MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Reachability in graph timelines
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Improved Deterministic Algorithms for Decremental Reachability and Strongly Connected Components
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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In this paper we consider the problem of dynamic transitive closure with lookahead. We present a randomized one-sided error algorithm with updates and queries in O(n ω(1,1,ε)−ε ) time given a lookahead of n ε operations, where ω(1,1,ε) is the exponent of multiplication of n×n matrix by n×n ε matrix. For ε≤0.294 we obtain an algorithm with queries and updates in O(n 2−ε ) time, whereas for ε=1 the time is O(n ω−1). This is essentially optimal as it implies an O(n ω ) algorithm for boolean matrix multiplication. We also consider the offline transitive closure in planar graphs. For this problem, we show an algorithm that requires $O(n^{\frac{\omega}{2}})$time to process $n^{\frac{1}{2}}$operations. We also show a modification of these algorithms that gives faster amortized queries. Finally, we give faster algorithms for restricted type of updates, so called element updates. All of the presented algorithms are randomized with one-sided error. All our algorithms are based on dynamic algorithms with lookahead for matrix inverse, which are of independent interest.