@inproceedings{11866,
  abstract     = {We present a deterministic (1+o(1))-approximation O(n1/2+o(1)+D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o(1))-approximation Õ(n1/2D1/4+D)-time algorithm of Nanongkai [STOC 2014] by a factor of as large as n1/8, and (ii) the O(є−1logє−1)-approximation factor of Lenzen and Patt-Shamir’s Õ(n1/2+є+D)-time algorithm [STOC 2013] within the same running time. Our running time matches the known time lower bound of Ω(n1/2/logn + D) [Das Sarma et al. STOC 2011] modulo some lower-order terms, thus essentially settling the status of this problem which was raised at least a decade ago [Elkin SIGACT News 2004]. It also implies a (2+o(1))-approximation O(n1/2+o(1)+D1+o(1))-time algorithm for approximating a network’s weighted diameter which almost matches the lower bound by Holzer et al. [PODC 2012].

In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the “hitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (no(1), o(1))-hop set of size O(n1+o(1)). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [Goldberg et al. STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+o(1))-approximation O(no(1))-time algorithm on congested cliques, and (ii) a (1+o(1))-approximation O(no(1)logW)-pass O(n1+o(1)logW)-space streaming algorithm, when edge weights are in {1, 2, …, W}. The first result answers an open problem in [Nanongkai, STOC 2014]. The second result partially answers an open problem raised by McGregor in 2006 [<pre>sublinear.info</pre>, Problem 14].},
  author       = {Henzinger, Monika H and Krinninger, Sebastian and Nanongkai, Danupon},
  booktitle    = {48th Annual ACM SIGACT Symposium on Theory of Computing},
  isbn         = {978-145034132-5},
  issn         = {0737-8017},
  location     = {Cambridge, MA, United States},
  pages        = {489 -- 498},
  publisher    = {Association for Computing Machinery},
  title        = {{A deterministic almost-tight distributed algorithm for approximating single-source shortest paths}},
  doi          = {10.1145/2897518.2897638},
  year         = {2016},
}

@inproceedings{11867,
  abstract     = {We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2+є)-approximate maximum matching in general graphs with O(poly(logn, 1/є)) update time. (2) An algorithm that maintains an αK approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1≤ αK < 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.},
  author       = {Bhattacharya, Sayan and Henzinger, Monika H and Nanongkai, Danupon},
  booktitle    = {48th Annual ACM SIGACT Symposium on Theory of Computing},
  isbn         = {978-145034132-5},
  issn         = {0737-8017},
  location     = {Cambridge, MA, United States},
  pages        = {398 -- 411},
  publisher    = {Association for Computing Machinery},
  title        = {{New deterministic approximation algorithms for fully dynamic matching}},
  doi          = {10.1145/2897518.2897568},
  year         = {2016},
}