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Fully Dynamic All Pairs All Shortest Paths [article]

Matteo Pontecorvi, Vijaya Ramachandran
2022 arXiv   pre-print
We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph G=(V,E) with a positive real weight on each edge  ...  Our method is a generalization and a variant of the fully dynamic algorithm of Demetrescu and Italiano [DI04] for unique shortest path, and it builds on our recent decremental APASP [NPR14].  ...  McQueeney for working on implementing the FULLY-DYNAMIC algorithm described in this paper.  ... 
arXiv:1412.3852v4 fatcat:7eget5wtrbeqbmm2yzwf4ckkea

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

Ittai Abraham, Shiri Chechik, Sebastian Krinninger
2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms  
We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph.  ...  Introduction In the all-pairs shortest paths (APSP) problem we are interested in computing the distance matrix of a given graph.  ...  Fully dynamic algorithms. The study of fully dynamic APSP algorithms for general directed graphs was initiated by King [Kin99] .  ... 
doi:10.1137/1.9781611974782.28 dblp:conf/soda/AbrahamCK17 fatcat:4qvphbqv6zfchjpywqht656h6y

Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds [chapter]

Maximilian Probst Gutenberg, Christian Wulff-Nilsn
2020 Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms  
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  ...  These are the first exact dynamic algorithms with truly-subcubic update time and space usage.  ...  Further, we present the first algorithm for the fully-dynamic All-Pairs Shortest Paths problem (even amortized) in weighted graphs that obtains truly sub-cubic time and space usage at the same time 1 .  ... 
doi:10.1137/1.9781611975994.156 dblp:conf/soda/GutenbergW20b fatcat:qugtpkckuva2tilhqcbeluk4xi

Fully dynamic all pairs shortest paths with real edge weights

Camil Demetrescu, Giuseppe F. Italiano
2006 Journal of computer and system sciences (Print)  
We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with real-valued edge weights.  ...  No previous fully dynamic algorithm was known for this problem.  ...  Next, in Section 5 we use this data structure to solve fully dynamic all pairs shortest paths in its generality.  ... 
doi:10.1016/j.jcss.2005.05.005 fatcat:uf2gpex2hrhcpirotvyypnejbm

Average update times for fully-dynamic all-pairs shortest paths

Tobias Friedrich, Nils Hebbinghaus
2011 Discrete Applied Mathematics  
We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary nonnegative edge weights.  ...  For a random update we show that the expected time per update is bounded by O(n 4/3+ε ) for all ε > 0. If the graph is outside the critical window, we prove even smaller bounds.  ...  In this article, we consider the fully-dynamic all-pairs shortest path problem (APSP) for undirected graphs, which is one of the most fundamental problems in dynamic graph algorithms.  ... 
doi:10.1016/j.dam.2011.02.007 fatcat:uxqljdpy3fexxokix7pode35rq

Average Update Times for Fully-Dynamic All-Pairs Shortest Paths [chapter]

Tobias Friedrich, Nils Hebbinghaus
2008 Lecture Notes in Computer Science  
We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary non-negative edge weights.  ...  For a random update we show that the expected time per update is bounded by O(n 4/3+ε ) for all ε > 0.  ...  In this article, we consider the fully-dynamic all-pairs shortest path problem (APSP) for undirected graphs, which is one of the most fundamental problems in dynamic graph algorithms.  ... 
doi:10.1007/978-3-540-92182-0_61 fatcat:vowyqjlnh5at7mdfdzldqqecw4

Fully Dynamic All-Pairs Shortest Paths: Breaking the O(n) Barrier

Ittai Abraham, Shiri Chechik, Kunal Talwar, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
In this paper we break a longstanding barrier in the design of fully dynamic all-pairs approximate distance oracles.  ...  A fully dynamic approximate distance oracle is a distance reporting data structure that supports dynamic insert edge and delete edge operations.  ...  Our Fully Dynamic All-Pairs Shortest Paths As mentioned earlier, we use the decremental distance oracle Dec from Section 4.  ... 
doi:10.4230/lipics.approx-random.2014.1 dblp:conf/approx/AbrahamCT14 fatcat:wd6y6tvkxvgylpie4qrdiuxuwi

