Minimizing randomness in minimum spanning tree, parallel connectivity, and set maxima algorithms
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
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There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing connected components and (minimum) spanning trees in parallel, and the problem of performing sensitivity analysis on shortest path trees and minimum spanning trees. However, while each of these problems has a randomized algorithm whose performance meets a known lower bound, all of these randomized algorithms use a number of random bits which is linear in the number of operations they perform. We address the issue of reduucing the number of random bits used in these randomized algorithms. For each of the problems listed above, we present randomized algorithms that have optimal performance but use only a polylogarithmic number of random bits; for some of the problems our optimal algorithms use only log* n random bits. Our results represent an exponential savings in the amount of randomness used to achieve the same optimal performance as in the earlier algorithms. Our techniques are general and could likely be applied to other problems.