# hw_sol_03 - CS 473 Fundamental Algorithms Spring 2011 HW 3...

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Unformatted text preview: CS 473: Fundamental Algorithms, Spring 2011 HW 3 Homework is due by Monday, 23:59:59, February 14 Problem 1 is due by Sunday, 23:59:59, February 13 This homework contains four problems. Read the instructions for submitting homework on the course webpage . Collaboration Policy: For this homework, Problems 2–4 can be worked in groups of up to three students. Problem 1 should be answered in Compass as part of the assessment HW3-Online and should be done individually. 1. (30 pts.) Short questions to be answered on compass individually. 2. (30 pts.) You are given an array A with n distinct numbers in it, and another array B of ranks i 1 < i 2 < ... < i k . An element x of A has rank u if there are exactly u- 1 numbers in A smaller than it. Design an algorithm that outputs the k elements in A that have the ranks i 1 ,i 2 ,...,i k . (A) (5 pts.) As a warm-up exercise describe how to solve this problem in O ( nk ) time. Solution : For each 1 ≤ j ≤ k , output the i j th smallest number in A using the O ( n ) time selection algorithm. Clearly the total running time is O ( nk ). Rubrik : • 3pts: If the writeup gives a correct algorithm. If the algorithm is incomplete but gives a key idea that could lead to a correct algorithm, it gets 2pts. • 2pts: If the algorithm has running time O ( nk ). 1 (B) (20 pts.) Describe a O ( n log k ) recursive algorithm for this problem. Prove the bound on the running time of the algorithm. Solution : We give the following algorithm k-Select . It takes two arrays A and B as input. Here B is an array of distinct positive integers, and the numbers are sorted in increasing order. Algorithm k-Select ( A,B,μ ): If | B | > | A | then B ← the smallest | A | elements in B . Let b 1 < b 2 < ... < b | B | be the integers in B If | B | = 1 and | A | ≤ b 1- μ then output the ( b 1- μ )th smallest number in A . Else Let μ = b b| B | / 2 c . If μ- μ > | A | then k-Select ( A, { b 1 ,b 2 ,...,b b| B | / 2 c } ,μ ). Else p ← the ( μ- μ ) th smallest element in A . Let A l := { a ∈ A | a ≤ p } and A r := { a ∈ A | a > p } . k-Select ( A l , { b 1 ,b 2 ,...,b b| B | / 2 c } ,μ ). k-Select ( A r , { b b| B | / 2 c +1 ,...,b | B | } ,μ } . We run k-Select ( A,B, 0) to answer the problem. Let T ( x,y ) denote the running time of the algorithm when | A | = x and | B | = y . Note that T ( x,y ) = T ( | A l | , b y/ 2 c ) + T ( | A r | , d y/ 2 e )+ O ( x ) or T ( x,y ) = T ( | A | , b y/ 2 c )+ O ( x ); T ( x, 1) = O ( x ). It is easy to see that the depth of the recursion tree is O (log k ). Further, we know that the total work at every level is O ( x ) since x = | A l | + | A r | . Hence the running time follows. Rubrik : • 15pts: If the writeup gives a correct algorithm with running time O ( n log k ). If the the algorithm is incomplete but gives a key idea that could lead to a correct algorithm, it gets 10pts....
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## This note was uploaded on 04/18/2011 for the course CS 473 taught by Professor Chekuri,c during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw_sol_03 - CS 473 Fundamental Algorithms Spring 2011 HW 3...

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