By Herbert S. Wilf
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Additional resources for Algorithms and Complexity (Internet edition, 1994)
Well, what ‘trivialthing’ shall we do, in this ‘trivial case’ ? The fact is that there isn’t any way of finding the maximum independent set in a graph where all vertices have ≤ 3 neighbors that’s any faster than the general methods that we’ve already discussed. In fact, if one could find a fast method for that restricted problem it would have extremely important consequences, because we would then be able to do all graphs rapidly, not just those special ones. 41 Chapter 2: Recursive Algorithms We will learn more about this phenomenon in Chapter 5, but for the moment let’s leave just the observation that the general problem of maxset turns out to be no harder than the special case of maxset in which no vertex has more than 3 neighbors.
11), and is ∼ 2n log n (n → ∞). Quicksort is, on average, a very quick sorting method, even though its worst case requires a quadratic amount of labor. 2 1. Write out an array of 10 numbers that contains no splitter. Write out an array of 10 numbers that contains 10 splitters. 2. Write a program that does the following. Given a positive integer n. Choose 100 random permutations of [1, 2, . , n],* and count how many of the 100 had at least one splitter. Execute your program for n = 5, 6, . , 12 and tabulate the results.
This economy is much appreciated by those who sort, because sorting applications can be immense and time consuming. One popular sorting application is in alphabetizing lists of names. It is easy to imagine that some of those lists are very long, and that the replacement of Θ(n2 ) by an average of O(n log n) comparisons is very welcome. An insurance company that wants to alphabetize its list of 5,000,000 policyholders will gratefully notice the difference between n2 = 25, 000, 000, 000, 000 comparisons and n log n = 77, 124, 740 comparisons.
Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf