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Chapter 2: 18 E (page 85)

Consider the task of searching a sorted array A[1,鈥�,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 鈥渋s A[i]鈮�z 0?鈥�), must take 惟(logn) steps.

Short Answer

Expert verified

Yes, any algorithm that accesses the array only via comparisons take 惟logn 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,鈥�,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 惟logn steps

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Most popular questions from this chapter

Question: On page 66 there is a high-level description of the quicksort algorithm.

(a) Write down the pseudocode for quicksort.

(b) Show that its worst - case running time on an array of size n is 螛(n)2.

(c) Show that its expected running time satisfies the recurrence relation.

T(n)鈮�O(n)+1n鈭�i=1n-1(Ti+Tn-i)

Then, show that the solution to this recurrence is O(nlogn).

Find the unique polynomial of degree 4 that takes on values p(1)=2,p(2)=1,p(3)=0,p(4)=4,andp(5)=0. Write your answer in the coefficient representation.

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

Practice with polynomial multiplication by FFT.

(a) Suppose that you want to multiply the two polynomials x + 1 and x2+1using the FFT. Choose an appropriate power of two, find the FFT of the two sequences, multiply the results component wise, and compute the inverse FFT to get the final result.

(b) Repeat for the pair of polynomials 1+x+2x2and 2 + 3x.

Professor F. Lake tells his class that it is asymptotically faster to square an -bit integer than to multiply two n-bit integers. Should they believe him?
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