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# A Deterministic Parallel APSP Algorithm and its Applications

TL;DR: In this article, a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs was presented, which has O(n/d)$-work and O(tilde{O}(d)depth.

Abstract: In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $\tilde{O}(nm+(n/d)^3)$ work and $\tilde{O}(d)$ depth for any depth parameter $d\in [1,n]$. To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP'17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has $\tilde{O}(nm+n^3/d^2)$ work and $\tilde{O}(d)$ depth [Ullman and Yannakakis, SIAM J. Comput. '91]. Our APSP algorithm turns out to be a powerful tool for designing efficient planar graph algorithms in both parallel and sequential regimes.
One notable ingredient of our parallel APSP algorithm is a simple deterministic $\tilde{O}(nm)$-work $\tilde{O}(d)$-depth procedure for computing $\tilde{O}(n/d)$-size hitting sets of shortest $d$-hop paths between all pairs of vertices of a real-weighted digraph. Such hitting sets have also been called $d$-hub sets. Hub sets have previously proved especially useful in designing parallel or dynamic shortest paths algorithms and are typically obtained via random sampling. Our procedure implies, for example, an $\tilde{O}(nm)$-time deterministic algorithm for finding a shortest negative cycle of a real-weighted digraph. Such a near-optimal bound for this problem has been so far only achieved using a randomized algorithm [Orlin et al., Discret. Appl. Math. '18].

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01 Oct 1992TL;DR: This book provides an introduction to the design and analysis of parallel algorithms, with the emphasis on the application of the PRAM model of parallel computation, with all its variants, to algorithm analysis.

Abstract: Written by an authority in the field, this book provides an introduction to the design and analysis of parallel algorithms. The emphasis is on the application of the PRAM (parallel random access machine) model of parallel computation, with all its variants, to algorithm analysis. Special attention is given to the selection of relevant data structures and to algorithm design principles that have proved to be useful. Features *Uses PRAM (parallel random access machine) as the model for parallel computation. *Covers all essential classes of parallel algorithms. *Rich exercise sets. *Written by a highly respected author within the field. 0201548569B04062001

1,577 citations

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TL;DR: Algorithms for finding shortest paths are presented which are faster than algorithms previously known on networks which are relatively sparse in arcs, and a class of “arc set partition” algorithms is introduced.

Abstract: Algorithms for finding shortest paths are presented which are faster than algorithms previously known on networks which are relatively sparse in arcs. Known results which the results of this paper extend are surveyed briefly and analyzed. A new implementation for priority queues is employed, and a class of “arc set partition” algorithms is introduced. For the single source problem on networks with nonnegative arcs a running time of O(min(n1+1/k + e, n + e) log n)) is achieved, where there are n nodes and e arcs, and k is a fixed integer satisfying k > 0. This bound is O(e) on dense networks. For the single source and all pairs problem on unrestricted networks the running time is O(min(n2+1/k + ne, n2 log n + ne log n).

1,124 citations

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TL;DR: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log 2 + 10(n - 1) using processors which can independently perform arithmetic operations in unit time.

Abstract: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log2n + 10(n - 1)/p using p ≥ 1 processors which can independently perform arithmetic operations in unit time. This bound is within a constant factor of the best possible. A sharper result is given for expressions without the division operation, and the question of numerical stability is discussed.

864 citations

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TL;DR: A simple characterization of λ∗, as well as an algorithm for computing it efficiently, is given, which is called the minimum cycle mean.

Abstract: Let C = (V,E) be a digraph with n vertices. Let f be a function from E into the real numbers, associating with each edge e ∈ E a weight ƒ(e) . Given any sequence of edges σ = e1,e2,…,ep define w(σ), the weight of σ, as ∑ i = 1 p ƒ(e i ) , and define m(σ), the mean weight of σ, as w(σ)⧸p. Let λ ∗ = min C m(C) where C ranges over all directed cycles in G; λ∗ is called the minimum cycle mean. We give a simple characterization of λ∗, as well as an algorithm for computing it efficiently.

807 citations

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TL;DR: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only O(√n) vertices, and this separator theorem in combination with a divide-and-conquer strategy leads to many new complexity results for planar graphs problems.

Abstract: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only $O(\sqrt n )$ vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results.

767 citations