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Chapter 2: Q16E (page 85)

Question: You are given an infinite array A[路]in which the first n cells contain integers in sorted order and the rest of the cells are filled with 鈭�. You are not given the value of n. Describe an algorithm that takes an integer x as input and finds a position in the array containing x, if such a position exists, in O(log n) time. (If you are disturbed by the fact that the array A has infinite length, assume instead that it is of length n, but that you don鈥檛 know this length, and that the implementation of the array data type in your programming language returns the error message 鈭�whenever elements A[i]withi>n are accessed.)

Short Answer

Expert verified

An algorithm exist which finds the position of input inter x in array A, in time bound of O(logn).

Step by step solution

01

Algorithm  

1. Check A[1],A[2],A[4],A[8],, and so on, doubling its indexing each time until infinity or the value x is discovered. Let q be the most recent index to be examined.

2. If x=A[q]after back q.

3. If not, conduct a binary find in A[q/2]...A[q]for x. Give the index if x is found; else, return FALSE.

02

Binary Search Pseudocode

Each sub-array will be subjected to something like a binary search algorithm A[q/2]...A[q]

binary_search (A,x) :

low = 1, high = size(A)

while low 鈮�high:

mid = low + (high - low)/2

if A[mid] == x:

return mid

else if A[mid] < x:

low = mid + 1

else:

high = mid - 1

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Most popular questions from this chapter

What is the sum of the nth roots of unity? What is their product if n is odd? If n is even?

This problem illustrates how to do the Fourier Transform (FT) in modular arithmetic, for example, modulo .(a) There is a number such that all the powers 蝇,蝇2,...,蝇6 are distinct (modulo ). Find this role="math" localid="1659339882657" 蝇, and show that 蝇+蝇2+...+蝇6=0. (Interestingly, for any prime modulus there is such a number.)

(b) Using the matrix form of the FT, produce the transform of the sequence (0,1,1,1,5,2) modulo 7; that is, multiply this vector by the matrix M6(蝇), for the value of 蝇you found earlier. In the matrix multiplication, all calculations should be performed modulo 7.

(c) Write down the matrix necessary to perform the inverse FT. Show that multiplying by this matrix returns the original sequence. (Again all arithmetic should be performed modulo 7.)

(d) Now show how to multiply the polynomials and using the FT modulo 7.

Given a sorted array of distinct integersA[1,...,n] , you want to find out whether there is an indexi for which A[i]=i. Give a divide-and-conquer algorithm that runs in time O(logn).

Suppose you are choosing between the following three algorithms: 鈥� Algorithm A solves problems by dividing them into five sub-problems of half the size, recursively solving each sub-problem, and then combining the solutions in linear time. 鈥�

Algorithm B solves problems of size n by recursively solving two sub-problems of size n-1and then combining the solutions in constant time. 鈥� Algorithm C solves problems of size n by dividing them into nine sub-problems of size n/3, recursively solving each sub-problem, and then combining the solutions in O(n2)time.

What are the running times of each of these algorithms (in big- O notation), and which would you choose?

A k鈭�way merge operation. Suppose you have ksorted arrays, each with nelements, and you want to combine them into a single sorted array ofkn elements.

(a)Here鈥檚 one strategy: Using the merge procedure from Section 2.3, merge the first two arrays, then merge in the third, then merge in the fourth, and so on. What is the time complexity of this algorithm, in terms of kand n?

(b) Give a more efficient solution to this problem, using divide-and-conquer.
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