wed Apr 23 9:10:33 2008設計 automaton2.1.8 b Find dfa\'s for the language
{w: every run of a\'s has length either two or three} on {a, b}.formal languages_27038111.odp習題2.1.8b)formal languagesmathematical induction曾愷銘Wed Apr 23 9:03:13 2008證明A.1-3 Prove by mathematical induction that 10n + 3•4n + 5 is divisible by 9 for all natural numbers n.formal languages_27037671.odp習題A1-3formal languagesregular languagesDFA謝棋安Wed Apr 23 8:54:01 2008設計 automaton2.1.9 b) Construct a DFA to accept all strings on {0, 1} containing 00 but not 000.formal languages_27037119.odp習題2.1.9b)formal languagesregular languagesDFA謝岦陞Wed Apr 23 8:44:11 2008設計 automaton2.1.7 b) Find a dfa for the language L on Σ = {a, b}. L = {w: |w| mod 5 ≠ 0}.formal languages_27036529.odp習題2.1.7b)formal languagesregular languagesDFA陳柏緯Wed Apr 23 8:37:15 2008設計 automatonA2.9 Construct a DFA that accepts the language of strings over the alphabet {0,1,2} which are the ternary expansions (base-3 representations) of positive integers which are congruent to 2 modulo 7.formal languages_27036112.odp習題A2.9formal languagesregular languagesgrammar鄭翰鴻Tue Apr 22 12:33:13 2008設計 grammars1.2.15(c) Find grammars for the language L = {w : |w| mod 3 ≠ |w| mod 2} on Σ = {a}.formal languages_26963871.odp習題1.2.15(c)formal languagessets黃弘碩Tue Apr 22 12:29:03 2008證明1.1.2 Show that if S1 and S2 are finite sets with |S1| = n and |S2| = m, then |S1 ∪ S2| ≤ n+m.formal languages_26963621.odp習題1.1.2formal languagesBig OTheta廖本宇Tue Apr 22 12:22:53 2008證明1.1.21 Show that if f(n) = (log2 n), then f(n) = (log10 n).formal languages_26963251.odp習題1.1.21formal languagesregular languagesDFA蔡學仁Tue Apr 22 12:13:34 2008證明2.1.15 Show that the language L = {an: n is a multiple of three, but not a multiple of 5} is regular.formal languages_26962692.odp習題2.1.15formal languagesregular languagesgrammar陳羿亘Tue Apr 22 11:40:25 2008設計 grammars1.2.11-b. Find grammars for Σ = {a, b} that generate the sets of all strings with at least one a.formal languages_26960703.odp習題1.2.11-bformal languagesregular languagesDFA陳羿亘Tue Apr 22 11:35:13 2008設計 automaton2.1.7 e) Find a dfa for the language L on ∑ = {a, b}. L = {w: (na(w) – nb(w)) mod 3 > 0}.formal languages_26960391.odp習題2.1.7 e)formal languagesrelation柯衍慶Tue Apr 22 11:30:31 2008找relationA.1-2 Give an example of a relation that is symmetric and transitive but not reflexive.formal languages_26960109.odp習題A1-2formal languagesrational number謝岦陞Tue Apr 22 11:26:31 2008證明1.1.31 Show that √3 is irrational.formal languages_26959869.odp習題1.1.31formal languagesregular languagesgrammar張家銘Tue Apr 22 11:19:58 2008設計 grammars1.2.15(d) Find grammars for the language
L = {w : |w| mod 3 ≥|w| mod 2} on Σ = {a}.formal languages_26959476.odp習題1.2.15(d)formal languagesregular languagesNFA賴映潔Tue Apr 22 11:12:04 2008設計 automaton2.2.10 a) Find an nfa with three states that accepts the language L = {an : n ≥ 1} ∪ { bmak : m ≥ 0, k ≥ 0}.
b) Do you think the language in part a) can be accepted by an nfa with fewer than three states?formal languages_26959002.odp習題2.2.10formal languagesregular languagesNFA鄭翰鴻Tue Apr 22 11:06:53 2008設計 automatonA2.3 Construct an NFA that accepts the set of binary strings that contain either a substring 010 or a substring 101, and end with 111 or 000.formal languages_26958691.odp習題A2.3formal languagesinfinite number賴映潔Tue Apr 22 11:03:02 2008證明1.1.34 Prove that the set of all prime numbers is infinite.formal languages_26958460.odp習題1.1.34formal languagesBig O林祐誼Tue Apr 22 10:54:25 2008證明1.1.18 Prove that if f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n)).formal languages_26957943.odp習題1.1.18formal languagesmathematical induction林柏存Tue Apr 22 10:48:50 2008證明A1-4Prove by mathematical induction that 10n + 3•4n + 5 is divisible by 9 for all natural numbers n.formal languages_26957608.odp習題A1.4formal languagesregular languagesDFA洪喜音Tue Apr 22 10:42:23 2008設計 automatonA2.8 Construct a DFA that accepts the language of strings over the alphabet {1,2,3} in which the sum of all symbols is divisible by 5.formal languages_26957221.odp習題A2.8formal languagescontext-free languages胡偉諄Tue Apr 22 10:38:50 2008設計 grammars1.2.14-c. Find a grammar that generates
L3 = {an+2bn: n ≥ 1}.formal languages_26957008.odp習題1.2.14-cformal languagesmathematical induction謝棋安Tue Apr 22 10:34:07 2008證明A.1-5 Prove by mathematical induction that log72 + log714 + log798 + . . . + log7(2 * 7n ) = 1/2(n + 1)(n + log74). [Hint： logdxn = n * logdx]formal languages_26956725.odp習題A.1-5formal languagescontext-free languagescontext-free grammarsspringTue Apr 22 8:39:48 2008設計 grammars1.2.14-g 之變形 Find a grammar that generates L33
L3 = {an+2 bn: n ³ 1}.formal languages_26949866.odp習題1.2.14-g之變形formal languagesBig OThetaspringTue Apr 22 8:34:14 2008證明1.1.15 Show that if f(n) = O(g(n)) and g(n) = O(f(n)), then f(n) = (g(n)).formal languages_26949532.odp習題1.1.15algorithmsdata structurelinked listspringTue Apr 22 8:14:59 2008翻譯請寫出下列英文的中文大意
A linked list is a data structure in which the objects are arranged in a linear order. Unlike an array, though, in which the linear order is determined by the array indices, the order in a linked list is determined by a pointer in each object. Linked lists provide a simple, flexible representation for dynamic sets.algorithms_26948377.odp2008期中考algorithmsselectionrandomized algorithmsspringTue Apr 22 8:02:25 2008執行演算法Suppose we use Randomized-Select to select the minimum element of the array A = 〈3, 2, 9, 0, 7, 5, 4, 8, 6, 1〉. Describe a sequence of partitions that results in a worst-case performance of Randomized-Select.algorithms_26947623.odp習題 9.2-4algorithmssortingcounting sortstablein placespringTue Apr 22 7:58:00 2008設計演算法Suppose that we have an array of n data records to sort and that the key of each record has the value 0 or 1. An algorithm for sorting such a set of records might possess some subset of the following three desirable characteristics:
1. The algorithm runs in O(n) time.
2. The algorithm is stable.
3. The algorithm sorts in place.
a. Give an algorithm that satisfies criteria 1 and 2 above.
b. Give an algorithm that satisfies criteria 1 and 3 above.
c. Give an algorithm that satisfies criteria 2 and 3 above.algorithms_26947358.odp習題 problem 8-2algorithmstime complexityfunctionspringTue Apr 22 7:55:21 2008證明Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures.
a) f(n)=O(g(n)) implies g(n) = O(f(n)).
b) f(n)=O(g(n)) implies 2f(n) = O(2g(n)).
c) f(n) = Θ(f(n/2)).algorithms_26947199.odp2008期中考algorithmstime complexityrecursive functionmaster theoremspringTue Apr 22 7:53:26 2008計算Give asymptotic upper and lower bounds for T(n) in each of the following recurrences.
a. T(n) = T(n-1) + 1/n.
b. T(n) = T(n/2)+T(n/4)+T(n/8) + n.
c. T(n) = 5T(n/2) + n2.
d. T(n) = 27T(n/3) + n3.
e. T(n) = 2T(n-1) + 1.
f. T(n) = T(n/2) + n½.algorithms_26947084.odp2008期中考algorithmsheappriority queuespringTue Apr 22 7:51:20 2008執行演算法Perform the operation of Max-Heap-Insert(A, 20) on the heap A = 〈15, 13, 9, 5, 12, 8, 7, 4, 0, 6, 2, 1〉.algorithms_26946958.odp2008期中考algorithmsselectionsortingspringTue Apr 22 7:47:56 2008設計演算法Suppose that you have a \"black-box\" worst-case linear-time median subroutine. Give a simple, linear-time algorithm that solves the selection problem for an arbitrary order statistic.algorithms_26946754.odp習題9.3-5algorithmssortingselectionmedianspringTue Apr 22 7:41:51 2008分析演算法Given a set of n numbers, we wish to find the i largest in sorted order using a comparison-based algorithm. Find the algorithm that implements each of the following methods with the best asymptotic worst-case running time, and analyze the running times of the algorithms in terms of n and i.
a. Sort the numbers, and list the i largest.
b. Build a max-priority queue from the numbers, and call Extract-Max i times.
c. Use an order-statistic algorithm fo find the ith largest number, partition around that number, and sort the i largest numbers.algorithms_26946389.odp習題 problem 9-1algorithmstime complexitydata structuresortingspringTue Apr 22 7:38:00 2008是非題是非題
n0.001 + lg1000 n = (lg1000n).
1.001n + n1000 = O(n1000).
The time complexity of worst case of quicksort is (n lg n).
The time complexity of the best case of insertion sort is O(n).
