Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)

Sankowski, Piotr

doi:10.1109/focs.2004.25

Cited by 84 publications

(115 citation statements)

References 19 publications

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“…1 Our results are tight in one of the following ways: (1) the query time of the existing algorithms cannot be improved without significantly increase the update time, (2) the update time of the existing algorithms cannot be improved without significantly increase the query time, (3) the update and query time of the existing algorithms cannot be improved simultaneously, and (4) the approximation guarantee cannot be improved without significantly increasing both query and update time. 2 For the s-t reachability problem, our result does not subsume the result based on the Boolean matrix multiplication (BMM) conjecture because the latter result holds only for combinatorial algorithms, and it is in fact larger than an upper bound provided by the non-combinatorial algorithm of Sankowski [40] (see Section 1.2 for a discussion). Also note that the result based on the triangle detection problem which is not subsumed by our result holds only for a more restricted notion of amortization (see Section 1.2).…”

Section: Omv-hardness For Dynamic Algorithmsmentioning

confidence: 75%

“…For example, it was shown in [1] that the combinatorial BMM conjecture implies that there is no combinatorial algorithm with n 3− preprocessing time, n 2− update time, and n 2− query time for the fully-dynamic s-t reachability and bipartite perfect matching problems. However, we can break these bounds using Sankowski's algebraic algorithms [40,43] which requires n ω preprocessing time, n 1.495 worst-case update time, and O(1) worst case query time, where ω is the exponent of the best known matrix multiplication algorithm (currently, ω < 2.3728639 [19]). …”

Section: Discussionmentioning

confidence: 99%

“…This leaves the open problem whether we can develop a faster exact algorithm for this problem using algebraic techniques (e.g. by adapting Sankowski's techniques [40,41,42]). Our OMv conjecture implies that this is not possible: there is no exact algorithm with polynomial preprocessing and mn 1− total update time if we needÕ(1) query time.…”

Section: Omv-hardness For Dynamic Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Henzinger

Krinninger

Nanongkai

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

239

327

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Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n × n matrix M and will receive n column-vectors of size n, denoted by v 1 , . . . , v n , one by one. After seeing each vector v i , we have to output the product M v i before we can see the next vector. A naive algorithm can solve this problem using O(n 3 ) time in total, and its running time can be slightly improved to O(n 3 / log 2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n 3− )) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, dfailure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "Strassenlike algorithms" [Ballard et al. SPAA'11].The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures -such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase -thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fullydynamic densest subgraph and diameter problems.

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Section: Omv-hardness For Dynamic Algorithmsmentioning

confidence: 75%

Section: Discussionmentioning

confidence: 99%

Section: Omv-hardness For Dynamic Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Henzinger

Krinninger

Nanongkai

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

239

327

View full text Add to dashboard Cite

show abstract

“…Our algorithms work on directed acyclic digraphs, and are randomized with one-sided error. After our work, Sankowski [16] has presented subquadratic algorithms for fully dynamic transitive closure on general graphs.…”

Section: Discussionmentioning

confidence: 99%

Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n ² barrier

Demetrescu

Italiano

2005

J. ACM

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We present an algorithm for directed acyclic graphs that breaks through the O(n 2 ) barrier on the single-operation complexity of fully dynamic transitive closure, where n is the number of edges in the graph. We can answer queries in O(n ) worst-case time and perform updates in O(n ω(1, ,1)− + n 1+ ) worst-case time, for any ∈ [0, 1], where ω(1, , 1) is the exponent of the multiplication of an n × n matrix by an n × n matrix. The current best bounds on ω(1, , 1) imply an O(n 0.575 ) query time and an O(n 1.575 ) update time in the worst case. Our subquadratic algorithm is randomized, and has one-sided error. As an application of this result, we show how to solve single-source reachability in O(n 1.575 ) time per update and constant time per query.

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“…The former maintains the inverse of a matrix under updates, and is a critical routine for many graph algorithms [45,33,22]. However, this often leads to dense matrices, and we combine it with techniques from graph sparsification [49,1,30,35] to obtain our nearlylinear running times.…”

Section: Introductionmentioning

confidence: 99%

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Li¹,

Zhang²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Estimating the relative importance of vertices and edges is a fundamental issue in the analysis of complex networks, and has found vast applications in various aspects, such as social networks, power grids, and biological networks. Most previous work focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called θ-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter θ. We define two equivalent measures for θ-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the θ-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its -approximation for all the m edges in nearly linear time in m. The proposed θ-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the θ-Kirchhoff edge centrality, we present a θ-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute -approximate Kirchhoff vertex centrality for all the n vertices in nearly linear time in m.

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Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)

Cited by 84 publications

References 19 publications

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n ² barrier

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

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Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)

Cited by 84 publications

References 19 publications

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n 2 barrier

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

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Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n ² barrier