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1 CS5800 : Algorithms Fall 015 Nov, 015 Quiz Practice Total Points: 0. Duration: 1hr 1. (7,8) points Binary Heap. (a) The following is a sequence of elements presented to you (in order from left to right): 19,,,, 4, 11, 1. Show the result of forming a binary min heap by adding each new element to the end and heapifying up. (b) Give an optimal algorithm for finding the largest (not smallest) element in a binary min heap, along with proof of correctness and runtime analysis. Argue that your algorithm is optimal. (a) First heap from inserting 19: 19 Insert and heapify up: 19 Insert and heapify up: 19 Insert and heapify up: 19 Insert 4 and heapify up: 1

2 19 4 Insert 11 and heapify up: Insert 1 and heapify up: Note that the heap is actually an array, and the binary tree is an interpretation of it. As we add elements, we fill out the implicit binary tree one level at a time, from left to right. (b) Every element in the heap is larger than it s parent. Thus we should check all the leaves, or if we view it as an array with the heap property, the last half of the elements. We may have to check as many as half the elements, so the run time is O(n). It is optimal because we must check every leaf of the heap. If we do not, then it is always possible that the leaf we did not check was the actual maximum (pretend an adversary chooses the values of the entries right as we query them, subject only to satisfying the heap property).. (,1) points Alice and Bob, the world s most cryptic cat burglars, have just pulled off an unprecedented heist at the Museum of Finite Automatons, stealing priceless computing artifacts! They have stolen k computers of worth $v 1, $v,..., $v k. They now wish to divide up the computers, so that each of them has the same total dollar worth of computers. (a) Give an example of a value sequence $v 1, $v,..., $v k such that

3 k > 1, and the sum of the dollar amounts is an even integer, but it is not possible to partition the values into two sets of equal worth. (b) Give an algorithm which takes as input $v 1, $v,..., $v k, and returns whether or not it is possible to partition the values into two sequences of equal worth. Give a proof of correctness of your algorithm, and an analysis of the run time. (a) Consider v 1 = 10, v = 4. Then no matter how we partition them, it is not possible to get two evenly valued sets. (b) Let S be the sum of all the values. We wish to know if a subset of the v i values sums to S/. If such a subset exists, then the remaining elements must also sum to S/, yielding the desired partition. Let OP T (i, x) be the sub problem of whether or not a subset of {v 1, v, v i } sums to x. If x = 0, it is true. If x < 0, it is false. Else, assume it is possible, and it uses some element v i. Then O(i 1, x v i ) must also be true. If it is possible, but does not use the element v i, then O(i 1, x) must still be true. Thus we get the recurrence: OP T (i, x) = {OP T (i 1, x v i ), OP T (i 1, x)} This yields the algorithm: S = v 1 + v + + v k if S 1 mod then return False end if O(i, 0) True O(0, x) False, if x > 0 for i = 1,, k do for x = 1,, S/ do OP T (i, x) = {OP T (i 1, x v i ), OP T (i 1, x)} return OP T (k, S/)

4 Computing OP T (i, x) takes time O(1), since it is an OR operation over two values. Thus the entirety of the for loop takes time O(kS). The final return value is a constant time return. Thus the entire algorithm is O(kS). (1, ) points Floyd-Warshall. (a) Give pseudo-code for finding all-pairs shortest paths using Floyd- Warshall. Give a proof of correctness and runtime analysis. (b) Does Floyd-Warshall work with negative weight edges? Explain. If the answer is sometimes, explain when it does and does not work. (a) Let our source node be v. The algorithm is as follows: Initialize dist[][] for u V do dist[u][u] = 0 for (u, v) E do dist[u][v] = w(e uv ) for k = 1,, V do for i = 1,, V do for j = 1,, V do if dist[u] +w < dist[v] then dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]) end if for k = 1,, V do if dist[k][k] 0 then return error, negative cycle end if return dist 4

5 Initialization of dist takes O( V ) time. Each for loop has V iterations, and the inner most line takes O(1) time, so the triple nesting takes O( V ) time. THe final check for negative cycles takes time O( V ). Thus the total run time is O( V ) (b) Yes, but only if there are no negative cycles. If there is a negative cycle, then a notion of shortest path does not exist. If there is no negative cycle, then the algorithm is guaranteed to return the shortest possible path lengths between all nodes. 4. (5,10) points Median. (a) Let A and B be any two sets of numbers of odd cardinality. Let m S denote the median of any odd-sized set S. Prove that min(m A, m B ) m A B max(m A, m B ). (b) Give pseudo-code for the linear-time median finding algorithm. There is no need for a proof of correctness. Give a run-time analysis. (a) Note, we assume that we are working with multisets / arrays. Thus duplicate elements remain in the union of A and B. WLOG assume m A m B. Let the sizes of A and B be x+1, y+1 respectively. Then there are x elements greater than m A and y elements greater than m B in A and B. The items in B are also greater than m A by our assumption. Thus in A B, assuming m A m B we have that there can be at most x + y elements less than m A and at most x+y elements greater than m B which means the median of the union lies between them. (b) The pseudocode acts as follows. Formally, our algorithm Rank(A, k) searchs array A for the element of rank k. Our first call will be Rank(A, n/). Split the array into N/5 groups of 5 items each. Find the median of each group. Recursively find the median of these medians using this algorithm. 5

6 Partition A into B and C, using the median of medians as a pivot. Here B is all elements smaller, and C is all elements larger. If B >= N/, return Rank(B, N / ). Else return Rank(C, N/ B ) The recursive median of medians will be an input of size N/5, while the median of medians pivot is guarenteed to split the array such that the largest is no more than 7N/10. Thus our run time recurrence is no more than: T (n) = T (n/5) + T (7n/10) + O(n) Which can be shown to be no larger than O(n).

Total Points: 60. Duration: 1hr

Total Points: 60. Duration: 1hr CS800 : Algorithms Fall 201 Nov 22, 201 Quiz 2 Practice Total Points: 0. Duration: 1hr 1. (,10) points Binary Heap. (a) The following is a sequence of elements presented to you (in order from left to right):