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Chapter 9: Problem 22

Can an algorithm be a polynomial-time algorithm for a problem using one encoding scheme, and an exponential-time algorithm for the same problem using another encoding scheme? Justify your answer.

Short Answer

Expert verified
Yes, it's possible for an algorithm to be a polynomial-time algorithm using one encoding scheme and an exponential-time algorithm using another encoding scheme. This is because the encoding scheme represents the problem for the algorithm and the way it represents it can change the size of the input, hence impacting the time complexity.

Step by step solution

01

Understand Polynomial-time Algorithms

A polynomial-time algorithm is an algorithm whose time complexity can be bounded by a polynomial expression in the size of the input for the algorithm. So, for a given input size n, the time taken can be expressed as \(k*n^m\) where k and m are constants.
02

Understand Exponential-time Algorithms

An exponential-time algorithm is an algorithm that grows exponentially with the size of the input. This means that for an increase in the size of the input, the time taken by the algorithm increases exponentially. This can be expressed as \(k*m^n\) where k and m are constants and n is the size of the input.
03

Effect of Encoding on Time Complexity

Whether an algorithm functions in polynomial time or exponential time can also depend on the encoding of the problem that the algorithm is attempting to solve. The way we choose to represent the problem as an input to the algorithm can affect the size of the input (n), and therefore impact the time complexity.
04

Final Explanation

Therefore, it's possible for an algorithm to be polynomial-time using one encoding but exponential-time using another because the encoding might change the size of the input substantially. If the encoding scheme represents the problem more concisely, the algorithm may function in polynomial time. If the encoding scheme represents the problem in a more expanded form, the algorithm may function in exponential time.

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Most popular questions from this chapter

Suppose that problem \(A\) and problem \(B\) are two different decision problems. Furthermore, assume that problem \(A\) is polynomial-time many-one reducible to problem \(B\), If problem \(A\) is \(N P\) -complete, is problem \(B\) NP-complete? Justify your answer.

Is the Towers of Hanoi Problem an NP-complete problem? Is it an \(N P\) -easy problem? Is it an \(N P\) -hard problem? Is it an \(N P\) -equivalent problem? Justify your answers. This problem is presented in Exercise 17 in Chapter 2.

Can you develop an approximation algorithm for the CNF-Satisfiability Problem by stating it as an optimization problem- that is, by finding a truth assignment of the literals in the expression that makes the maximum possible number of clauses true?

Show that a problem is \(N P\) -easy if and only if it reduces to an \(N P\) -complete problem.

Given a list of \(n\) positive integers \((n\) even), divide the list into two sublists such that the difference between the sums of the integers in the two sublists is minimized. Is this problem an \(N P\) -complete problem? Is this problem an \(N P\) -hard problem?
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