A stack is the data structure based on the principle of “First In First Out”.algorithms_26946158.odp2008期中考algorithmsall-pairs shortest pathsspringTue Dec 25 12:50:31 2007翻譯請寫出下列英文的中文大意
We can solve an all-pairs shortest-paths problem by running a single-source shortest-paths algorithm |V| times, once for each vertex as the source. If all edge weights are nonnegative, we can use Dijkstra\'s algorithm. If we use the linear-array implementation of the min-priority queue, the running time is O(V3+VE) = O(V3). The binary min-heap implementation of the min-priority queue yields a running time of O(VE lg V), which is an improvement if the graph is sparse. Alternatively, we can implement the min-priority queue with a Fibonacci heap, yielding a running time of O(V2 lg V + VE).algorithms_16683309.odp2007期中考題algorithmsshortest pathsDijkstra\'s algorithmspringTue Dec 25 12:47:43 2007證明Suppose that we are given a weighted, directed graph G = (V, E) in which edges that leave the source vertex s may have negative weights, all other edge weights are nonnegative, and there are no negative-weight cycles. Argue that Dijkstra\'s algorithm correctly finds shortest paths from s in this graph.algorithms_16683141.odp2007期中考題algorithmsshortest pathstreespringTue Dec 25 12:44:19 2007設計演算法The diameter of a tree T = (V, E) is given by
that is, the diameter is the largest of all shortest-path distances in the tree. Given an efficient algorithm to compute the diameter of a tree, and analyze the running time of your algorithm.algorithms_16682937.odp2007期中考題algorithmsshortest pathslinear programmingspringTue Dec 25 12:42:16 2007執行演算法Find a feasible solution or determine that no feasible solution exists for the following system of difference constraints:
x1 – x2 ≤ 1, x1 – x4 ≤ -4,
x2 – x3 ≤ 2, x2 – x5 ≤ 7,
x2 – x6 ≤ 5, x3 – x6 ≤ 10,
x4 – x2 ≤ 2, x5 – x1 ≤ -1,
x5 – x4 ≤ 3, x6 – x3 ≤ -8。algorithms_16682814.odp2007期中考題algorithmsshortest pathsJohnson\'s algorithmspringTue Dec 25 12:38:09 2007執行演算法Use Johnson\'s algorithm to find the shortest paths between all pairs of vertices in the following graph. Show the values of h and w\' computed by the algorithm.algorithms_16682567.odp2007期中考題algorithmsshortest pathsFloyd-WarshallspringTue Dec 25 9:02:51 2007執行演算法Run the Floyd-Warshall algorithm on the following weighted, directed graph. Show the matrix D(k) that results for each iteration of the outer loop.algorithms_16669649.odp2007期中考題algorithmsdisjoint setsunion and findspringTue Dec 25 9:00:08 2007執行演算法0~12的數字，假設一開始每個數字都是一個集合，用union by rank與path compression作以下的運作，請畫出最後的結果。Union(0, 1), Union(2, 3), Union(6, 7), Union(0, 12), Union(8, 9), Union(10, 11), Union(0, 2), Union(8, 10), Union(0, 4), Union(4, 5), Find-set(6), Union(0, 8), Union(4, 6), Find-set(8)。algorithms_16669486.odp2007期中考題algorithmsshortest pathsBellman-Ford algorithmspringTue Dec 25 8:58:41 2007執行演算法Run the Bellman-Ford algorithm on the following directed graph, using vertex z as the source.algorithms_16669399.odp2007期中考題data structurebinary treetree traversalspringMon Dec 24 19:42:05 2007翻譯請寫出下列英文的中文大意
Suppose we have the preorder sequence A B C D E F G H I and the inorder sequence B C A E D G H F I of the same binary tree. Does such a pair of sequences uniquely define a binary tree ? Put another way, can this pair of sequences come from more than one binary tree ?data structure_16621603.odp2007平常測驗二data structureselection treewinner treeloser treespringMon Dec 24 19:40:54 2007基本定義用 8, 7, 6, 10, 5, 18, 7, 11 建一個 winner tree 與一個 loser tree.data structure_16621532.odp2007平常測驗二data structuredisjoint setsunion and findspringMon Dec 24 19:39:11 2007證明假設一開始每棵 tree 都只有一個點，之後用 weighting rule 的方式作 union，若最後有一棵 tree T 的點數為 m，請證明 T 的高度不會超過 ⌊log2 m⌋ + 1。data structure_16621429.odp2007平常測驗二data structureheapspringMon Dec 24 19:37:40 2007執行演算法將下圖當做 max-heap，先刪掉最大值、再 insert 10, 畫出每次運算完成之後的heap.data structure_16621338.odp2007平常測驗二data structurebinary treetree traversalspringMon Dec 24 19:36:00 2007執行演算法Write down the preorder, inorder, postorder, and level-order traversal of the following binary tree.data structure_16621238.odp2007平常測驗二data structurebinary search treespringMon Dec 24 19:34:36 2007執行演算法在右圖這個 binary search tree中連續作下面的運算、畫出每個運算之後的結果。
a. insert 12,
b. delete 3,
c. delete 15,
d. insert 13,
e. delete 8。data structure_16621153.odp2007平常測驗二data structurebinary treeinorder sequencepreorder sequencespringMon Dec 24 19:30:07 2007執行演算法令 binary tree T 的 inorder sequence 為 B D C E A F G I H J，而T 的 preorder sequence 為 A B C D E F G H I J，請畫出 T。data structure_16620885.odp2007平常測驗二data structurelinked listsgeneralized listsspringMon Dec 24 19:28:43 2007畫圖請用 generalized list 的方式畫出多項式 3x2y6z2 + 8x3y5z3 + 2x4y2z3 + 4x4y6z2 + 6xy2z3 的 list。data structure_16620801.odp2007平常測驗二data structurebinary treespringMon Dec 24 19:26:59 2007計算請問5個點的binary tree 有幾個、6個點的binary tree 又有幾個呢？data structure_16620697.odp2007平常測驗二data structurelinked listspolynomialspringMon Dec 24 19:25:50 2007畫圖畫出下列多項式相加之後的結果。data structure_16620628.odp2007平常測驗二data structurelinked liststreespringMon Dec 24 19:24:36 2007是非題是非題
a)complete binary tree中不可能有degree為1的點。
b)要將binary tree每個點都拜訪過一次一定要使用 stack 才行。
c)每個 tree 的點數至少為 1。
d)每個 binary tree的點數至少為1。
e)點數至少為1的binary tree都一定有左子樹和右子樹。
f)在 heap 中作 insert 與刪掉最大值的時間都與 heap 的高度成正比。
g)在binary search tree中每個點的值都不能相同。
h)要將k個已經排序好的數列merge成一個排序好的數列，worst case一定要在 (nk)的時間複雜度才作得到，其中n是k個數列的元素個數總合。
i)10個點的 binary tree，其高度至少是5，最多是10。
j)存在兩個 binary trees T1 與 T2，它們的 inorder sequence相同、level-order sequence 也相同。data structure_16620554.odp2007平常測驗二data structurebinary treespringMon Dec 24 19:19:42 2007翻譯請寫出下列英文的中文大意
Binary trees are an important type of tree structure that occurs very often. Binary trees are characterized by the fact that any node can have at most two branches (i.e., there is no node with degree greater than two). For binary trees we distinguish between the subtree on the left and that on the right, whereas for trees the order of the subtrees is irrelevant. Also, a binary tree may have zero nodes. Thus, a binary tree is really a different object from a tree.data structure_16620260.odp2007平常測驗二data structureselection treewinner treeloser treespringMon Dec 24 19:18:35 2007基本定義用 3, 9, 6, 16, 5, 8, 7, 11 建一個 winner tree 與一個 loser tree.data structure_16620193.odp2007平常測驗二data structuredisjoint setsunion and findspringMon Dec 24 19:16:51 2007執行演算法假設0~12 共13個元素，一開始每個元素都是一個集合,使用 weighting rule 與 collapsing rule，做下列的運算，畫出最後的結果。 union(0, 1), union(0,2), union(3,4), union(5, 6), union(7, 8), union(0, 3), union(11, 0), Find(4), union(10, 12), union(5, 7), union(10, 0), union(0, 5), Find(8) 。data structure_16620089.odp2007平常測驗二data structurebinary treeheapspringMon Dec 24 19:14:40 2007執行演算法將第7題的圖當做 max-heap，先刪掉最大值、再 insert 18, 畫出每次運算完成之後的heap.data structure_16619958.odp2007平常測驗二data structurebinary treetree traversalspringMon Dec 24 19:12:53 2007執行演算法Write down the preorder, inorder, postorder, and level-order traversal of the following binary tree.data structure_16619851.odp2007平常測驗二data structuretreebinary treebinary search treespringMon Dec 24 19:09:21 2007執行演算法在右圖這個 binary search tree中連續作下面的運算、畫出每個運算之後的結果。
a. insert 6,
b. delete 3,
c. delete 4,
d. insert 12,
e. delete 11。data structure_16619639.odp2007平常測驗二data structuretreebinary treespringMon Dec 24 19:07:23 2007證明Prove the following statement. For any nonempty binary tree, T, if n0 is the number of leaf nodes and n2 the number of nodes of degree 2, then n0 = n2 + 1.data structure_16619521.odp2007平常測驗二data structurelinked listsgeneralized listsspringMon Dec 24 19:05:48 2007畫圖請用 generalized list 的方式畫出多項式 3x2y8z2 + 3x3y5z2 + 2x4y2z + 3x4y5z2 + 2xy2z 的 list。data structure_16619426.odp2007平常測驗二data structuredisjoint setsunion and findspringMon Dec 24 19:03:56 2007執行演算法令S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, pairs of equivalence relation: 0 ≡ 1, 4 ≡ 13, 3 ≡ 9, 1 ≡ 10, 11 ≡ 12, 4 ≡ 5, 2 ≡ 8, 0 ≡ 9, 1 ≡ 11, 7 ≡ 10, 2 ≡ 5, 請輸出其 equivalence class 。data structure_16619314.odp2007平常測驗二data structurelinked listspringWed Dec 19 19:31:14 2007執行演算法畫出下列多項式相加之後的結果。data structure_16188952.odp2007平常測驗二data structuretreespringWed Dec 19 19:28:31 2007是非題是非題
