Question

Consider the task of searching a sorted array $A[1 \ldots n]$ for a given element $x:$ a task we usually perform by binary search in time $O(\log n)$. Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form "is $A[i] \leq z ?$ ), must take $\Omega(\log n)$ steps.

Name: Problem 18, Chapter 1: Algorithms
Uploaded: 2025-09-02T22:38:35-08:00
Duration: 2 min 9 s
Channel: James Kiss
Description: 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 ? ), must take Ω(logn) steps.

   Consider the task of searching a sorted array $A[1 \ldots n]$ for a given element $x:$ a task we usually perform by binary search in time $O(\log n)$. Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form "is $A[i] \leq z ?$ ), must take $\Omega(\log n)$ steps.

Algorithms

Sanjoy Dasgupta,… 1st Edition

Chapter 1, Problem 18

Instant Answer

Solved by Expert James Kiss

09/02/2025

Step 1

Suppose we have an algorithm that accesses the array only via comparisons and it takes less than $\log n$ steps to find the element $x$. Show more…

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Consider the task of searching a sorted array $A[1 \ldots n]$ for a given element $x:$ a task we usually perform by binary search in time $O(\log n)$. Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form "is $A[i] \leq z ?$ ), must take $\Omega(\log n)$ steps.