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Chapter 2: 18 E (page 85)

Consider the task of searching a sorted array A[1,…,n] for a given element x: a task we usually perform by binary search in time O(logn) . Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form “is A[i]≤z 0?”), must take Ω(logn) steps.

Short Answer

Expert verified

Yes, any algorithm that accesses the array only via comparisons take Ωlogn steps.

Step by step solution

01

Explain Binary Search

Binary search is implemented on a sorted array. Binary search always starts with the mid element. Mid element compared with every other element and the element to be searched will be found.

02

Show that any algorithm that accesses the array via comparisons takes Ω(logn) steps.

Knowing that a query produces two kinds of results that is yes or no. The arrayA1,…,ncan be divided into two parts based on the result of the query.

In the best case, the two divided parts will be equal and the search can be scaled regardless of the query result.

Therefore, To locate a unique element by comparisons, it takes Ωlogn steps

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Most popular questions from this chapter

Practice with the fast Fourier transform.

(a) What is the FFT of (1,0,0,0)? What is the appropriate value of ωin this case? And of which sequence is (1,0,0,0)the FFT?

(b)Repeat for (1,0,1,-1).

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 ω,ω2,...,ω6 are distinct (modulo ). Find this role="math" localid="1659339882657" ω, and show that ω+ω2+...+ω6=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(ω), for the value of ωyou 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.

Question: You are given an infinite array A[·]in which the first n cells contain integers in sorted order and the rest of the cells are filled with ∞. 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’t know this length, and that the implementation of the array data type in your programming language returns the error message ∞whenever elements A[i]withi>n are accessed.)

You are given an array of nelements, and you notice that some of the elements are duplicates; that is, they appear more than once in the array. Show how to remove all duplicates from the array in time O(nlogn) .

How many lines, as a function of n (in ⊝(.)form), does the following program print? Write a recurrence and solve it. You may assume is a power of . function f (n) if n > 1:

print_line (‘‘still going’’)

f (n/2)

f (n/2)
See all solutions

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