Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds
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Given a directed weighted graph G=(V,E) undergoing vertex insertions and deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of G. In two breakthrough results, Italiano and Demetrescu [STOC '03] presented an algorithm that requires Õ(n^2)amortized update time, and Thorup showed in [STOC '05] that worst-case update time Õ(n^2+3/4) can be achieved. In this article, we make substantial progress on the problem. We present the following new results: (1) We present the first deterministic data structure that breaks the Õ(n^2+3/4) worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to Õ(n^2+5/7) = Õ(n^2.71..) and to Õ(n^2+3/5) = Õ(n^2.6) for unweighted graphs. (2) We present a simple deterministic algorithm with Õ(n^2+3/4) worst-case update time (Õ(n^2+2/3) for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time Õ(n^2+2/3) (Õ(n^2 + 1/2) for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space Õ(n^2). These are the first exact dynamic algorithms with truly-subcubic update time and space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC '03, ICALP'04, Encyclopedia of Algorithms '08] and is critical to algorithms in practice [TALG '06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA '17].
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