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Chapter 2: Problem 22

Verify the following identity \(\sum_{p=1}^{n}[A(p-1)+A(n-p)]=2 \sum_{p=1}^{n} A(p-1)\) This result is used in the discussion of the average-case time complexity analysis of Algorithm 2.6 (Quicksort).

Short Answer

Expert verified
Both sides of the equation have been found to be equal, hence the identity has been verified.

Step by step solution

01

Understanding the Problem

The given identity includes two summations \(\sum_{p=1}^{n}[A(p-1)+A(n-p)]=2 \sum_{p=1}^{n} A(p-1)\). We need to simplify the left hand side and show that it equates to the right hand side.
02

Advice on Approach

A good way to approach this type of problem is by starting from the more complex side, which in this case is the left side of the equation, and simplifying it to potentially end with the right side of the equation. Instead of directly proving the identity, it would be useful to remember that we can rearrange the terms in a summation without affecting the sum and properties of summation which should be useful in the process.
03

Start with the left side.

Let's start by simplifying the left side: \(\sum_{p=1}^{n}[A(p-1)+A(n-p)]\). The terms inside the summation can be broken down into two separate summations: \(\sum_{p=1}^{n}A(p-1) + \sum_{p=1}^{n}A(n-p)\).
04

Apply the property of summation

Now apply the property of summation: changing the order of summation does not change the sum. Rewrite the second summation by replacing 'p' with 'n - p', then the limits of the series change and the summation becomes \(\sum_{p=0}^{n-1}A(p)\). But since \(A(-1)\) equals to zero, the limits of both series are unified which become from 0 to n-1.
05

Bring the sums together

Now that the bounds of the sums are the same, they can be combined: \(\sum_{p=0}^{n-1}A(p) + \sum_{p=0}^{n-1}A(p) = 2\sum_{p=0}^{n-1}A(p)\).
06

Final Step

The final step is merely a matter of rewriting the solution found. The solution is \(2\sum_{p=0}^{n-1}A(p)\), therefore the original statement \(\sum_{p=1}^{n}[A(p-1)+A(n-p)]=2 \sum_{p=1}^{n} A(p-1)\) has been verified.

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Most popular questions from this chapter

Show that the worst-case time complexity for Binary Search (Algorithm 2.1) is given by \(W(n)=[\lg n]+1\) When \(n\) is not restricted to being a power of \(2 .\) Hint. First show that the recurrence equation for \(W(n)\) is given by \(W(n)=1+W\left(\left\lfloor\frac{n}{2}\right\rfloor\right) \quad\) for \(n>1\) \(W(1)=1\) To do this, consider even and odd values of \(n\) separately. Then use induction to solve the recurrence equation.

Write an algorithm that searches a sorted list of \(n\) items by dividing it into three sublists of almost \(n / 3\) items. This algorithm finds the sublist that might contain the given item and divides it into three smaller sublists of almost equal size. The algorithm repeats this process until it finds the item or concludes that the item is not in the list. Analyze your algorithm and give the results using order notation.

Write a divide-and-conquer algorithm for the Towers of Hanoi problem. The Towers of Hanoi problem consists of three pegs and \(n\) disks of different sizes. The object is to move the disks that are stacked, in decreasing order of their size, on one of the three pegs to a new peg using the third one as a temporary peg. The problem should be solved according to the following rules: (1) when a disk is moved, it must be placed on one of the three pegs; (2) only one disk may be moved at a time, and it must be the top disk on one of the pegs; and (3) a larger disk may never be placed on top of a smaller disk. a. Show for your algorithm that \(S(n)=2^{n}-1\). (Here \(S(n)\) denotes the number of steps (moves), given an input of \(n\) disks.) b. Prove that any other algorithm takes at least as many moves as given in part (a).

Write a recursive \(\Theta(n \lg n)\) algorithm whose parameters are three integers \(x, n,\) and \(p,\) and which computes the remainder when \(x^{n}\) is divided by \(p\) For simplicity, you may assume that \(n\) is a power of \(2-\) that is, that \(n=2^{k}\) for some positive integer \(k\)

Use Binary Search, Recursion (Algorithm 2.1) to search for the integer 120 in the following list (array) of integers. Show the actions step by step. \(\begin{array}{lllllllll}12 & 34 & 37 & 45 & 57 & 82 & 99 & 120 & 134\end{array}\)
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