Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 15.1, Problem 2E
Program Plan Intro
To give a counter example so that the following greedy strategy does not always determine an optimal way to cut rods.
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Exercise 4
20= 10+4+6
The rod-cutting problem consists of a rod of n units long that can be cut into integer-length
pieces. The sale price of a piece i units long is Pi for i = 1,...,n. We want to apply dynamic
programming to find the maximum total sale price of the rod. Let F(k) be the maximum price
for a given rod of length k.
1. Give the recurrence on F(k) and its initial condition(s).
2. What are the time and space efficiencies of your algorithm?
Now, consider the following instance of the rod-cutting problem: a rod of length n=5, and the
following sale prices P1=2, P2=3, P3-7, P4=2 and P5=5.
Let's say there are n villages, {X1, . . . , Xn} on the country-road and we aim to build K < n restaurants to cover them. Each restaurant has to be built in a village, and we hope to minimize the average distance from each village to the closest restaurant. Please give an algorithm to compute the optimal way to place these K restaurants. The algorithm should run in O(k * n^2) time. Solutions with slightly higher time complexity also accepted.
An electrician has wired n lights, all initially on, so that:
1) light 1 can always be turned on/off, and
2) for k > 1, light k cannot be turned either on or off unless light k – 1 is on and all preceding lights
are off for k > 1.
The question we want to explore is the following: how many moves are required to turn all n lights
off? For n = 5, a solution sequence has been worked out below. Fill in the missing entries. The
lights are counted from left to right, so the first bit is the first light, and so on.
11111
01111
11011
10011
00010
10010
11010
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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