# hw4(10) - CS 573 Homework 4(due November 1 2010 Fall 2010...

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CS 573 Homework 4 (due November 1, 2010) Fall 2010 CS 573: Graduate Algorithms, Fall 2010 Homework 4 Due Monday, November 1, 2010 at 5pm (in the homework drop boxes in the basement of Siebel) 1. Consider an n -node treap T . As in the lecture notes, we identify nodes in T by the ranks of their search keys. Thus, ‘node 5’ means the node with the 5th smallest search key. Let i , j , k be integers such that 1 i j k n . (a) What is the exact probability that node j is a common ancestor of node i and node k ? (b) What is the exact expected length of the unique path from node i to node k in T ? 2. Let M [ 1.. n ,1. . n ] be an n × n matrix in which every row and every column is sorted. Such an array is called totally monotone . No two elements of M are equal. (a) Describe and analyze an algorithm to solve the following problem in O ( n ) time: Given indices i , j , i 0 , j 0 as input, compute the number of elements of M smaller than M [ i , j ] and larger than M [ i 0 , j 0 ] . (b) Describe and analyze an algorithm to solve the following problem in O ( n ) time: Given indices i , j , i 0 , j 0 as input, return an element of M chosen uniformly at random from the elements smaller than M [ i , j ] and larger than M [ i 0 , j 0 ] . Assume the requested range is always non-empty. (c) Describe and analyze a randomized algorithm to compute the median element of

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hw4(10) - CS 573 Homework 4(due November 1 2010 Fall 2010...

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