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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

This paper presents parallel, distributed, and quantum algorithms for single-source shortest paths when edges can have negative integer weights (negative-weight SSSP). We show a framework that reduces negative-weight SSSP in all these settings to n^{o(1)} calls to any SSSP algorithm that works on inputs with non-negative integer edge weights (non-negative-weight SSSP) with a virtual source. More specifically, for a directed graph with m edges, n vertices, undirected hop-diameter D, and polynomially bounded integer edge weights, we show randomized algorithms for negative-weight SSSP with
- W_{SSSP}(m,n)n^{o(1)} work and S_{SSSP}(m,n)n^{o(1)} span, given access to a non-negative-weight SSSP algorithm with W_{SSSP}(m,n) work and S_{SSSP}(m,n) span in the parallel model, and
- T_{SSSP}(n,D)n^{o(1)} rounds, given access to a non-negative-weight SSSP algorithm that takes T_{SSSP}(n,D) rounds in CONGEST, and
- Q_{SSSP}(m,n)n^{o(1)} quantum edge queries, given access to a non-negative-weight SSSP algorithm that takes Q_{SSSP}(m,n) queries in the quantum edge query model. This work builds off the recent result of Bernstein, Nanongkai, Wulff-Nilsen [Bernstein et al., 2022], which gives a near-linear time algorithm for negative-weight SSSP in the sequential setting.
Using current state-of-the-art non-negative-weight SSSP algorithms yields randomized algorithms for negative-weight SSSP with
- m^{1+o(1)} work and n^{1/2+o(1)} span in the parallel model, and
- (n^{2/5}D^{2/5} + √n + D)n^{o(1)} rounds in CONGEST, and
- m^{1/2}n^{1/2+o(1)} quantum queries to the adjacency list or n^{1.5+o(1)} quantum queries to the adjacency matrix. Up to a n^{o(1)} factor, the parallel and distributed results match the current best upper bounds for reachability [Jambulapati et al., 2019; Cao et al., 2021]. Consequently, any improvement to negative-weight SSSP in these models beyond the n^{o(1)} factor necessitates an improvement to the current best bounds for reachability. The quantum result matches the lower bound up to an n^{o(1)} factor [Aija Berzina et al., 2004].
Our main technical contribution is an efficient reduction from computing a low-diameter decomposition (LDD) of directed graphs to computations of non-negative-weight SSSP with a virtual source. Efficiently computing an LDD has heretofore only been known for undirected graphs in both the parallel and distributed models, and been rather unstudied in quantum models. The directed LDD is a crucial step of the sequential algorithm in [Bernstein et al., 2022], and we think that its applications to other problems in parallel and distributed models are far from being exhausted.
Other ingredients of our results include altering the recursion structure of the scaling algorithm in [Bernstein et al., 2022] to surmount difficulties that arise in these models, and also an efficient reduction from computing strongly connected components to computations of SSSP with a virtual source in CONGEST. The latter result answers a question posed in [Bernstein and Nanongkai, 2019] in the negative.

Vikrant Ashvinkumar, Aaron Bernstein, Nairen Cao, Christoph Grunau, Bernhard Haeupler, Yonggang Jiang, Danupon Nanongkai, and Hsin-Hao Su. Parallel, Distributed, and Quantum Exact Single-Source Shortest Paths with Negative Edge Weights. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ashvinkumar_et_al:LIPIcs.ESA.2024.13, author = {Ashvinkumar, Vikrant and Bernstein, Aaron and Cao, Nairen and Grunau, Christoph and Haeupler, Bernhard and Jiang, Yonggang and Nanongkai, Danupon and Su, Hsin-Hao}, title = {{Parallel, Distributed, and Quantum Exact Single-Source Shortest Paths with Negative Edge Weights}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.13}, URN = {urn:nbn:de:0030-drops-210849}, doi = {10.4230/LIPIcs.ESA.2024.13}, annote = {Keywords: Parallel algorithm, distributed algorithm, shortest paths} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced.
We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation.
To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest.

Vikrant Ashvinkumar, Sepehr Assadi, Chengyuan Deng, Jie Gao, and Chen Wang. Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ashvinkumar_et_al:LIPIcs.APPROX/RANDOM.2023.58, author = {Ashvinkumar, Vikrant and Assadi, Sepehr and Deng, Chengyuan and Gao, Jie and Wang, Chen}, title = {{Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {58:1--58:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.58}, URN = {urn:nbn:de:0030-drops-188830}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.58}, annote = {Keywords: Streaming algorithms, structural balance, pseudo-randomness generator} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no O(1)-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Bose and Morin, 2004] showed that there exists an online routing algorithm that is O(1)-competitive. However, a Delaunay triangulation can have Omega(n) vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree.
We show a simple construction, given a set V of n points in the Euclidean plane, of a planar geometric graph on V that has small weight (within a constant factor of the weight of a minimum spanning tree on V), constant degree, and that admits a local routing strategy that is O(1)-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an O(1)-competitive routing strategy.

Vikrant Ashvinkumar, Joachim Gudmundsson, Christos Levcopoulos, Bengt J. Nilsson, and André van Renssen. Local Routing in Sparse and Lightweight Geometric Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ashvinkumar_et_al:LIPIcs.ISAAC.2019.30, author = {Ashvinkumar, Vikrant and Gudmundsson, Joachim and Levcopoulos, Christos and Nilsson, Bengt J. and van Renssen, Andr\'{e}}, title = {{Local Routing in Sparse and Lightweight Geometric Graphs}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {30:1--30:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.30}, URN = {urn:nbn:de:0030-drops-115269}, doi = {10.4230/LIPIcs.ISAAC.2019.30}, annote = {Keywords: Computational geometry, Spanners, Routing} }

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