Search problem: The search problem is to take a sorted array A[1,...,n] and an element x and either find a location of x in A or report that one does not exist. Notice that we are indexing the array starting from 1. For the sake of simplicity, you may assume that n is a power of 2. For example, if A = (4,9,11,57) and x = 9, then the answer to the search problem is 2 because A[2] = 9. Recall that the binary search CMPSC 465, Spring 2023, HW 3 1 algorithm for this problem compares x to A[(n + 1)/2] and, based on that, recurses down one of the halves. (a) (4pt) The execution of the binary search algorithm can be viewed as a tree. Each non-leaf node of this tree corresponds to a pair of values being compared. Each non-leaf node has three children, corresponding to the results of the comparison: either less than, equal, or greater than. Each leaf node contains an output of the algorithm: either NF, which means not found, or an index i such that A[i] = x. Draw this tree for n = 8 (you may use handwriting for this part). The drawing below will help you get started by showing part of the tree. (b) (2pt) Next, consider any comparison-based algorithm to solve the search problem and its corresponding decision tree. By comparison-based, we mean that all the decisions of the algorithm must stem from the result of comparing x to some element of the array. Let d be the maximum depth of the decision tree and let be the number of leaves. What is the maximum number of leaves that this tree can have, as a function of d? (c) (2pt) What is the minimum number of leaves that this tree needs, as a function of n, in order to be able to output every possible location? (d) (4pt) Argue that any comparison-based algorithm must take (log n) time.

3. (12 pts.) Search problem: The search problem is to take a sorted array A[1,,n] and an element x and either find a location of x in A or report that one does not exist. Notice that we are indexing the array starting from 1. For the sake of simplicity, you may assume that n is a power of 2 . For example, if A(4,9,11,57) and x=9, then the answer to the search problem is 2 hecause A[2]=9. Recall that the binary search CMPSC 465, Spring 2023, HW 3 algorithm for this problem compares x to A[(n+1)/2]] and, based on that, recurses down one of the halves. (a) (4pt) The cxecution of the binary scarch algorithm can be vicwed as a trec. Each non-leaf node of this tree corresponds to a pair of values being compared. Each non-leaf node has three children, corresponding to the results of the comparison; either less than, equal, or greater than. Each leaf node contains an output of the algorithm: either NF, which means not found, or an index i such that A[i]=x. Draw this tree for n=8 (you may use handwriting for this part). The drawing below will help you get started by showing part of the tree. (b) (2pt) Next, consider any comparison-based algorithm to sol ve the search problem and its corresponding algorithm for this problem compares x to A[(n+1)/2] and, based on that, recurses down one of the halves. (a) (4pt) The execution of the binary search algorithm can be viewed as a tree. Each non-leaf node of this tree corresponds to a pair of values being compared. Each non-leaf noxle has three children, corresponding to the results of the comparison: either less than, equal, or greater than. Each leaf node oontains an output of the algorithm: either NF, which means not found, or an index i such that A[i]=x. Draw this trec for n=8 (you may use handwriting for this part). The drawing below will help you get started by showing part of the tree. (b) (2pt) Next, consider any comparison-based algorithm to solve the scarch problem and its corresponding decision tree. By "comparison-based", we mean that all the decisions of the algorithm must stern from the result of comparing x to some element of the array. Let d be the maximum depth of the decision tree and let be the number of leaves. What is the maximum number of leaves that this tree can have, as a function of d? (c) (2pt) What is the minimum number of leaves that this tree needs, as a function of n, in order to be able to output every possible location? (d) (4pt) Argue that any comparison-based algorithm must take (logn) time