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Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time [article]

Jan van den Brand, Danupon Nanongkai
2019 arXiv   pre-print
Williams, STOC'13], we also obtain the first subquadratic worst-case update time for (5/3+ϵ)-approximating the eccentricities and (1.5+ϵ)-approximating the diameter and radius for unweighted graphs (with  ...  This query is rather general and captures several versions of the dynamic shortest paths problem.  ...  ., n s }, so for any 1 ≤ k ≤ n s we can query for all pairs, if the distance is at most k inÕ(sn ω(1,µ+s,1) ) time. After every edge update, we query this information for k = n s .  ... 
arXiv:1909.10850v2 fatcat:xecognte6jbupk2r6slt4za25u

Hardness of Approximation in P via Short Cycle Removal: Cycle Detection, Distance Oracles, and Beyond [article]

Amir Abboud, Karl Bringmann, Seri Khoury, Or Zamir
2022 arXiv   pre-print
The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths.  ...  Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4- or 5-cycles in a worst-case instance had been the obstacle  ...  We also thank Thatchaphol Saranurak for references on dynamic shortest paths.  ... 
arXiv:2204.10465v1 fatcat:mh277wuu7zccpewp6marj6usqy

A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond [article]

Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak
2020 arXiv   pre-print
We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an n-vertex graph is n^o(1), thus resolving a major open problem  ...  In particular, we obtain a (log m)^1/ϵ-approximation in time m^1+O(1/√(ϵ)) for any constant ϵ, and a (log m)^f(m)-approximation in time m^1+o(1), for any slowly growing function m.  ...  Gao and Peng were supported in part by NSF grant CCF-1718533.  ... 
arXiv:1910.08025v2 fatcat:mpgj4cb635antd4hsnrutgo3pa

New Techniques and Fine-Grained Hardness for Dynamic Near-Additive Spanners [article]

Thiago Bergamaschi, Monika Henzinger, Maximilian Probst Gutenberg, Virginia Vassilevska Williams, Nicole Wein
2021 arXiv   pre-print
Our new algebraic techniques and spanner algorithms allow us to also obtain (1) a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) with update and path query time O(n^1.9); (2) a fully dynamic  ...  For any constant ϵ∈ (0,1], there is a fully dynamic algorithm with worst-case update time O(n^1.529) that whp maintains an n^1+o(1) edge (1+ϵ,n^o(1))-spanner.  ...  There exists an algorithm that maintains (1 + )approximate all-pairs shortest-path in worst-case time O(n 1.843+o (1) ) per edge update, in worst-case time O(n 1+o (1) ) per path reporting query, and in  ... 
arXiv:2010.10134v3 fatcat:7fkwbkabsjbm3mwgnjibrywjwa

Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

Amir Abboud, Virginia Vassilevska Williams
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n 3−ε ) time for some ε > 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle?  ...  The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph  ...  ACKNOWLEDGMENTS We would like to thank Liam Roditty, Ryan Williams, and Uri Zwick for valuable discussions.  ... 
doi:10.1109/focs.2014.53 dblp:conf/focs/AbboudW14 fatcat:ayagllj5svejblnqsmsenj64ge

Popular conjectures imply strong lower bounds for dynamic problems [article]

Amir Abboud, Virginia Vassilevska Williams
2014 arXiv   pre-print
Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n^3-ϵ) time for some ϵ>0? 4. Is there a linear time algorithm that detects whether a given graph contains a triangle? 5.  ...  The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph  ...  We would like to thank Liam Roditty, Ryan Williams, and Uri Zwick for valuable discussions.  ... 
arXiv:1402.0054v1 fatcat:bgdvvk4e2naetd5taffv2ilovy

Algorithms and Hardness for Diameter in Dynamic Graphs

Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, Nicole Wein, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
Shortest Paths (APSP), which is very computationally intensive.  ...  While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs  ...  Our conditional lower bounds for the fully dynamic setting also apply to incremental and decremental algorithms that have worst case update and query time guarantees.  ... 
doi:10.4230/lipics.icalp.2019.13 dblp:conf/icalp/AnconaHRWW19 fatcat:bg2j3huworg7jbf3lt2clk2clm

Finding Shortest Paths With Computational Geometry [chapter]

