Deterministic Decremental SSSP and Approximate Min-Cost Flow in Almost-Linear Time

Bernstein, Aaron; Gutenberg, Maximilian Probst; Saranurak, Thatchaphol

doi:10.48550/arxiv.2101.07149

Cited by 5 publications

(19 citation statements)

References 36 publications

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Section: The Full Constructionmentioning

confidence: 99%

“…A data-structure with the above requirements indeed exists by the recent work of Bernstein et al [BGS21]. Since this exact formulation was not explicitly stated in [BGS21], let us point out how to obtain it. In Section III.5 of [BGS21], there are log γ nW distance scales of the data structures where γ ∈ (2, n o(1) ).…”

Section: The Full Constructionmentioning

confidence: 99%

“…For this reason, the data-structures (e.g. [HKN14,BGS21]) typically use a hopset: by adding a set E hopset of m • n o(1) edges to the graph, it guarantees that there exist a path from s to any v that is (1+ )-shortest-path and that only uses n o(1) edges; namely, it has only n o(1) hops. A hopset-edge e ∈ E hopset typically has a length that corresponds to the shortest path distance between its endpoints, and therefore the distance in the new graph is never shorter than the original distance.…”

Section: The Basic Constructionmentioning

confidence: 99%

“…Since this exact formulation was not explicitly stated in [BGS21], let us point out how to obtain it. In Section III.5 of [BGS21], there are log γ nW distance scales of the data structures where γ ∈ (2, n o(1) ). For each scale i, the objects in Lemma 4.6, such as the estimates di (v) for all v, are maintained by ApxBall π data structure described in Sections II.5 and III.3 of [BGS21].…”

Section: The Full Constructionmentioning

confidence: 99%

“…In Section III.5 of [BGS21], there are log γ nW distance scales of the data structures where γ ∈ (2, n o(1) ). For each scale i, the objects in Lemma 4.6, such as the estimates di (v) for all v, are maintained by ApxBall π data structure described in Sections II.5 and III.3 of [BGS21]. Hi is called an emulator in Definition II.5.2 in [BGS21].…”

Section: The Full Constructionmentioning

confidence: 99%

See 4 more Smart Citations

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

Abboud¹,

Krauthgamer²,

Li³

et al. 2021

Preprint

Self Cite

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In 1961, Gomory and Hu showed that the max-flow values of all n 2 pairs of vertices in an undirected graph can be computed using only n − 1 calls to any max-flow algorithm, and succinctly represented them in a tree (called the Gomory-Hu tree later). Even assuming a lineartime max-flow algorithm, this yields a running time of O(mn) for this problem; with current max-flow algorithms, the running time is Õ(mn + n 5/2 ). We break this 60-year old barrier by giving an Õ(n 2 )-time algorithm for the Gomory-Hu tree problem, already with current max-flow algorithms. For unweighted graphs, our techniques show equivalence (up to poly-logarithmic factors in running time) between Gomory-Hu tree (i.e., all-pairs max-flow values) and a singlepair max-flow. * The first version of this paper (in November 2021) showed a slower time bound of Õ(n 23/8 ) for computing a Gomory-Hu tree. The only technical changes are in Section 2.

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Section: The Full Constructionmentioning

confidence: 99%

Section: The Full Constructionmentioning

confidence: 99%

Section: The Basic Constructionmentioning

confidence: 99%

Section: The Full Constructionmentioning

confidence: 99%

Section: The Full Constructionmentioning

confidence: 99%

See 3 more Smart Citations

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

Abboud¹,

Krauthgamer²,

Li³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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Geometric bounds on the fastest mixing Markov chain

Olesker-Taylor,

Zanetti

2024

Probab. Theory Relat. Fields

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In the Fastest Mixing Markov Chain problem, we are given a graph $$G = (V, E)$$ G = ( V , E ) and desire the discrete-time Markov chain with smallest mixing time $$\tau $$ τ subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time $$\tau _\textsf {RW}$$ τ RW of the lazy random walk on G is characterised by the edge conductance $$\Phi $$ Φ of G via Cheeger’s inequality: $$\Phi ^{-1} \lesssim \tau _\textsf {RW} \lesssim \Phi ^{-2} \log |V|$$ Φ - 1 ≲ τ RW ≲ Φ - 2 log | V | . Analogously, we characterise the fastest mixing time $$\tau ^\star $$ τ ⋆ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance $$\Psi $$ Ψ of G: $$\Psi ^{-1} \lesssim \tau ^\star \lesssim \Psi ^{-2} (\log |V|)^2$$ Ψ - 1 ≲ τ ⋆ ≲ Ψ - 2 ( log | V | ) 2 . This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only $$\varepsilon $$ ε -close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time $$\tau \lesssim \varepsilon ^{-1} ({\text {diam}} G)^2 \log |V|$$ τ ≲ ε - 1 ( diam G ) 2 log | V | . Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.

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Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

Dey¹,

Zhang²

2021

Preprint

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Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the 1-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain an approximation. First, taking advantage of the proximity of PD points, we condense them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called 1 PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

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Deterministic Decremental SSSP and Approximate Min-Cost Flow in Almost-Linear Time

Cited by 5 publications

References 36 publications

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

Geometric bounds on the fastest mixing Markov chain

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

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