A complexity theoretic approach to incremental computation

Abstract

We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We give problems that are complete for P, NLOGSPACE, and LOGSPACE under incremental reductions. We introduce a restricted notion of completeness called NRP-completeness and show that problems which are NRP-complete for P are also incr-POLYLOGTIME-complete for P. We also look at the incremental space-complexity of circuit value and network stability problems restricted to comparator gates. We show that the dynamic version of the comparator circuit value problem is in LOGSPACE while the dynamic version of the network stability problem can be solved in LOGSPACE given an NLOGSPACE oracle. This shows that problems like the Lex-First Maximal Matching problem and the Man-Optimal Stable Marriage problem can be quickly updated in parallel even though there are no known NC algorithms to solve them from scratch.