Po-Shen Loh
2006 Graph Algorithms and Applications 4  
We present a heuristic search algorithm for the R d Manhattan shortest path problem that achieves front-to-front bidirectionality in subquadratic time.  ...  ) time and O(n log d−1 n) space, where n is the number of visited vertices.  ...  Acknowledgements Special thanks to Alain Martin and Mika Nyström for introducing this problem to the author, and to Charles Leiserson for providing pointers toward related literature.  ... 
doi:10.1142/9789812773296_0013 fatcat:mqosy3hfdndubaieajnih7a7nq

Finding Shortest Paths With Computational Geometry

Po-Shen Loh
2003 Journal of Graph Algorithms and Applications  
We present a heuristic search algorithm for the R d Manhattan shortest path problem that achieves front-to-front bidirectionality in subquadratic time.  ...  ) time and O(n log d−1 n) space, where n is the number of visited vertices.  ...  Acknowledgements Special thanks to Alain Martin and Mika Nyström for introducing this problem to the author, and to Charles Leiserson for providing pointers toward related literature.  ... 
doi:10.7155/jgaa.00071 fatcat:6der3ge3vrfxncwsvptboggodm

Chapter 7 A survey of computational geometry [chapter]

Joseph S.B. Mitchell, Subhash Suri
1995 Handbooks in Operations Research and Management Science  
We brie y mention most of the major developments in path planning research over the last two decades, but to a large extent limit ourselves to issues related to shortest paths in a planar domain.  ...  The topic of path planning is a vast one, with problems ranging from nding shortest paths in a discrete graph to deciding the feasible motion of a complex robot in an environment full of complex obstacles  ...  Nevertheless, the underlying geometry can be exploited to compute an MST in subquadratic worst-case time.  ... 
doi:10.1016/s0927-0507(05)80124-0 fatcat:nogmdqmv6bflni3waox5dxrsiy

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs [article]

Maximilian Probst Gutenberg, Virginia Vassilevska Williams, Nicole Wein
2020 arXiv   pre-print
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph G=(V,E) subject to edge insertions and deletions and a source vertex s∈ V, and the goal is to maintain the distance d(s,  ...  [STOC'15] showed that under the OMv Hypothesis, the partially dynamic exact s-t Shortest Path problem in undirected graphs requires amortized update or query time m^1/2-o(1), given polynomial preprocessing  ...  Unless otherwise stated, queries take worst-case constant time. Algorithms for partially dynamic directed SSSP.  ... 
arXiv:2001.10751v1 fatcat:3na6ck346varzprcoumpfamaki

On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter [article]

Søren Dahlgaard
2016 arXiv   pre-print
While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental  ...  Furthermore no algorithm with amortized update time O(n^1-ε) exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time O(n^1/2 - ε) exists  ...  Acknowledgements I would like to thank Amir Abboud and Virginia Vassilevska Williams for helpful discussions and observations.  ... 
arXiv:1602.06705v2 fatcat:os3gntola5bcnea5wl67dxdjpa

Algorithms and Hardness for Diameter in Dynamic Graphs [article]

Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, Nicole Wein
2019 arXiv   pre-print
Shortest Paths (APSP), which is very computationally intensive.  ...  While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs  ...  Our conditional lower bounds for the fully dynamic setting also apply to incremental and decremental algorithms that have worst case update and query time guarantees.  ... 
arXiv:1811.12527v3 fatcat:6unt2p5gf5hapmwbzwlsrobqni

An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions

Yevgeny Schreiber, Micha Sharir
2007 Discrete & Computational Geometry  
The algorithm constructs a dynamic version of Mount's data structure [16] that implicitly encodes the shortest paths from s to all other points of the surface.  ...  The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11] , and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront"  ...  paper, and to Joe Mitchell for his comments of Kapoor's paper.  ... 
doi:10.1007/s00454-007-9031-0 fatcat:d67mebu4m5d2xcwaeqdghd7mfq

An optimal-time algorithm for shortest paths on a convex polytope in three dimensions

Yevgeny Schreiber, Micha Sharir
2006 Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06  
The algorithm constructs a dynamic version of Mount's data structure [16] that implicitly encodes the shortest paths from s to all other points of the surface.  ...  The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11] , and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront"  ...  paper, and to Joe Mitchell for his comments of Kapoor's paper.  ... 
doi:10.1145/1137856.1137862 dblp:conf/compgeom/SchreiberS06 fatcat:4nbrn6574zatbjaf7c5jgfsedm
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