a. complete binary tree中不可能有degree為1的點。
b. 要將binary tree每個點都拜訪過一次一定要使用 stack 才行。
c. 點數為10的tree每個點都用4個指標指向children，共有31個指標指向0。
d. 每個 binary tree的點數至少為1。
e. 點數至少為1的binary tree都一定有左子樹和右子樹。
f. 在 heap 中作 insert 與刪掉最大值的時間都與 heap 的高度成正比。
g. 在binary search tree中每個點的值都不能相同。
h. 要將k個已經排序好的數列merge成一個排序好的數列，worst case一定要在 (nk)的時間複雜度才作得到，其中n是k個數列的元素個數總合。
i. 20個點的 binary tree，其高度至少是5，最多是15。
j. 在equivalence classes的問題中，若有m個元素與n個相等的pairs，其演算法的時間複雜度為 (mn)。data structure_16188789.odp2007平常測驗二data structuretree traversalinorderpreorderlevel-orderspringFri Dec 14 9:00:56 2007執行演算法Write down the preorder, inorder, postorder, and level-order traversal of the binary tree.data structure_15719134.odp2007隨堂測驗data structureheapspringFri Dec 14 8:57:59 2007執行演算法將下面的 complete binary tree 當做 max-heap，先 insert 23、再刪掉最大值，畫出每次運算之後的 heap。data structure_15718957.odp2007隨堂測驗data structurebinary search treebinary treespringFri Dec 14 8:52:34 2007執行演算法在下圖這個 binary search tree中連續作下面的運算、畫出每個運算之後的結果。a. insert 7, b. delete 14, c. delete 4, d. delete 8。data structure_15718631.odp2007隨堂測驗data structureselection treewinner treeloser treespringFri Dec 14 8:51:10 2007基本定義用 8, 9, 6, 12, 5, 13, 7, 11 建一個 winner tree 與一個 loser tree.data structure_15718548.odp2007隨堂測驗data structuredisjoint setsunion and findspringThu Dec 13 20:34:19 2007執行演算法假設0~8 共9個元素，一開始每個元素都是一個集合，使用 weighting rule 與 collapsing rule，做下列的運算，畫出最後的結果。union(0,1) ,union(0, 2), union(3,4), union(5, 6), union(7,8), union(0, 3), Find(4), union(5,7), union(0, 5), Find(8)。data structure_15674337.odp2007隨堂測驗data structureheapspringThu Dec 13 20:11:35 2007執行演算法在下面的 max heap 中連續作下列兩個運算、並畫出每個運算後的結果。
i. delete maximum
ii. insert 18data structure_15672973.odp2007隨堂測驗data structureselection treewinner treeloser treespringThu Dec 13 19:59:49 2007基本定義用 10, 9, 6, 12, 5, 13, 7, 11 建一個 winner tree 與一個 loser tree.data structure_15672267.odp2007隨堂測驗data structurebinary search treebinary treespringThu Dec 13 19:55:17 2007執行演算法在下圖這個 binary search tree中連續作下面的運算、畫出每個運算之後的結果。a. insert 6, b. delete 10, c. delete 11, d. delete 9。data structure_15671995.odp2007隨堂測驗algorithmstopological sortdirected acyclic graphsdepth-first search習題22.4-1springWed Nov 21 6:36:33 2007執行演算法Show the ordering of vertices produced by TOPOLOGICAL-SORT when it is run on the following dag.algorithms_13723271.odp2007期中考題data structure英文翻譯springWed Nov 21 6:11:41 2007翻譯請寫出下列英文的中文大意
What are the virtues of postfix notation that enable easy evaluation of expressions? To begin with, the need for parentheses is eliminated. Second, the priority of the operators is no longer relevant. The expression may be evaluated by making a left to right scan, stacking operands, and evaluating operators using as operands the correct number from the stack and finally placing the result onto the stack. This evaluation process is much simpler than attempting direct evaluation from infix notation.data structure_13721779.odp2007期中考題data structurelinked listspringWed Nov 21 6:08:45 2007基本性質a).請畫出右表的 linked list, 假設第一個元素在第 8 個位置。
b).若要將 CAT 從 linked list 中刪掉，請問表中的資料要如何修改？data structure_13721603.odp2007期中考題data structurepostfixprefixspringWed Nov 21 6:03:00 2007計算Compute the value of the following expressions.
a). Postfix: 7 8 2 – 6 3 / * +
b). Prefix: / + * 6 – 7 4 8 2data structure_13721258.odp2007期中考題data structureinfixprefixpostfixspringWed Nov 21 6:00:39 2007計算Write the postfix form and prefix form of the following expressions (照C++的優先順序)data structure_13721117.odp2007期中考題data structuremazespringWed Nov 21 5:57:08 2007maze 設計請設計一個 6 × 7 的迷宮，以課本的演算法來執行時，所找到的路徑愈長愈好。data structure_13720906.odp2007期中考題data structuremazespringWed Nov 21 5:51:27 2007執行演算法下面 7× 8 的迷宮用課本老鼠走迷宮的演算法來走，請畫出所走出來的路徑。左上方為入口、右下方為出口。data structure_13720565.odp2007期中考題interconnection networksKautz networkinternally disjoint pathsroutingspringTue Nov 20 17:24:18 2007基本性質請畫出 Kautz network B(6, 5) 裡面從 63154 到 54542 之間的六條 internally disjoint paths。interconnection networks_13675736.odp2007隨堂測驗data structurefailure functionspringTue Nov 20 12:24:23 2007計算以下列 failure function 的定義，計算下面兩個字串的 failure functions。
a) xyzyxyzyxxyz;
b) aabbcaabaaba;data structure_13657741.odp2007期中考題data structuretime complexityspringTue Nov 20 12:22:44 2007證明Prove or disprove the following equalities.
a. n4 + 200n3 = (n4)
b. 2n3 + 5n2 = O(n2)data structure_13657642.odp2007期中考題data structurehanoi towerrecursiveprogrammingspringTue Nov 20 12:16:39 2007執行程式用 H(3, \'X\', \'Y\', \'Z\') 呼叫下列 C++ 的函式，其執行結果為何？data structure_13657277.odp2007期中考題data structuretime complexityinfix formpostfix formqueuespringTue Nov 20 12:14:08 2007是非題是非題
a).32n + 23n = (23n)
b).n2.0001 + n2 log2 n = (n2 log2 n)
c).KMP algorithm為作 string pattern matching 的演算法，其時間複雜度為 Q(LengthP * LengthS)，LengthP與LengthS為兩個字串的長度。
d).將 infix 轉成postfix的表示方式，其演算法為 (n)，n 是運算式的 token 數。
e).Queue 是 Last-In-First-Out、stack 是 First-In-First-Out 的資料結構。data structure_13657126.odp2007期中考題data structurestackpostfix formspringTue Nov 6 8:42:14 2007計算計算 6 3 2 + * 2 / 5 – 的值。data structure_12434812.odp2007隨堂測驗data structurestackinfix formpostfix formprefix formspringTue Nov 6 8:40:51 2007計算將 (A + B) * (C + E) / (F + A) * (D + C) 轉成 postfix 與 prefix 形式 。data structure_12434729.odp2007隨堂測驗data structurestackpostfix formspringTue Nov 6 8:39:04 2007計算計算 6 3 2 – * 2 / 5 + 的值。data structure_12434622.odp2007隨堂測驗data structurestackinfix formpostfix formspringTue Nov 6 8:37:47 2007計算將 A + (B * C + E) / (F + A) * D + C 轉成 postfix 。data structure_12434545.odp2007隨堂測驗algorithmsgraphsdepth-first searchspringMon Nov 5 18:03:32 2007給反例Give a counterexample to the conjecture that if there is a path from u to v in a directed graph G, and if d[u] ≤ d[v] in a depth-first search of G, then v is a descendant of u in the depth-first forest produced.algorithms_12382090.odp習題22.3-7algorithmsshortest pathsdirected acyclic graphspringMon Nov 5 17:58:56 2007設計演算法Give an efficient algorithm to count the total number of paths in a directed acyclic graph. Analyze your algorithm.algorithms_12381813.odp習題24.2-4algorithmsall-pairs shortest pathsJohnson\'s algorithmspringMon Nov 5 17:56:04 2007基本性質Suppose that w(u, v) ≥ 0 for all edges (u, v) ∈ E. What is the relationship between the weight functions w and w\'?algorithms_12381642.odp習題25.3-3algorithmsall-pairs shortest pathstransitive closurespringMon Nov 5 17:54:00 2007設計演算法Give an O(VE)-time algorithm for computing the transitive closure of a directed graph G(V, E).algorithms_12381518.odp習題25.2-8algorithmsall-pairs shortest pathsnegative-weight cyclespringMon Nov 5 17:52:06 2007修改演算法Modify FASTER-ALL-PAIRS-SHORTEST-PATHS so that it can detect the presence of a negative-weight cycle.algorithms_12381404.odp習題25.1-9algorithmsshortest pathsBellman-Ford algorithmnegative-weight cyclespringMon Nov 5 17:49:14 2007證明Let G = (V, E) be a weighted, directed graph with source vertex s, and let G be initialized by INITIALIZE-SINGLE-SOURCE(G, s). Prove that if a sequence of relaxation steps sets π[s] to a non-NIL value, then G contains a negative-weight cycle.algorithms_12381232.odp習題24.5-4algorithmsshortest pathslinear programmingspringMon Nov 5 17:46:48 2007執行演算法Find a feasible solution or determine that no feasible solution exists for the following system of difference constraints:
x1 – x2 ≤ 4, x1 – x5 ≤ 5,
x2 – x4 ≤ -6, x3 – x2 ≤ 1,
x4 – x1 ≤ 3, x4 – x3 ≤ 5,
x4 – x5 ≤ 10, x5 – x3 ≤ -4,
