Complexity Models for Incremental Computation

We present a new complexity theoretic approach to incremental computation. We de ne complexity classes that capture the intuitive notion of incremental e ciency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have small incremental time complexity. We show that all common LOGSPACE-complete problems for P are also incr -POLYLOGTIME-complete for P. 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 give incrementally complete problems for NLOGSPACE, LOGSPACE, and non-uniform NC. We show that under certain restrictions problems which have e cient dynamic solutions also have e cient parallel solutions. We also consider a non-uniform model of incremental computation and show that in this model most problems have almost linear complexity. In addition, we present some techniques for lower bounding the complexity of explicitly de ned problems. We also look at the time complexity of circuit value and network stability problems restricted to comparator gates. We show that the comparator-circuit value problem and the \Lex-First Maximal Matching" problem are in incr -LOGSPACE while the comparator-network stability and the \Man-Optimal Stable Marriage Problem" are in rincr -LOGSPACE(NLOGSPACE). This shows that the dynamic versions of these problems are solvable quickly in parallel even though there are no known NC algorithms to solve them from scratch. This paper reports the combined work of research e orts that have appeared in shortened form in \A complexity theoretic approach to incremental computation," by S. Sairam, J. S. Vitter, and R. Tamassia (STACS '93), and \On-line reevaluation of functions," by P. B. Miltersen (Tech-report, Aarhus university). Supported in part by the ESPRIT II Basic Research Actions Program of the European Community under contract No. 3075 (project ALCOM). Email: [email protected] Supported in part by an NSF PYI award CCR{9047466 with matching funds from IBM, by NSF research grant CCR{9007851, by Army Research O ce grant DAAL03{91{G{0035, and by the O ce of Naval Research and the Defense Advanced Research Projects Agency under contract N00014{91{J{4052, ARPA order 8225. Email: [email protected], [email protected] Supported in part by an NSF research grant CCR{9007851, by Army Research O ce grant DAAL03{ 91{G{0035, and by the O ce of Naval Research and the Defense Advanced Research Projects Agency under contract N00014{91{J{4052, ARPA order 8225. Email: [email protected]