Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 16

Show that to fully parenthesize an expression having \(n\) matrices we need \(n-1\) pairs of parentheses.

Short Answer

Expert verified
We have shown through the process of mathematical induction that to fully parenthesize an expression having \(n\) matrices, we need \(n-1\) pairs of parentheses.

Step by step solution

01

Define Full Parenthesization

When we say full parenthesization, it simply means that we place parentheses in the expression in such a way to indicate the order of multiplication. For instance, if we have three matrices \(A\), \(B\), \(C\), then a full parenthesization would look like either \((AB)C\) or \(A(BC)\).
02

Base Case

To start proving the exercise, the base case is the starting point. This is where we have only one matrix, hence \(n=1\). Since we have just one matrix, there is no need for any set of parentheses, so 0 pair is needed. Therefore, it follows that we need \(n-1 = 1 - 1 = 0\) pairs of parentheses which verifies the base case.
03

Inductive Hypothesis

Let's assume the statement is true for \(k\) matrices, that is we need \(k-1\) pairs of parentheses for \(k\) numbers. This is our inductive hypothesis.
04

Inductive Step

We need to verify the statement is true for \(k+1\) matrices assuming it's true for \(k\) matrices. When we have \(k+1\) matrices, we divide these matrices into two groups, the first group having \(i\) matrices (\(1 \leq i \leq k\)) and the second group having \(k+1-i\) matrices. Each group will need to be fully parenthesized which by our inductive hypothesis, will require \(i-1\) and \(k+1-i-1=k-i\) pairs of parentheses respectively. Additionally, one more pair of parentheses is needed for the total product. Therefore, in total, we need \((i-1) + (k-i) + 1 = k-1+1=k\) pairs of parentheses for \(k+1\) matrices, hence the proof.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an optimization problem in which the principle of optimality does not apply, and therefore the optimal solution cannot be obtained using dynamic programming, Justify your answer.

Can Floyd's Algorithm for the Shortest Paths Problem 2 (Algorithm 3.4 ) be modificd to give just the shortest path from a given vertex to another specificd vertex in a graph? Justify your answer.

Implement Floyd's Algorithm for the Shortest Paths Problem 2 (Algorithm 3.4) on your system, and study its performance using different graphs.

Show that the number of binary search trees with \(n\) keys is given by the formula \\[\frac{1}{(n+1)}\left(\begin{array}{c}2 n \\\n\end{array}\right)\\]

Write a more detailed version of the dynamic programming algorithm for the Traveling Salesperson Problem (Algorithm 3.11 ).
See all solutions

Recommended explanations on Computer Science Textbooks

Data Representation in Computer Science

Read Explanation

Issues in Computer Science

Read Explanation

Data Structures

Read Explanation

Databases

Read Explanation

Computer Network

Read Explanation

Functional Programming

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free