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Course: CS 231, Fall 2009
School: Bowdoin College
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Word Count: 11340

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Bowdoin College - CS - 231
CSci 231 Homework 2 SolutionGrowth of Functions, Summations and Recurrences CLRS Chapter 3, 4 and Appendix A1. Solve the recurrence: T (n) = Hint: use1 if n = 1 T (n - 1) + n(n - 1) if n 2 n n(n+1)(2n+1) 2 . i=1 i = 6 T (n) = T (n - 2) + (n - 1
Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
CSci 231 Homework 3 SolutionsQuicksort and Linear Time Sorting CLRS Chapter 7 and 81. (CLRS 7.2-2) What is the running time of Quicksort when all elements of arrary A have the same value? Solution: The partition algorithm puts all equal elements o
Bowdoin College - CS - 231
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Bowdoin College - CS - 231
CSci 231 Homework 4 SolutionsSelection CLRS Chapter 81. (CLRS 9.3-5) Suppose that you have a &quot;black-box&quot; worst-case linear-time median subroutine. Give a simple, linear-time algorithm that solves the selection problem for an arbitrary order statis
Bowdoin College - CS - 231
CSci 231 Homework 5 SolutionsHeaps CLRS Chapter 61. (CLRS 6.1-1) What are the minimum and maximum number of elements in a heap of height h? Solution: The minimum number of elements is 2h and the maximum number of elements is 2h+1 - 1. 2. (CLRS 6.1
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CSci 231 Homework 8Amortized Analysis CLRS Chapter 17Use a (single) separate sheet of paper for each problem. Be concise.1. (CLRS 17.2-1) A sequence of stack operations is performed on a stack whose size never exceeds k. After every k operatio
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Bowdoin College - CS - 231
CLRS 22-3 (a) Prove that a directed graph has an Euler circuit if and only of for all v in G, indeg(v) = outdeg(v). Solution: First note that the proof must have two parts: =: If G has an Euler circuit C, then C is either a simple cyle (does not inte
Bowdoin College - CS - 231
CSci 231 Homework 9Graph Algorithms CLRS Chapter 22, 241. [CLRS 22.1-5] Give and analyse an algorithm for computing the square of a directed graph G given in (a) adjacency-list representation and (b) adjacency-matrix representation. Solution: To c
Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
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Bowdoin College - CS - 231
15610 11 12 1315 16
Bowdoin College - CS - 231
\$120 \$100 \$6010 20 3030= \$2202020 10= \$160Items in value per pound orderOptimal solution for knapsack of size 50Greedy solution for knapsack of size 50
Bowdoin College - CS - 231
Local minima12n
Bowdoin College - CS - 231
(4,2)(3,0)(4,1)
Bowdoin College - CS - 231
(6,4)(0,2)(15, -3)(-1,0)(2,0) (4,6)(10,7)(0,1)(7,0)
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Bowdoin College - CS - 231
CSci 231 Homework 10 SolutionsBasic Graph Algorithms1. [CLRS 22.1-1] Describe how to compute the in-degree and out-degree of the vertices of a graph given its (1) adjacency -list representation and (b) adjacency-matrix representation. Solution:
Bowdoin College - CS - 231
CSci 231 Homework 11 Partial SolutionsShortest Paths, CLRS Chapter 24.0, 24.31. Consider a directed weighted graph with non-negative weights and V vertices arranged on a rectangular grid. Each vertex has an edge to its southern, eastern and southe
Bowdoin College - CS - 231
1. Consider an array A of length n for which we know that A[1] A[2] and A[n - 1] A[n]. We say that A[x] is a local minimum if A[x - 1] A[x] and A[x] A[x + 1]. Note that A must have at least one local minimum.Local minima12nWe can obviousl
Bowdoin College - CS - 231
CSci 231 Homework 3Quicksort, Linear Time Sorting CLRS Chapter 7 and 8Write and justify your answers on this sheet in the space provided.1 1. (CLRS 7.2-2) What is the running time of Quicksort when all elements of arrary A have the same value? Sol
Bowdoin College - CS - 231
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Bowdoin College - CS - 231
CSci 231 Homework 4 SolutionsSelection and Heaps CLRS Chapter 8 and 6Write and justify your answers on this sheet in the space provided.1 1. (CLRS 9.3-5) Suppose that you have a &quot;black-box&quot; worst-case linear-time median subroutine. Give a simple,
Bowdoin College - CS - 231
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Bowdoin College - CS - 231
CSci 231 Homework 5 SolutionsSelection and Heapsort CLRS Chapter 6 and 91. (CLRS 6.1-1) What are the minimum and maximum number of elements in a heap of height h? Solution: The minimum number of elements is 2h and the maximum number of elements is
Bowdoin College - CS - 231
CSci 231 Homework 5review Exam 11. Show using the substitution method (induction) that the recurrence T (n) = n T ( n) + n if n &gt; 2 1 otherwise has solution T (n) = O(n log log n).22. Recall that in a worst-case efficient O(n log n) time v
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CSci 231 Homework 6 SOLUTIONSBinary Search Trees and Hashing CLRS Chapter 11.1-11.3 and 12.1-12.3Write and justify your answers on this sheet in the space provided.1 1. (CLRS 12.2-5) Show that if a node in a binary search tree has two children, t
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