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 $\Theta\left(n^{2}\right)$ (c) Show that its expected running time satisfies the recurrence relation \[ T(n) \leq O(n)+\frac{1}{n} \sum_{i=1}^{n-1}(T(i)+T(n-i)) \] Then, show that the solution to this recurrence is $O(n \log n)$.

Name: Problem 24, Chapter 1: Algorithms
Uploaded: 2025-09-02T22:07:50-08:00
Duration: 2 min 24 s
Channel: James Kiss
Description: 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 Θ(n^2) (c) Show that its expected running time satisfies the recurrence relation T(n) ≤ O(n)+(1)/(n)∑i=1^n-1(T(i)+T(n-i)) Then, show that the solution to this recurrence is O(n logn).

   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 $\Theta\left(n^{2}\right)$
(c) Show that its expected running time satisfies the recurrence relation
\[
T(n) \leq O(n)+\frac{1}{n} \sum_{i=1}^{n-1}(T(i)+T(n-i))
\]
Then, show that the solution to this recurrence is $O(n \log n)$.

Algorithms

Sanjoy Dasgupta,… 1st Edition

Chapter 1, Problem 24

Instant Answer

Solved by Expert James Kiss

09/02/2025

Step 1

Upper bound: In the worst case, the pivot chosen is always the smallest or largest element in the array. This leads to a partitioning where one partition has size 0 and the other partition has size n-1. Show more…

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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 $\Theta\left(n^{2}\right)$ (c) Show that its expected running time satisfies the recurrence relation \[ T(n) \leq O(n)+\frac{1}{n} \sum_{i=1}^{n-1}(T(i)+T(n-i)) \] Then, show that the solution to this recurrence is $O(n \log n)$.