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NP-complete problems - People

Ndingshortestpathsandminimumspanningtree singraphs, matchingsinbipartitegraphs, maximumincreasingsub-sequences, maximum owsinnetworks, cient, becauseineach casetheirtimerequirementgrowsasa polynomialfunction(such asn,n2, orn3) ofthesizeof betterappreciatesuch ef cientalgorithms, considerthealternative:Inalltheseprob-le mswearesearchingfora solution(path,tree, matching, etc.)fromamonganexponentialpopulationofp ossibilities. Indeed,nboyscanbematchedwithngirlsinn!di fferentways, agraphwithnverticeshasnn 2spanningtrees, anda typicalgraphhasanexponentialnum-berof pathsfromstot. Alltheseproblemscouldinprinciplebesolved inexponentialtimebycheckingthroughallcan didatesolutions, onebyone.

nential algorithms make polynomially slow progress, while polynomial algorithms advance exponentially fast! For Moore’s law to be reected in the world we need efcient algorithms. As Sissa and Malthus knew very well, exponential expansion cannot be sustained in-denitely in our nite world. Bacterial colonies run out of food; chips hit the ...

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