x5 – x4 ≤ -8.algorithms_12381086.odp習題24.4-2algorithmsshortest pathsDijkstra\'s algorithmspringMon Nov 5 17:45:16 2007給反例Give a simple example of a directed graph with negative-weight edges for which Dijkstra\'s algorithm produces incorrect answers. Why doesn\'t the proof of Theorem 24.6 go through when negative-weight edges are allowed?algorithms_12380994.odp習題24.3-2algorithmsshortest pathsBellman-Ford algorithmspringMon Nov 5 17:41:06 2007執行演算法Modify the Bellman-Ford algorithm so that it sets d[v] to -∞ for all vertices v for which there is a negative-weight cycle on some path from the source to v.algorithms_12380744.odp習題24.1-4algorithmsminimum cost spanning treeKruskal\'s algorithmspringMon Nov 5 17:30:45 2007分析演算法Suppose that all edge weights in a graph are integers in the range from 1 to |V|. How fast can you make Kruskal\'s algorithm run? What if the edge weights are integers in the range from 1 to W for some constant W?algorithms_12380123.odp習題23.2-4algorithmsminimum cost spanning treelight edgespringMon Nov 5 17:26:44 2007證明Show that a graph has a unique minimum spanning tree if, for every cut of the graph, there is a unique light edge crossing the cut. Show that the converse is not true by giving a counterexample.algorithms_12379882.odp習題23.1-6algorithmsgraphsstrongly connected componentsdirected graphspringMon Nov 5 17:19:35 2007基本性質How can the number of strongly connected components of a graph change if a new edge is added?algorithms_12379453.odp習題22.5-1algorithmsgraphsdepth-first searchcyclespringMon Nov 5 17:16:58 2007設計演算法Give an algorithm that determines whether or not a given undirected graph G = (V, E) contains a cycle. Your algorithm should run in O(V) time, independent of |E|.algorithms_12379296.odp習題22.4-3algorithmsgraphsbreadth-first searchspringMon Nov 5 17:10:56 2007執行演算法Show the d and π values that result from running breadth-first search on the directed graph of Figure 22.2(a), using vertex 3 as the source.algorithms_12378934.odp習題22.2-1algorithmsdisjoint setsunion and findspringMon Nov 5 17:09:01 2007證明Professor Dante reasons that because node ranks increase strictly along a path to the root, node levels must monotonically increase along the path. In other words, if rank(x) > 0 and p[x] is not a root, then level(x) ≤ level(p[x]). Is the professor correct?algorithms_12378819.odp習題21.4-5algorithmsdisjoint setsunion and findspringMon Nov 5 17:07:14 2007設計演算法Suggest a simple change to the Union procedure for the linked-list representation that removes the need to keep the tail pointer to the last object in each list. Whether or not the weighted-union heuristic is used, your change should not change the asymptotic running time of the Union procedure. (Hint: Rather than appending one list to another, splice them together.)algorithms_12378712.odp習題21.2-5data structuremazing problemspringFri Oct 26 15:17:21 2007設計迷宮請設計一個 4 × 5 的迷宮，以課本的演算法來執行時，所找到的路徑愈長愈好。data structure_11508119.odp2007隨堂測驗data structuremazing problemspringFri Oct 26 15:13:53 2007執行演算法下面 6 × 8 的迷宮用課本老鼠走迷宮的演算法來走，請畫出所走出來的路徑。左上方為入口、右下方為出口。data structure_11507911.odp2007隨堂測驗interconnection networksdiameterHamiltonianCartesian productspringFri Oct 26 15:10:39 2007畫圖關於下列圖形 (a) 請問其 diameter 為何？ (b) 請畫出一個 Hamiltonian cycle 。interconnection networks_11507717.odp2007隨堂測驗data structuremazing problemspringThu Oct 25 14:48:21 2007執行演算法下面 6 × 7 的迷宮用課本老鼠走迷宮的演算法來走，請畫出所走出來的路徑。左上方為入口、右下方為出口。data structure_11419979.odp2007隨堂測驗data structure英文翻譯springWed Oct 24 15:09:47 2007翻譯請寫出下列英文的中文大意。
The concept of an algorithm is fundamental to computer science. Algorithms exist for many common problems, and designing efficient algorithms plays a crucial role in developing large-scale computer systems. Therefore, before we proceed further, we discuss this concept more fully.data structure_11334865.odp2007平常測驗一data structurefailure functionspringWed Oct 24 15:08:40 2007計算若 failure function 改成下列定義，上面兩個字串的 failure functions 的值會是多少？
a) zzyxzzxzzyxz;
b) ccababccabac;data structure_11334798.odp2007平常測驗一data structurefailure functionspringWed Oct 24 15:07:28 2007計算以下列 failure function 的定義，計算下面兩個字串的 failure functions。
a) zzyxzzxzzyxz;
b) ccababccabac;data structure_11334725.odp2007平常測驗一data structurearrayarray representationspringWed Oct 24 15:06:27 2007計算How many values can be held by the array A[3][4][3][2] in C++ ?Assume that the address of A[0][0][0][0][0] is 1000 and every element is one byte. What is the address of A[1][1][2][0] ?data structure_11334665.odp2007平常測驗一data structuretime complexityBig OThetaspringWed Oct 24 15:05:24 2007證明Prove or disprove the following equalities.
a. n3 + 106n2 = (n3) b. 33n3 + 4n2 = O(n2)data structure_11334602.odp2007平常測驗一data structurerecursive functionAckermann\'s functionspringWed Oct 24 15:01:16 2007程式設計Write a C++ recursive function for the following function.data structure_11334354.odp2007平常測驗一data structurearraymatrix transposefast transposespringWed Oct 24 14:59:52 2007計算Write the transpose of the following 65 sparse matrix using the FastTranspose algorithm. You must also write down the two arrays RowSize and RowStart.data structure_11334270.odp2007平常測驗一data structurerecursive functionHanoi TowerspringWed Oct 24 14:58:12 2007程式設計寫下搬動河內塔(Hanoi tower)的遞迴函式，令三個塔分別是\'A\', \'B\', \'C\'，若剛開始的 8 個圓盤都放在 \'C\'，最後要搬到 \'B\'，寫下呼叫您函式的方法。data structure_11334170.odp2007平常測驗一data structuretime complexityThetaspringWed Oct 24 14:56:58 2007計算What is the time complexity of the following algorithm? 以與 n 來表示。data structure_11334096.odp2007平常測驗一data structurerecursive functionspringWed Oct 24 14:55:24 2007簡答請描述下列函式 f 的功能data structure_11334002.odp2007平常測驗一data structuretime complexityBig OOmegaThetaspringWed Oct 24 14:53:54 2007是非題是非題data structure_11333912.odp2007平常測驗一data structure英文翻譯springWed Oct 24 14:49:58 2007翻譯請寫出下列英文的中文大意
Assume that we have two strings, s and pat, where pat is a pattern to be searched for in s. We will determine if pat is in s by using the function Find. The invocation s.Find(pat) returns an index i such that pat matches the substring of s that begins at position i. It returns -1 if and only if pat is either empty or is not a substring of s. Let us examine how a function Find may be implemented.data structure_11333676.odp2007平常測驗一data structurefailure functionspringWed Oct 24 14:48:26 2007計算若 failure function 改成下列定義，下面兩個字串的 failure functions 的值會是多少？
a) abcaabcab;
b) bacbabcba;data structure_11333584.odp2007平常測驗一data structurefailure functionspringWed Oct 24 14:44:52 2007計算以下列 failure function 的定義，計算下面兩個字串的 failure functions。
a) abcaabcab;
b) bacbabcba;data structure_11333370.odp2007平常測驗一data structurearrayrow majorspringWed Oct 24 14:42:53 2007計算How many values can be held by the array A[2][4][3][2][4] in C++ ?Assume that the address of A[0][0][0][0][0] is 10000 and every element is one byte. What is the address of A[1][3][0][1][2] ?data structure_11333251.odp2007平常測驗一data structurearraymatrix transposefast transposespringWed Oct 24 14:41:18 2007執行演算法Write the transpose of the following 6x5 sparse matrix using the FastTranspose algorithm. You must also write down the two arrays RowSize and RowStart.data structure_11333156.odp2007平常測驗一data structuretime complexityBig OThetaspringWed Oct 24 14:39:25 2007證明Prove or disprove the following equalities.
a. 20n2 + 4n = (n2) b. 2n3 + 5n = O(n2)data structure_11333043.odp2007平常測驗一data structurerecursive functionselectionspringWed Oct 24 14:36:23 2007程式設計Write a C++ recursive function for the following function.data structure_11332861.odp2007平常測驗一data structurerecursive functionHanoi TowerspringWed Oct 24 14:33:39 2007程式設計寫下搬動河內塔的遞迴函式，令三個塔分別是\'X\', \'Y\', \'Z\'，若剛開始的 5 個圓盤都放在 \'Y\'，最後要搬到 \'Z\'，寫下呼叫您函式的方法。data structure_11332697.odp2007平常測驗一data structuretime complexityprogramspringWed Oct 24 14:28:43 2007計算
What is the time complexity of the following algorithm? 以與 n1, n2, n3 來表示。
data structure_11332401.odp2007平常測驗一data structuretime complexityBig OOmegaThetaspringWed Oct 24 14:26:40 2007是非題是非題
n! = O(nn)
C++ 的 class 中之 constructor 與 destructor 都不能有參數。
n2.00001 + n2 log n = Ө(n2log n)
The worst case of selection sort for n elements is Ө(n2).