Fully Dynamic (2 + ε) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time

Aaron Bernstein
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
However, little progress has been made for fully dynamic approximate algorithms.  ...  For any pair x, y ∈ V let π(x, y) be the shortest x − y path, and let d(x, y) be the weight of π(x, y).  ... 
doi:10.1109/focs.2009.16 dblp:conf/focs/Bernstein09 fatcat:larkpeoodrgghprv7rk6v4o5bm

Fully dynamic all pairs shortest paths with real edge weights

C. Demetrescu, G.F. Italiano
Proceedings 2001 IEEE International Conference on Cluster Computing  
We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with real-valued edge weights.  ...  No previous fully dynamic algorithm was known for this problem.  ...  Next, in Section 5 we use this data structure to solve fully dynamic all pairs shortest paths in its generality.  ... 
doi:10.1109/sfcs.2001.959900 dblp:conf/focs/DemetrescuI01 fatcat:pg2n6btemjgrfgpxbvyzfjeaou

Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs

V. King
40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)  
This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b.  ...  For exact shortest paths the amortized update time is O(n 2:5 p b log n).  ...  For exact all-pairs shortest paths, the amortized update time is O(n 2:5 p b log n). There are no previously known fully dynamic algorithms for general graphs.  ... 
doi:10.1109/sffcs.1999.814580 dblp:conf/focs/King99 fatcat:xnyzfujzfrdfbepvmekmprqonm

Triangulations [chapter]

Seiya Yokohama
2013 Discrete Mathematics and Its Applications  
Italiano: Experimental analysis of dynamic all pairs shortest path algorithms. ACM Transactions on Algorithms 2 (4): 578-601 (2006).  ...  Italiano A New Approach to Dynamic All Pairs Shortest Paths Journal of the Association for Computing Machinery (JACM), 51(6), pp. 968-992, November 2004 Experimental study of dynamic NAPSP algorithms  ...  Maintaining shortest paths under deletions in weighted directed graphs. In STOC, 725-734, 2013.  ... 
doi:10.1201/b16132-52 fatcat:wpkdrbdlubhjrocxna6h34nfyi

Algorithmic Techniques for Maintaining Shortest Routes in Dynamic Networks

Camil Demetrescu, Giuseppe F. Italiano
2007 Electronical Notes in Theoretical Computer Science  
In this paper, we survey algorithms for shortest paths in dynamic networks.  ...  Dynamic Shortest Paths In this section we survey the best known algorithms for fully dynamic shortest paths. We start with a formal definition of the fully dynamic all pairs shortest paths problem.  ...  Since by Lemma 2.6 locally historical paths include shortest paths, this yields the fastest known algorithm for fully dynamic all pairs shortest paths.  ... 
doi:10.1016/j.entcs.2006.11.006 fatcat:ss7kxtmk4fah5a5qmpxieln2be

Fully Dynamic Betweenness Centrality [chapter]

Matteo Pontecorvi, Vijaya Ramachandran
2015 Lecture Notes in Computer Science  
For graphs with ν * = O(n), our algorithms match the fully dynamic all pairs shortest paths (APSP) bounds of Demetrescu and Italiano [8] and Thorup [28] for unique shortest paths, where ν * = n − 1.  ...  We present fully dynamic algorithms for maintaining betweenness centrality (BC) of vertices in a directed graph G = (V, E) with positive edge weights.  ...  Both of our BC algorithms are obtained through fully dynamic all pairs all shortest paths.  ... 
doi:10.1007/978-3-662-48971-0_29 fatcat:yh7hl4aapvgijgcajrvbejnqym

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

Camil Demetrescu, Giuseppe F. Italiano
2006 Journal of Discrete Algorithms  
In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths.  ...  This is an annotated bibliography on fully dynamic algorithms for path problems on general directed graphs.  ...  Dynamic shortest paths For dynamic shortest paths, King [27] presented a fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with positive integer weights less than C:  ... 
doi:10.1016/j.jda.2005.12.003 fatcat:dyr5qbndajfqzlppmiojqhen4e

A new approach to dynamic all pairs shortest paths

Camil Demetrescu, Giuseppe F. Italiano
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems.  ...  Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in O(n 2 ) 1 amortized time per update and  ...  DYNAMIC ALL PAIRS SHORTEST PATHS In this section we show how to exploit the combinatorial properties of uniform paths presented in Section 2 to design a fully dynamic algorithm for maintaining all pairs  ... 
doi:10.1145/780542.780567 dblp:conf/stoc/DemetrescuI03 fatcat:wvuqqytvfbg65gwnrbgzzl22b4
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