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Homework Two
Design and Analysis of Algorithms
Due: Wednesday, Feb. 11, 2009, in class
1(a)
(b)
Design an O(nlgn) algorithm to determine if the n numbers in A[1..n] are distinct.
Prove that any comparison-based algorithm for part (a) needs (nlgn) time.
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Page 85, Problem 4-2. Finding the missing integer.
An array A[1..n] contains all the integers from 0 to n except one integer. It would be easy to
determine the missing integer in O(n) time by using an auxiliary array B[0..n] to record which
numbers appear in A. In this problem, however, we cannot access an entire integer in A with a
single operation. The elements of A are represented in binary, and the only operation we can use
to access them is “fetch the jth bit of A[i],” which takes constant time. Show that if we use only
this operation, we can still determine the missing integer in O(n) time. You may assume that n =
2k-1 and every number is a k-bit number.
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Problem 9.3-8 (p193)
Let X[1..n] and Y[1..n] be two arrays, each containing n distinct numbers already in sorted
order. Give an O(lgn)-time algorithm to find the median of all 2n elements in array X and Y.
(Hint: if X[k] is the median, what conditions the X[k] must satisfy?)
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Given an array A[1..n] of n numbers, the Min-first binary tree for the n numbers is constructed
as follows.
(1) The root holds the smallest number in A[1..n].
(2) Let A[r] be the root. Then, its left subtree is recursively constructed from A[1..r-1], and
its right subtree is recursively constructed from A[r+1..n].
(3) If a subtree contains no number, it is a leaf and construction stops here.
A small example is given below.
A[1] = 6, A[2] = 10, A[3] = 2, A[4] = 7, A[5] = 9, A[6] = 5
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A[1] = 6,
A[3]
A[1]
A[1]
= 6,
5 A[6]
6
A[2] A[1]
= 6,
A[4]
10
A[1]
= 6,
7
A[1]
A[5] = 6,
9
A[1]
= 6,
A[1]
= 6,
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(a) Draw the min-order binary tree for the following sequence.
6, 5, 2, 9, 7, 1, 3, 10, 9.
(b) Design a divide and conquer algorithm to construct the min-order binary tree for numbers
stored in A[1, n]. Pseudo-code is required. Also, you need to analyze the complexity of your
algorithm for the best case, worst case and average case. (The average complexity must be
O(nlgn). The worst complexity can be O(n2).
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