若 f(n) = (g(n)) 則 f(n) = O(g(n)) 且 f(n) = (g(n))。data structure_11332278.odp2007平常測驗一algorithmsshortest pathssingle source shortest pathsdirected acyclic graphstopological sortspringWed Oct 24 14:18:29 2007執行演算法Run DAG-SHORTEST-PATHS on the directed graph of Figure 24.5, using vertex r as the source.algorithms_11331787.odp習題24.2-1algorithmsshortest pathssingle source shortest pathsBellman-Fordnegative edgespringWed Oct 24 14:14:54 2007執行演算法Run the Bellman-Ford algorithm on the directed graph of Figure 24.4, using vertex z as the source.algorithms_11331572.odp習題24.1-1algorithmsminimum spanning treePrim\'s algorithmspringWed Oct 24 6:21:34 2007計算Is the Fibonacci-heap implementation of Prim\'s algorithm asymptotically faster than the binary-heap implementation for a sparse graph G = (V, E), where |E| = Θ(V)? What about for a dense graph, where |E| = Θ(V2)? How must |E| and |V| be related for the Fibonacci-heap implementation to be asymptotically faster than the binary-heap implementation?algorithms_11303172.odp習題23.2-32007隨堂測驗algorithmsminimum spanning treespringWed Oct 24 6:19:17 2007證明Let (u, v) be a minimum-weight edge in a graph G. Show that (u, v) belongs to some minimum spanning tree to G.algorithms_11303035.odp習題23.1-32007隨堂測驗data structurestringfailure functionspringWed Oct 24 6:06:55 2007計算若 failure function 改成下列定義，上面兩個字串的 failure functions 的值會是多少？
(a).01210101201;
(b).aabbaabbbab;data structure_11302293.odp2007隨堂測驗data structurestringfailure functionspringWed Oct 24 6:04:44 2007計算以下列 failure function 的定義，計算下面兩個字串的 failure functions。
(a).01210101201;
(b).aabbaabbbab;data structure_11302162.odp2007隨堂測驗data structurestringfailure functionspringWed Oct 24 6:01:22 2007計算若 failure function 改成下列定義，上面兩個字串的 failure functions 的值會是多少？
(a).abaaab;
(b).ababaa;data structure_11301960.odp2007隨堂測驗data structurestringfailure functionspringWed Oct 24 5:58:38 2007計算以下列 failure function 的定義，計算下面兩個字串的 failure functions。
(a) abaaab;
(b) ababaa;data structure_11301796.odp2007隨堂測驗data structurecomplexityclassspringTue Oct 9 21:37:01 2007是非題是非題
A. The worst case of binary search for n elements is Ө(n).
B. C++ 的 class 中 constructor 與 destructor 都不能有傳回值資料型態。
C. n2.0001+ n2log n = Ө(n2log n)
D. C++ 的 class 中，放在 public 後面的 data member 或 member function 都不能被類別以外的程式呼叫。data structure_10062099.odp2007年隨堂測驗data structurecomplexityThetaspringTue Oct 9 21:34:07 2007證明題證明或反證下列的式子， n2/ log n = (n2)。data structure_10061924.odp2007年隨堂測驗data structurecomplexityclassspringTue Oct 9 21:32:04 2007是非題是非題
A. The worst case of selection sort for n elements is Ө(n).
B. n1.001 + n log n = Ө(n log n)
C. C++ 的 class 中 constructor 與 destructor 都不能有傳回值資料型態。
D. C++ 的 class 中 constructor 與 destructor 都不能有參數。data structure_10061802.odp2007年隨堂測驗data structurecomplexitybig OspringTue Oct 9 21:29:40 2007證明證明或反證下列的式子，n2log n = O(n2)data structure_10061658.odp2007年隨堂測驗interconnection networksminimum degreeconnectivityedge connectivityspringTue Oct 9 21:22:31 2007畫圖請畫一個滿足下列性質的圖形 G，κ(G) = 4, λ(G) = 6, δ(G) = 7。interconnection networks_10061229.odp2007隨堂測驗interconnection networksvertex cutedge cutspringTue Oct 9 21:19:29 2007畫圖請分別畫出一個下列圖形的 κ-cut 與 λ-cut。interconnection networks_10061047.odp2007隨堂測驗algorithmsbreadth-first searchbipartite graphspringWed Oct 3 17:35:26 2007設計演算法There are two types of professional wrestlers(摔角選手): \"good guys\" and \"bad guys\". Between any pair of professional wrestlers, there may or may not be a rivalry(競賽). Suppose we have n professional wrestlers and we have a list of r pairs of wrestlers for which there are rivalries. Given an O(n+r)-time algorithm that determines whether it is possible to designate some of the wrestlers as good guys and the remainder as bad guys such that each rivalry is between a good guy and a bad guy. If it is possible to perform such a designation, your algorithm should produce it.algorithms_9529204.odp習題22.2-6algorithmsbreadth-first searchspringWed Oct 3 17:25:59 2007執行演算法Show the d and π values that result from running breadth-first search on the undirected graph of Figure 22.3, using vertex u as the source.algorithms_9528637.odp習題22.2-2algorithmsgraphsadjacency listadjacency matrixsquare of a graphspringWed Oct 3 16:09:44 2007基本性質The square of a directed graph G = (V, E) is the graph G2 = (V, E2) such that (u, w) ∈ E2 if and only if for some v ∈ V, both (u, v) ∈ E and (v, w) ∈ E. That is, G2 contains an edge between u and w whenever G contains a path with exactly two edges between u and w. Describe efficient algorithms for computing G2 from G for both the adjacency-list and adjacency-matrix representations of G. Analyze the running times of your algorithms.algorithms_9524062.odp習題22.1-5algorithmsgraphs基本性質multigraphspringWed Oct 3 15:44:43 2007基本性質Given an adjacency-list representation of a multigraph G = (V, E), describe an O(V + E)-time algorithm to compute the adjacency-list representation of the \"equivalent\" undirected graph G\' = (V, E\'), where E\' consists of the edges in E with all multiple edges between two vertices replaced by a single edge and with all self-loops removed.algorithms_9522561.odp習題22.1-4algorithmsgraphs基本性質indegreeoutdegreespringWed Oct 3 15:35:16 2007基本性質Given an adjacency-list representation of a directed graph, how long does it take to compute the out-degree of every vertex? How long does it take to compute the in-degrees?algorithms_9521994.odp習題22.1-1algorithmsgraphsbreadth-fire searchspringWed Oct 3 12:42:37 2007執行演算法Show the d and π values that result from running breadth-first search on the following directed graph, using vertex 3 as the source.algorithms_9511635.odp習題22.2-1algorithmsgraphssinkerspringWed Oct 3 12:40:26 2007設計演算法When an adjacency-matrix representation is used, most graph algorithms require time Ω(V2), but there are some exceptions. Show that determining whether a directed graph G contains a universal sink – a vertex with in-degree |V| - 1 and out-degree 0 – can be determined in time O(V), given an adjacency matrix for G.algorithms_9511504.odp習題22.1-6data structurecomplexitybig-Obig-OmegaspringTue Oct 2 21:15:31 2007證明Show that the following equalities are correct:
a) 5n2 + 6n = O(n2)
b) 33n3 – 20n2 = (n3)data structure_9456008.odp2007隨堂測驗data structurerecursive functionspringTue Oct 2 21:12:56 2007程式設計寫一個C++的遞迴函式來計算下列遞迴方程式的fi。data structure_9455854.odp2007隨堂測驗data structurerecursive functionlcmgcdspringMon Oct 1 11:33:14 2007程式設計如何求三個整數的 gcd 與 lcm 呢？data structure_9334672.odp2007隨堂測驗data structurerecursive functionlcmspringMon Oct 1 11:32:13 2007程式設計利用 gcd 的函式如何求最小公倍數？lcm(m,n)data structure_9334611.odp2007隨堂測驗data structurerecursive functiongcdspringMon Oct 1 11:30:53 2007程式設計寫一個遞迴函式、用輾轉相除法來計算最大公因數gcd，遞迴方程式如下：假設 m ≥ n。data structure_9334531.odp2007隨堂測驗data structurebinomial coefficientrecursive functionspringSat Sep 29 6:12:59 2007程式設計請寫一個遞迴函式來計算「從 n 個不同的東西挑出 m 個的方法數」，遞迴公式是：C(n, m) = C(n-1, m) + C(n-1, m-1)，而 C(n, 0) = C(n, n) = 1。data structure_9142657.odp習題algorithmsdisjoint setsunion and find習題springWed Sep 26 13:47:36 2007證明Show that any sequence of m Make-set, Find-set, and Union operations, where all the Union operations appear before any of the Find-set operations, takes only O(m) time if both path compression and union by rank are used. What happens in the same situation if only the path-compression heuristic is used?algorithms_8910734.odp習題21.3-4algorithmsdisjoint setsunion and findspringWed Sep 26 13:43:00 2007執行演算法0~11的數字，假設一開始每個數字都是一個集合，用union by rank與path compression作以下的運作，請畫出最後的結果。Union(0, 1), Union(2, 3), Union(6, 7), Union(8, 9), Union(10, 11), Union(0, 2), Union(8, 10), Union(0, 4), Union(4, 5), Find-set(6), Union(0, 8), Union(4, 6), Find-set(8)。algorithms_8910458.odpalgorithmsgraphstreediameterspringTue Sep 25 16:21:27 2007設計演算法Given an efficient algorithm to compute the diameter of a tree, and analyze the running time of your algorithm.algorithms_8833565.odp習題 22.2-7interconnection networksgraphs基本性質springTue Sep 25 13:59:26 2007畫圖請畫 出下列圖形 G 的 induced subgraph G[{1,2,3, 6,7, 8}]interconnection networks_8825044.odp2007隨堂測驗interconnection networksgraphs基本性質springTue Sep 25 13:57:26 2007證明Show that for any bipartite graph G, ε(G) ≤ 1/4 υ2 if υ is even and ε(G) ≤ 1/4 (υ – 1)(υ + 1) if υ is odd, where υ = υ(G).interconnection networks_8824924.odp2007隨堂測驗interconnection networksgraphsisomorphicspringTue Sep 25 13:55:39 2007證明請證明下列兩個圖形是 isomorphic。interconnection networks_8824817.odp2007隨堂測驗data structureshortest pathssingle source shortest pathsDijstra\'s algorithmgraphspringWed Sep 12 17:40:38 2007執行演算法請找出下圖中的 D 到其他所有點的 shortest paths。data structure_7715115.odp2006隨堂測驗data structurespanning treedepth-first searchbreadth-first searchspringWed Sep 12 15:55:45 2007執行演算法找出下圖中以 A 為起點的一個 DFS spanning tree 與一個 BFS spanning tree.(忽略邊的 weight)data structure_7708822.odp2006隨堂測驗data structureminimum cost spanning treePrim\'s algorithmKruskal\'s algorithmSollin\'s algorithmspringWed Sep 12 15:48:07 2007執行演算法請分別用Kruskal’s algorightm、Prim’s algorithm(以C為起點)與Sollin\'s algorithm找下圖的minimum cost spanning tree。請寫下加edges的順序。data structure_7708365.odp2006隨堂測驗data structuregraphsdegreedepth-first searchbreadth-first searchspringWed Sep 12 15:21:25 2007基本性質For the digraph in below obtain
(A) the in-degree and out-degree of each vertex
(B) its adjacency-matrix
(C) its adjacency-list representation
(D) its DFS(5) spanning tree
(E) its BFS(3) spanning treedata structure_7706763.odp2006隨堂測驗data structureAckermann\'s functionrecursive functionspringWed Sep 12 14:50:57 2007計算What are the values of the Ackermann\'s functions A(2, 2), A(2, 3), A(3, 1), and A(3, 2)?data structure_7704935.odp2006隨堂測驗data structurebinary treetree traversalpreorderlevel-orderspringWed Sep 12 14:44:34 2007執行演算法Draw two binary trees with the same preorder sequence and the same level-order sequence.data structure_7704552.odp2006隨堂測驗data structuretreesbinary treespringWed Sep 12 14:37:28 2007執行演算法How many are the numbers of distinct binary trees having 5 and 6 nodes?data structure_7704126.odp2006隨堂測驗data structurebinary treetree traversalpreorderinorderspringWed Sep 12 14:34:53 2007執行演算法Let T be a binary tree. The preorder sequence and inorder sequence of T are abdehklicfgj and dbaklheifcjg, respectively. Please draw the binary tree T.data structure_7703971.odp2006隨堂測驗data structuretreeheapspringWed Sep 12 10:55:45 2007執行演算法在右邊的 max heap 中連續作下列兩個運算、並畫出每個運算後的結果。A)delete maximum B) insert 23data structure_7690823.odp2005隨堂測驗data structureselection treewinner treeloser treespringWed Sep 12 10:27:47 2007執行演算法用 8, 9, 6, 12, 21, 13, 7, 15 建一個 winner tree 與一個 loser tree.data structure_7689145.odp2005隨堂測驗data structurebinary search treespringWed Sep 12 10:23:12 2007執行演算法在右圖這個 binary search tree中連續作下面的運算、畫出每個運算之後的結果。a. insert 3, b. delete 10, c. delete 12, d. delete 8。data structure_7688870.odp2005隨堂測驗data structuretreesbinary treespringWed Sep 12 9:59:37 2007基本性質Let T be a binary tree with 12 leaves. How many nodes of degree 2 does T have?data structure_7687455.odp2005隨堂測驗data structuretreesbinary treecomplete binary treespringWed Sep 12 9:57:54 2007計算Let T be a complete binary tree with 23 nodes. What is the height of T?data structure_7687352.odp2005隨堂測驗data structuretreesspringWed Sep 12 9:56:10 2007計算Let T be a tree with 20 nodes. Assume that every node of T has 4 pointers to its children. How many pointers of T are 0?data structure_7687248.odp2005隨堂測驗data structurelinked listsgeneralized listsspringWed Sep 12 9:52:01 2007計算請用 generalized list 的方式畫出多項式 3x2y8z2 + 4x3y2z2 + 2x4y4z + 3x3y4z2 + 2y2z 的 list。data structure_7686999.odp2005隨堂測驗data structurelinked listsgeneralized listsdepthspringWed Sep 12 9:14:27 2007計算What is the depth of the following generalized list ?data structure_7684745.odp2005隨堂測驗data structurelinked listsgeneralized listspolynomialspringWed Sep 12 9:11:18 2007計算請用 generalized list 的方式畫出下列兩個多項式的 lists。A. 3x3y ＋ 4x2y B. 3x7y3z2 + 4x6y2z2 + 2x4y4z + 3x3y4z2 + 2y2zdata structure_7684556.odp2005隨堂測驗data structurelinked listsequivalence classspringWed Sep 12 8:47:51 2007執行演算法令S = {0,1,2,3,4,5,6,7,8,9,10,11,12,13}， pairs of equivalence relation: 0 ≡ 1, 4 ≡ 13, 6 ≡ 9, 1 ≡ 10, 11 ≡ 12, 4 ≡ 5, 2 ≡ 8, 0 ≡ 9, 1 ≡ 11, 7 ≡ 10，請輸出其 equivalence classdata structure_7683149.odp2005隨堂測驗data structurelinked listspolynomialspringWed Sep 12 8:41:04 2007執行演算法請畫出下列兩個多項式相加之後的結果。data structure_7682742.odp2005隨堂測驗data structurelinked listsequivalence classspringWed Sep 12 8:37:28 2007執行演算法令S = {0,1,2,3,4,5,6,7,8,9,10,11,12}， pairs of equivalence relation: 0 ≡ 3, 4 ≡ 7, 6 ≡ 9, 10 ≡ 1, 12 ≡ 11, 5 ≡ 4, 8 ≡ 2, 9 ≡ 0, 11 ≡ 1, 7 ≡ 10，請輸出其 equivalence classdata structure_7682526.odp2005隨堂測驗data structurelinked listspolynomialspringWed Sep 12 8:24:59 2007執行演算法請畫出下列兩個多項式相加之後的結果。data structure_7681777.odp2005隨堂測驗data structureprefix formpostfix formspringTue Sep 11 15:25:12 2007計算將 + – / * a + b c d e f 轉成 postfix formdata structure_7620590.odp2005隨堂測驗data structureevaluateprefix formspringTue Sep 11 15:22:29 2007計算計算 + / * 4 – 3 2 2 5 的值。data structure_7620427.odp2005隨堂測驗data structureinfix formprefix formpostfix formspringTue Sep 11 15:20:44 2007執行演算法將 A + (B * C + E) / (F + A) * D + C 轉成 postfix 與 prefix form。data structure_7620322.odp2005隨堂測驗data structuretree traversalinorderpreorderspringTue Sep 11 15:12:53 2007執行演算法請畫出符合inorder sequence為 f d g b e a h c j i k，而preorder sequence為a b d f g e c h i j k 的binary tree。data structure_7619851.odp2005隨堂測驗data structuredisjoint setsunion and findspringTue Sep 11 15:07:26 2007執行演算法0~11的數字，假設一開始每個數字都是一個集合，用weighting rule for union與Collapsing rule for find作以下的運作，請畫出最後的結果。union(0, 1), union(2, 3), union(4, 5), union(6, 7), union(8, 9), union(10, 11), union(0, 2), union(4, 6), union(8, 10), union(0, 4), find(7), union(0, 8), find(11)。data structure_7619524.odp2005隨堂測驗data structuretree traversalforestinorderpreorderspringTue Sep 11 14:56:33 2007執行演算法請寫出下列forest的preorder與inorder traversal, tree的順序是由左到右。data structure_7618871.odp2005隨堂測驗data structureselection treewinner treeloser treespringTue Sep 11 14:51:47 2007執行演算法十六個非遞減數列的第一個數字分別為12, 25, 6, 8, 10, 2, 13, 17, 22, 11, 3, 24, 5, 15, 19, 23，請畫出其winner tree與loser tree。data structure_7618585.odp2005隨堂測驗data structurebinary search treespringTue Sep 11 14:41:07 2007執行演算法請對下圖這個binary search tree連續作下列的運算：a. Insert 28, b. delete 15, c. delete 30, d. insert 36,e. delete 20data structure_7617945.odp2005隨堂測驗data structureheapspringTue Sep 11 14:13:22 2007執行演算法請對右上圖這個max heap連續作下列的運算：a. Insert 18, b. insert 13, c. delete max, d. insert 11data structure_7616280.odp2005隨堂測驗data structurebinary treetree traversalspringMon Sep 10 16:49:06 2007執行演算法請對下圖分別作inorder, preorder, postorder, level-order traversal，請寫出每一種輸出的結果。data structure_7539224.odp2005隨堂測驗data structurebinary treespringMon Sep 10 16:42:54 2007基本性質請問高度為k的binary tree最多有幾個點？
若binary tree T有15個leaf，請問它有幾個degree為2的點？
T為一個32個點的complete binary tree，請問T的高度是多少？data structure_7538852.odp2005隨堂測驗data structurethreaded binary treespringMon Sep 10 16:38:49 2007基本性質Threaded binary trees與原來的binary trees有何不同？
Threaded binary trees的比原來的binary trees多了哪些部份？data structure_7538607.odp2005隨堂測驗data structurebinary treespringMon Sep 10 16:34:45 2007基本性質請問copy一個n個點的binary tree的時間複雜度是多少？
判斷兩個n個點的binary trees是否完全相等，其時間複雜度是多少？data structure_7538363.odp2005隨堂測驗data structurebinary treesatisfiablity problemspringMon Sep 10 16:32:41 2007基本性質請問下列satisfiability problem的答案是true還是false？(x1 or -x2) and (-x1 or x3) and -x3 and (x3 or x2)data structure_7538239.odp2005隨堂測驗data structurebinary treetree traversalspringMon Sep 10 16:27:13 2007執行演算法請對下圖分別作inorder, preorder, postorder, level-order traversal，請寫出每一種輸出的結果。請問前面幾種tree traversal的時間複雜度是多少？data structure_7537911.odp2005隨堂測驗data structurebinary treespringMon Sep 10 16:23:27 2007基本性質請畫一個6個點的complete binary tree。data structure_7537685.odp2005隨堂測驗data structurebinary treespringMon Sep 10 16:19:47 2007基本性質請問一個binary tree最少有幾個點？
請問n層的binary tree最多有幾個點？
若binary tree T有8個leaf，請問它有幾個degree為2的點？
T為一個18個點的complete binary tree，請問T的高度是多少？data structure_7537465.odp2005隨堂測驗data structureAckermann\'s functionrecursive functionspringSat Sep 8 19:13:58 2007程式設計Write a recursive function for computing the Ackermann\'s function A(m, n).data structure_7375116.odpdata structureequivalence classeslinked listspringSat Sep 8 10:56:51 2007執行演算法共有 0~14 的元素，輸入1 ≡ 2, 2 ≡ 8, 2 ≡ 13, 3 ≡ 6, 4 ≡ 7, 4 ≡ 10, 6 ≡ 7, 9 ≡ 12, 11 ≡ 13，請問其 equivalence classes 共有哪些？data structure_7345289.odpdata structureprefix formevaluatespringSat Sep 8 10:40:32 2007計算Evaluate the following prefix expression:
/ + / * 8 5 4 + 6 2 3data structure_7344310.odpdata structureprefix formpostfix formspringSat Sep 8 10:38:38 2007執行演算法Write the postfix form of the following prefix expression: + a * b * / c + + d e a bdata structure_7344196.odpdata structureprefix forminfix formspringSat Sep 8 10:30:58 2007執行演算法Write the prefix form of the following infix expressions:
