Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max

Bringmann, Karl; Künnemann, Marvin; Węgrzycki, Karol

doi:10.1145/3313276.3316373

Cited by 13 publications

(14 citation statements)

References 76 publications

(101 reference statements)

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“…The first example is the approximate positive APSP problem where the goal is to compute a 1 ± approximation of the distances for a graph with positive weights. A recent result by Bringmann, Künnemann, and Wegrzycki [2] shows that approximate positive APSP can be solved in Õ n poly( ) time 1 . The algorithm in [2] is obtained by showing an equivalance between approximate positive APSP and the (min,max)-product problem which is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

$\{-1,0,1\}$-APSP and (min,max)-Product Problems

Barr¹,

Kopelowitz²,

Porat³

et al. 2019

Preprint

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In the {−1, 0, 1}-APSP problem the goal is to compute all-pairs shortest paths (APSP) on a directed graph whose edge weights are all from {−1, 0, 1}. In the (min,max)-product problem the input is two n × n matrices A and B, and the goal is to output the (min,max)-product of A and B.This paper provides a new algorithm for the {−1, 0, 1}-APSP problem via a simple reduction to the target-(min,max)-product problem where the input is three n × n matrices A, B, and T , and the goal is to output a Boolean n × n matrix C such that the (i, j) entry of C is 1 if and only if the (i, j) entry of the (min,max)-product of A and B is exactly the (i, j) entry of the target matrix T . If (min,max)-product can be solved in T M M (n) = Ω(n 2 ) time then it is straightforward to solve target-(min,max)-product in O(T M M (n)) time. Thus, given the recent result of Bringmann, Künnemann, and Wegrzycki [STOC 2019], the {−1, 0, 1}-APSP problem can be solved in the same time needed for solving approximate APSP on graphs with positive weights.Moreover, we design a simple algorithm for target-(min,max)-product when the inputs are restricted to the family of inputs generated by our reduction. Using fast rectangular matrix multiplication, the new algorithm is faster than the current best known algorithm for (min,max)product.

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Section: Introductionmentioning

confidence: 99%

$\{-1,0,1\}$-APSP and (min,max)-Product Problems

Barr¹,

Kopelowitz²,

Porat³

et al. 2019

Preprint

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show abstract

“…The dominance product has algorithm with complexity O(n (3+ω)/2 ) = O(n 2.686 ) [13], and has been slightly improved by rectangular matrix multiplication [19]. The complexity of O(n (3+ω)/2 ), which is the exponential "middle point" between cubic time and FMM, is the current time complexity for many problems, such as all-pair bottleneck path [15,7], all-pair non-decreasing path [18,8], approximate APSP without scaling [4]. Recently, solving linear program is also shown to have current running time same as FMM [5,11].…”

Section: Related Workmentioning

confidence: 99%

Faster Algorithms for Bounded-Difference Min-Plus Product

Chi¹,

Duan²,

Xie³

2021

Preprint

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Min-plus product of two n × n matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a special class of matrices, called δ-bounded-difference matrices, in which the difference between any two adjacent entries is bounded by δ = O(1). Our algorithm runs in randomized time O(n 2.779 ) by the fast rectangular matrix multiplication algorithm [Le Gall & Urrutia 18], better than Õ(n 2+ω/3 ) = O(n 2.791 ) (ω < 2.373 [Alman & V.V.Williams 20]). This improves previous result of Õ (n 2.824 ) [Bringmann et al. 16]. When ω = 2 in the ideal case, our complexity is Õ(n 2+2/3 ), improving Bringmann et al.'s result of Õ(n 2.755 ).

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“…One can improve upon this for graphs with small integer weights using matrix multiplication [17,28,44]. A subcubic-time (1 + ǫ)-approximation can also be achieved this way [11,50]. Much of the recent work regarded approximating the minimum weight cycle within factor at least 2 [14,15,18].…”

Section: Related Workmentioning

confidence: 99%

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

Karczmarz

2021

Preprint

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We consider the directed minimum weight cycle problem in the fully dynamic setting. To the best of our knowledge, so far no fully dynamic algorithms have been designed specifically for the minimum weight cycle problem in general digraphs. One can achieve O(n 2 ) amortized update time by simply invoking the fully dynamic APSP algorithm of Demetrescu and Italiano [J. ACM '04]. This bound, however, yields no improvement over the trivial recompute-from-scratch algorithm for sparse graphs.Our first contribution is a very simple deterministic (1+ǫ)-approximate algorithm supporting vertex updates (i.e., changing all edges incident to a specified vertex) in conditionally nearoptimal O(m log (W )/ǫ) amortized time for digraphs with real edge weights in [1, W ]. Using known techniques, the algorithm can be implemented on planar graphs and also gives some new sublinear fully dynamic algorithms maintaining approximate cuts and flows in planar digraphs.Additionally, we show a Monte Carlo randomized exact fully dynamic minimum weight cycle algorithm with O(mn 2/3 ) worst-case update that works for real edge weights. To this end, we generalize the exact fully dynamic APSP data structure of Abraham et al. [SODA'17] to solve the multiple-pairs shortest paths problem, where one is interested in computing distances for some k (instead of all n 2 ) fixed source-target pairs after each update. We show that in such a scenario, O((m + k)n 2/3 ) worst-case update time is possible.

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Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max

Cited by 13 publications

References 76 publications

$\{-1,0,1\}$-APSP and (min,max)-Product Problems

$\{-1,0,1\}$-APSP and (min,max)-Product Problems

Faster Algorithms for Bounded-Difference Min-Plus Product

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

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