Programming Languages

Algorithm design / monograph by Jon Kleinberg; Eva Tardos

By Jon Kleinberg; Eva Tardos

Creation: a few consultant difficulties -- fundamentals of algorithms research -- Graphs -- grasping algorithms -- Divide and overcome -- Dynamic programming -- community movement -- NP and computational intractability -- PSPACE: a category of difficulties past NP -- Extending the boundaries of tractability -- Approximation algorithms -- neighborhood seek -- Randomized algorithms -- Epilogue: algorithms that run eternally

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This contrast can be formalized in the class of PSPACE-complete problems, of which Competitive Facility Location is an example. PSPACE-complete problems are believed to be strictly harder than NP-complete problems, and this conjectured lack of short “proofs” for their solutions is one indication of this greater hardness. The notion of PSPACE-completeness turns out to capture a large collection of problems involving game-playing and planning; many of these are fundamental issues in the area of artificial intelligence.

We then develop running-time bounds for some basic algorithms, beginning with an implementation of the Gale-Shapley algorithm from Chapter 1 and continuing to a survey of many different running times and certain characteristic types of algorithms that achieve these running times. In some cases, obtaining a good running-time bound relies on the use of more sophisticated data structures, and we conclude this chapter with a very useful example of such a data structure: priority queues and their implementation using heaps.

There would exist some stable matching M in which a good man m was married to a bad woman w. Now, let’s consider what the other pairs in M look like. There are k good men and k good women. Could it be the case that every good woman is married to a good man in this matching M? No: one of the good men (namely, m) is already married to a bad woman, and that leaves only k − 1 other good men. So even if all of them were married to good women, that would still leave some good woman who is married to a bad man.

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