A) a / (b * c + d)
B) – (a – b) * (c / d) || e / (a ＆＆ b – c) ＆＆ ddata structure_7343736.odpdata structureAckermann\'s functionrecursive functionspringSat Sep 8 8:44:25 2007計算The definition of Ackermann\'s function is as follows. What are the values of A (2, 1), A(2, 2), A(3,1) and A(4, 1)?data structure_7337342.odpdata structuredisjoint setunion and findspringSat Sep 8 8:38:37 2007執行演算法假設有0~9共十個數字，一開始每個數字都是一個集合，用 weighting rule 與 collapsing rule 作集合的union and find，請畫出作完下列運算之後的集合。union(0, 3), union(1, 5), union(2, 9), union(0, 2), union(8, 1), union(1, 0), union(4, 6), union(4, 0), find(8), find(6)data structure_7336995.odpdata structurebinary treespringSat Sep 8 8:26:44 2007證明請證明：For any nonempty binary tree T, if n0 is the number of leaf nodes and n2 is the number of nodes of degree 2, then n0 = n2 + 1.data structure_7336282.odpdata structureselection treewinner treeloser treespringSat Sep 8 8:21:47 2007執行演算法若有八個遞增數列要作merge，八個數列的第一個元素分別為：3, 9, 2, 7, 16, 4, 15, 6，請畫出一開始的winner tree與loser tree。若第三個數列在 2 之後的元素為8，請畫出輸出最小值、 補上另一個數字之後的 winner tree 與 loser tree。data structure_7335985.odpdata structureheapspringSat Sep 8 8:13:38 2007執行演算法請在圖二的 max heap 中連續作以下的運算：(a) delete max (b) insert 28 (c) insert 32。請畫出每次運算、調整結束後的heap。data structure_7335496.odpdata structuretree traversalinorderlevel-orderspringSat Sep 8 7:52:39 2007建 binary treeThe inorder sequence and level-order sequence of a binary tree T are ECFBGDHAIKMLNJ and ABICDJEFGHKLMN, respectively. Please draw the binary tree T.data structure_7334237.odpdata structurebinary search treespringSat Sep 8 7:16:53 2007執行演算法將圖一當作一個binary search tree來看待，請連續作下列的運算：(a). insert 10 (b). delete 4 (c) delete 15 (d) delete 8 (e) insert 9。畫出每個運算之後的binary search tree。data structure_7332090.odpdata structuretree traversalpreorderpostorderlevel-orderspringFri Sep 7 22:12:58 2007執行演算法Write the inorder, preorder, postorder, and level-order sequence of the tree in Figure 1.data structure_7299456.odpdata structuredisjoint setsweighting rulespringFri Sep 7 16:49:57 2007證明假設一開始每棵 tree 都只有一個點，之後用 weighting rule 的方式作 union，若最後有一棵 tree T 的點數為 m，請證明 T 的高度不會超過 ⌊log2 m⌋ + 1。data structure_7280075.odpdata structuredisjoint setsunion and findspringFri Sep 7 16:46:41 2007執行演算法假設有0~8共九個數字，一開始每個數字都是一個集合，用 weighting rule 與 collapsing rule 作集合的union and find，請畫出作完下列運算之後的集合。union(0, 1), union(2, 3), union(2, 4), union(5, 6), union(7, 0), union(2, 0), union(2, 8), union(5, 2), find(7)data structure_7279879.odpdata structurebinary treematrix multiplicationspringFri Sep 7 16:33:38 2007計算請問6個矩陣相乘共有幾種不同的乘法順序？6個點的binary tree 共有幾種？data structure_7279096.odpdata structureselection treewinner treeloser treespringFri Sep 7 16:29:45 2007執行演算法若有八個遞增數列要作merge，八個數列的第一個元素分別為：7, 8, 9, 3, 7, 6, 4, 2，請畫出一開始的winner tree與loser tree。假設最後一個數列中2之後為5，當輸出最小值之後的 winner tree 與 loser tree 會變成什麼樣子。data structure_7278863.odpdata structureheapspringFri Sep 7 16:13:52 2007執行演算法請在圖二的 max heap 中作以下的運算：(a) insert 32 (b) insert 23 (c) delete max。請畫出每次運算、調整結束後的heap。data structure_7277910.odpdata structurestack排列springFri Sep 7 15:42:02 2007計算將 1, 2, ..., n 的數字按照順序放到 stack 裡面並輸出，當 n = 3 與 n = 4 時，其輸出的順序各有哪些？data structure_7276000.odpdata structurepostfix formevaluatespringFri Sep 7 15:37:06 2007計算Compute the value of the following postfix form expressions:
a. 7 8 – 6 4 2 * / +
b. 3 8 1 2 * – + 4 5 6 – * +data structure_7275704.odpdata structurepostfix forminfix formspringFri Sep 7 15:33:42 2007執行演算法Write the postfix form of the following expressions:
A) A / (b * - (c + d))
B) (A + B) * D + E * (F + A / D) + Cdata structure_7275500.odpdata structuremazespringFri Sep 7 15:12:45 2007maze 設計請設計一個 7 × 5 的迷宮，使得課本的 maze 演算法執行的路徑最長。data structure_7274243.odpdata structuremazespringFri Sep 7 14:54:48 2007執行演算法下方 6 × 7 的迷宮用課本老鼠走迷宮的演算法來走，請畫出所走出來的路徑。左上方為入口、右下方為出口。data structure_7273166.odpdata structurefailure functionspringFri Sep 7 14:42:01 2007計算Compute the failure function for each of the following patterns
1). b a a b a b a a b b a b a a
2). b c d a a b c d a b c d a a adata structure_7272399.odpdata structuretime complexityspringFri Sep 7 9:12:42 2007證明Prove or disprove the following equalities
1) 24n2– 1000n = Ө(n2)
2) n2/ log n = Ө(n2)data structure_7252640.odpdata structuretree traversalinorderpostordertreespringSun Aug 26 6:06:10 2007執行演算法The inorder sequence and postorder sequence of a binary tree T are CEDFBAHKJLIG and EFDCBKLJIHGA, respectively. Please draw the binary tree T.data structure_6204648.odpdata structurebinary search treespringSun Aug 26 5:48:42 2007執行演算法將圖一當作一個binary search tree來看待，請連續作下列的運算：(a). insert 11 (b). delete 5 (c) delete 6 (d) delete 12 (e) insert 7。畫出每個運算之後的binary search tree。data structure_6203600.odpdata structurebinary treetree traversalinorderlevel-orderspringSun Aug 26 5:21:09 2007執行演算法Write the inorder, preorder, postorder, and level-order sequence of the tree in 圖一。data structure_6201947.odpdata structuredepth-first spanning treedepth first numberbiconnected componentsgraphspringSat Aug 25 17:44:08 2007執行演算法請畫出一個圖五以3為起點的 DFS spanning tree，算出每個點的 depth-first number，並計算每個點的 low，最後請寫出這個圖的 articulation points。data structure_6160126.odpdata structurepostfix formprefix formspringSat Aug 25 17:07:02 2007計算Write the postfix form of the following prefix expression:
a). * + a b / – c d + e f
b). || ＆＆ ＞ a b ＜ b c ＜ ─ a b ddata structure_6157900.odpdata structurebinary treetree traversalsgraphspringSat Aug 25 16:48:02 2007執行演算法一個binary tree的inorder與 level order sequence分別為：BHFIDAEJGKC與ABCDEFGHIJK，請畫出這個binary tree。data structure_6156760.odpdata structureshortest pathsall-pairs shortest pathsFloyd-Warshall algorithmgraphspringSat Aug 25 16:37:43 2007執行演算法請找出圖四的all pairs shortest paths，寫出長度即可。data structure_6156141.odpdata structureshortest pathssingle source shortest pathsDijstra\'s algorithmgraphspringSat Aug 25 16:20:29 2007執行演算法請問圖三中點A到其他點的shortest paths之長度？data structure_6155107.odpdata structureminimum cost spanning treePrim\'s algorithmKruskal\'s algorithmSollin\'s algorithmspringSat Aug 25 16:05:23 2007執行演算法請分別用Kruskal’s algorightm、Prim’s algorithm(以C為起點)與Sollin\'s algorithm找圖二的minimum cost spanning tree。請寫下加edges的順序。data structure_6154201.odpdata structuredepth first spanning treebreadth first spanning treetree traversalspringSat Aug 25 8:14:20 2007執行演算法請畫出圖一中由點4開始的一個DFS spanning tree與一個BFS spanning tree；並各給一個DFS與BFS的拜訪順序。data structure_6125938.odpdata structureBinomial heapspringFri Aug 24 22:02:09 2007執行演算法Assume that Figure 4 is a Binomial heap. Draw the result after insert 16 and then delete 42.data structure_6089207.odpdata structureFibonacci heapspringFri Aug 24 21:59:51 2007執行演算法Assume that Figure 4 is a Fibonacci heap. Draw the result after insert 16 and then delete 42.data structure_6089068.odpdata structureshortest pathsall-pairs shortest pathsspringFri Aug 24 21:58:50 2007執行演算法Find all-pairs of shortest paths of Figure 3.data structure_6089008.odpdata structuredepth-first spanning treebreadth-first spanning treespanning treespringFri Aug 24 21:57:38 2007執行演算法Draw a depth-first spanning tree and a breadth-first spanning tree for Figure 2 with root 5.data structure_6088936.odpdata structurebinary treetree traversalsspringFri Aug 24 21:56:08 2007執行演算法Write out the inorder, preorder, postorder, and level-order traversals for the binary tree of the Figure 1.data structure_6088846.odpdata structuretreebinary treespringFri Aug 24 21:53:17 2007計算How many distinct binary trees are there with 6 nodes?data structure_6088675.odpdata structureprefixpostfixinfixspringFri Aug 24 21:51:22 2007計算Write the postfix and prefix of the following expressions: A + ( B * C + E ) / ( F + A ) * D + C.data structure_6088560.odpdata structurefailure functionspringFri Aug 24 21:47:20 2007計算Compute the values of failure function for each of the following patterns:
a、a b a a b a a b b
b、b b b b a a b b bdata structure_6088318.odpalgorithmsdata structurefunctionrecursivespringFri Aug 24 21:44:51 2007程式設計The Fibonacci numbers are defined as: f0= 0, f1= 1, and fi= fi-1+ fi-2for i> 1. Write a recursive function to compute fi with C++ language.algorithms_6088169.odpdata structurealgorithmsdata structuretime complexityspringFri Aug 24 21:39:52 2007證明Prove or disprove the the following statements.
a、n! = O(nn).
b、10n2= Ω(n3).ds_6087870.odpalgorithmsgreedy algorithmtask-scheduling problemspringTue Jul 24 17:29:26 2007執行演算法Solve the instance of the scheduling problem given in the follows.algorithms_3394444.odpalgorithmsgreedy algorithmHuffman codespringTue Jul 24 17:21:50 2007執行演算法What is an optimal Huffman code for the following set of frequencies? a:1 b:5 c:7 d:10 e:15 f:23 g:31 h:40 。algorithms_3393988.odpalgorithmsgreedy algorithmknapsack0-1 knapsackspringTue Jul 24 17:08:47 2007執行演算法下表中代表每個物品的重點(wi)與價值(pi)，請問袋子最多可裝15公斤時，所能裝的物品價值最高為何？請求knapsack 與 0-1 knapsack 的答案。algorithms_3393205.odpalgorithmsgreedy algorithmactivity selectionspringTue Jul 24 16:52:45 2007執行演算法找出下表中所能參加最多的 activities。algorithms_3392243.odpalgorithmsdynamic programminglongest common subsequencespringTue Jul 24 16:18:47 2007執行演算法Determine an LCS of 〈1,0,0,1,0,1,0,1〉 and 〈0, 1, 0, 1, 1, 0, 1, 1, 0〉.algorithms_3390205.odpalgorithmsdynamic programmingMatrix-chain multiplicationspringTue Jul 24 8:17:53 2007執行演算法Find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is 〈5, 3, 12, 5, 6〉.algorithms_3361351.odpalgorithmsdynamic programmingMatrix-chain multiplicationspringTue Jul 24 7:52:31 2007執行演算法Find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is 〈5, 10, 3, 50, 6〉.algorithms_3359829.odpalgorithmsdynamic programmingspringTue Jul 24 7:34:31 2007執行演算法找出下列問題的最短路徑。algorithms_3358749.odpalgorithmsdynamic programmingspringTue Jul 24 7:15:00 2007執行演算法找出下列問題的最短路徑。algorithms_3357578.odpalgorithmsorder-statistic treedata structurespringMon Jul 23 7:40:50 2007設計演算法Show how to use an order-statistic tree to count the number of inversions (see Problem 2-4) in an array of size n in time O(n lg n).algorithms_3272728.odpalgorithmsred black treedata structurespringMon Jul 23 7:32:04 2007執行運算Show the red-black trees that result after successively inserting the keys 41, 38, 31, 12, 19, 8 into an initially empty red-black tree. And then show the red-black trees that result from the successive deletion of the keys in the order 8, 12, 19, 31, 38, 41.algorithms_3272202.odpalgorithmsbinary search treered black treespringSun Jul 22 20:16:43 2007計算Describe a red-black tree on n keys that realizes the largest possible ratio of red internal nodes to black internal nodes. What is this ratio? What tree has the smallest possible ratio, and what is the ratio?algorithms_3231681.odp習題13.1-7algorithmsred black treespringSun Jul 22 20:12:56 2007計算What is the largest possible number of internal nodes in a red-black tree with black-height k? What is the smallest possible number?algorithms_3231454.odp習題13.1-6algorithmsbinary search treesortingspringSun Jul 22 8:08:30 2007分析演算法We can sort a given set of n numbers by first building a binary search tree containing these numbers(using TREE_INSERT repreatedly to insert the numbers one by one) and then printing the numbers by an inorder tree walk. What are the worst-case and best-case running times for this sorting algorithm?algorithms_3187988.odp習題12.3-3algorithmsbinary search treespringSun Jul 22 8:00:59 2007選擇題Suppose that we have numbers between 1 and 1000 in a binary search tree and want to search for the number 363. Which of the following sequences could not be the sequence of nodes examined?
a. 2, 252, 401, 398, 330, 344, 397, 363.
b. 924, 220, 911, 244, 898, 258, 362, 363.
c. 925, 202, 911, 240, 912, 245, 363.
d. 2, 399, 387, 219, 266, 382, 381, 278, 363.
e. 935, 278, 347, 621, 299, 392, 358, 363.algorithms_3187537.odp習題12.2-1algorithmsbinary search treespringSat Jul 21 11:43:17 2007畫圖For the set of keys {1, 4, 5, 10, 16, 17, 21}, draw binary search trees of height 2, 3, 4, 5, and 6.algorithms_3114475.odp習題12.1-1algorithmshashing functionsspringSat Jul 21 11:35:28 2007計算Consider inserting the keys 10, 22, 31, 4, 15, 28, 17, 88, 59 into a hash table of length m = 11 using open addressing with the auxiliary hash function h\'(k) = k mod m. Illustrate the result of inserting these keys using linear probing, using quadratic probing with c1 = 1 and c2 = 3, and using double hashing with h2(k) = 1 + (k mod (m-1)).algorithms_3114006.odpalgorithmsdata structuretreespringSat Jul 21 11:21:28 2007畫圖Draw the binary tree rooted at index 6 that is represented by the follow fields.algorithms_3113166.odpalgorithmsdata structurestackqueuespringSat Jul 21 11:15:45 2007設計資料結構Show how to implement a queue using two stacks. Analyze the running time of the queue operations.algorithms_3112823.odp習題10.1-6algorithmsselectionrandomized algorithmsspringSat Jul 21 11:08:17 2007執行演算法Suppose we use Randomized-Select to select the minimum element of the array A = 〈3, 2, 9, 0, 7, 5, 4, 8, 6, 1〉. Describe a sequence of partitions that results in a worst-case performance of Randomized-Select.algorithms_3112375.odpalgorithmsquick sortrandomized algorithmsspringSat Jul 21 10:57:28 2007分析演算法During the running of the procedure Randomized-Quicksort, how many calls are made to the random-number generator Random in the worst case? How about in the best case? Give your answer in terms of Θ-notation.algorithms_3111725.odp習題7.3-2algorithmsquick sortspringSat Jul 21 10:41:23 2007修改演算法How would you modify Quicksort to sort into nonincreasing order?algorithms_3110761.odp習題7.1-4algorithmsquick sorttime complexityspringSat Jul 21 10:38:03 2007簡答What is the time complexity of Quicksort when the array A contains distinct elements and is sorted in decreasing order ?algorithms_3110561.odpalgorithmsdata structureheapspringSat Jul 21 10:32:25 2007計算令 H 是一個 48 個元素的 heap，請問 H 中有幾個高度為 3 的 nodes？algorithms_3110223.odpalgorithmsdata structureheapspringSat Jul 21 10:17:03 2007執行演算法Perform the operation of Build-Max-Heap on the array A = 〈5, 3, 17, 10, 84, 19, 6, 22, 9〉.algorithms_3109301.odp習題6.3-1algorithmsdata structureheapspringSat Jul 21 10:08:00 2007執行演算法Perform the operation of Max-Heapify(A, 3) on the array A = 〈27, 17, 3, 16, 13, 10, 1, 5, 7, 12, 4, 8, 9, 0〉.algorithms_3108758.odp習題6.2-1algorithmsdata structureheapspringSat Jul 21 10:01:20 2007計算What are the minimum and maximum numbers of elements in a heap of height h?algorithms_3108358.odp習題6.1-1algorithmsdata structureheapspringSat Jul 21 9:15:54 2007簡答Is the sequence 〈23, 17, 14, 6, 13, 10, 1, 5, 7, 12〉 a max-heap?algorithms_3105632.odpalgorithmsdata structureheapspringSat Jul 21 9:07:47 2007簡答Is an array that is in sorted order a min-heap?algorithms_3105145.odp習題6.1-5algorithmsdata structureheapspringSat Jul 21 9:03:58 2007簡答Where in a max-heap might the smallest element reside, assuming that all elements are distinct?algorithms_3104915.odpalgorithmsfunctionrecursivemaster theoremspringSat Jul 21 8:59:57 2007計算Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n ≤ 2.
A.T(n) = 2T(n/2) + n2.
B.T(n) = 16T(n/4) + n2.
C.T(n) = 3T(n/2) + n lg n.algorithms_3104675.odpalgorithmsfunctionrecursivespringSat Jul 21 8:49:32 2007計算Solve the recurrence T(n) = 2T(n1/2) + 1 by making a change of variables. We can assume that T(1) = θ(1).algorithms_3104050.odpalgorithmsfunctionintroductionspringSat Jul 21 8:39:37 2007證明Show that if f(n) and g(n) are monotonically increasing functions, then so are the functions f(g(n)).algorithms_3103455.odpalgorithmsfunctionintroductionspringSat Jul 21 8:34:51 2007證明Show that if f(n) and g(n) are monotonically increasing functions, then so are the functions f(n) + g(n).algorithms_3103169.odpalgorithmsGetting StartedinversionspringThu Jul 12 11:36:37 2007基本定義List the inversions of the array 〈3, 6, 8, 1, 5〉.algorithms_2336475.odpformal languagescontext-free languagescontext-free grammars曾俊淞Fri Jun 15 11:24:40 2007設計 grammars習題5.1.8(h)formal languages_2958.odpformal languagescontext-free languagescontext-free grammars曾凱煌Fri Jun 15 11:07:13 2007設計 grammars習題5.1.7(g)formal languages_1911.odpformal languagescontext-free languagessimple grammars蕭旻昆Fri Jun 15 11:01:33 2007設計 grammars習題5.2.3formal languages_1571.odpformal languagescontext-free languagesregular grammarsambiguous吳忠融Fri Jun 15 10:56:52 2007ambiguous 證明習題5.2.11formal languages_1290.odp