103s11chap6solutions

103s11chap6solutions - Math 103, Spring 2011, Solutions to...

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Math 103, Spring 2011, Solutions to Chapter 6 Homework Assignment Problem 24: (a) There are 25! Hamilton circuits in the complete graph K 26 , which is approximately 1.5511 x 10 25 . Dividing by a trillion, which equals 10 12 , we find that our supercomputer requires approximately 1.5511 x 10 13 seconds to list all of these Hamilton circuits. Dividing by 60, we find that the supercomputer requires approximately 2.5852 x 10 11 minutes. Dividing by 60, we find that the supercomputer requires approximately 4308669456 hours. Dividing by 24, we find that the supercomputer requires approximately 179527894 days. Dividing by 365 (neglecting leap years), we find that the supercomputer requires approximately 491,857 years to list all the Hamilton circuits in K 26 . (b) Very little extra work is now required to answer the same question for K 27 , which has 26! Hamilton circuits. Since 26! is just 26 times 25!, the supercomputer takes 26 times longer to list all the Hamilton circuits of K 27 as it does for K 26 . Thus it would take the supercomputer approximately 12,788,288 years. Comment: The point of this exercise is that what appears to be a reasonable task, such as finding the least expensive itinerary on a trip which visits each of 26 cities before returning home, the time required to apply brute force methods is quite unreasonable. "You mean, we can't just tell the computer to do all the work? We have to show some cleverness instead?" Yes, and the need to find more practical approaches than brute force enumeration is the reason for studying such methods as the repetitive nearest neighbor and cheapest link algorithms. Problem 26:
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103s11chap6solutions - Math 103, Spring 2011, Solutions to...

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