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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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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Miller and Reif’s FOCS'85 [Gary L. Miller and John H. Reif, 1989] classic and fundamental tree contraction algorithm is a broadly applicable technique for the parallel solution of a large number of tree problems. Additionally it is also used as an algorithmic design technique for a large number of parallel graph algorithms. In all previously explored models of computation, however, tree contractions have only been achieved in Ω(log n) rounds of parallel run time. In this work, we not only introduce a generalized tree contraction method but also show it can be computed highly efficiently in O(1/ε³) rounds in the Adaptive Massively Parallel Computing (AMPC) setting, where each machine has O(n^ε) local memory for some 0 < ε < 1. AMPC is a practical extension of Massively Parallel Computing (MPC) which utilizes distributed hash tables [MohammadHossein Bateni et al., 2017; Behnezhad et al., 2019; Raimondas Kiveris et al., 2014]. In general, MPC is an abstract model for MapReduce, Hadoop, Spark, and Flume which are currently widely used across industry and has been studied extensively in the theory community in recent years. Last but not least, we show that our results extend to multiple problems on trees, including but not limited to maximum and maximal matching, maximum and maximal independent set, tree isomorphism testing, and more.

MohammadTaghi Hajiaghayi, Marina Knittel, Hamed Saleh, and Hsin-Hao Su. Adaptive Massively Parallel Constant-Round Tree Contraction. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hajiaghayi_et_al:LIPIcs.ITCS.2022.83, author = {Hajiaghayi, MohammadTaghi and Knittel, Marina and Saleh, Hamed and Su, Hsin-Hao}, title = {{Adaptive Massively Parallel Constant-Round Tree Contraction}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {83:1--83:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.83}, URN = {urn:nbn:de:0030-drops-156790}, doi = {10.4230/LIPIcs.ITCS.2022.83}, annote = {Keywords: Adaptive Massively Parallel Computation, Tree Contraction, Matching, Independent Set, Tree Isomorphism} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

The densest subgraph problem, introduced in the 80s by Picard and Queyranne [Networks 1982] as well as Goldberg [Tech. Report 1984], is a classic problem in combinatorial optimization with a wide range of applications. The lowest outdegree orientation problem is known to be its dual problem. We study both the problem of finding dense subgraphs and the problem of computing a low outdegree orientation in the distributed settings.
Suppose G = (V,E) is the underlying network as well as the input graph. Let D denote the density of the maximum density subgraph of G. Our main results are as follows.
- Given a value D̃ ≤ D and 0 < ε < 1, we show that a subgraph with density at least (1-ε)D̃ can be identified deterministically in O((log n) / ε) rounds in the LOCAL model. We also present a lower bound showing that our result for the LOCAL model is tight up to an O(log n) factor.
In the CONGEST~ model, we show that such a subgraph can be identified in O((log³ n) / ε³) rounds with high probability. Our techniques also lead to an O(diameter + (log⁴ n)/ε⁴)-round algorithm that yields a 1-ε approximation to the densest subgraph. This improves upon the previous O(diameter /ε ⋅ log n)-round algorithm by Das Sarma et al. [DISC 2012] that only yields a 1/2-ε approximation.
- Given an integer D̃ ≥ D and Ω(1/D̃) < ε < 1/4, we give a deterministic, Õ((log² n) /ε²)-round algorithm in the CONGEST~ model that computes an orientation where the outdegree of every vertex is upper bounded by (1+ε)D̃. Previously, the best deterministic algorithm and randomized algorithm by Harris [FOCS 2019] run in Õ((log⁶ n)/ ε⁴) rounds and Õ((log³ n) /ε³) rounds respectively and only work in the LOCAL model.

Hsin-Hao Su and Hoa T. Vu. Distributed Dense Subgraph Detection and Low Outdegree Orientation. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{su_et_al:LIPIcs.DISC.2020.15, author = {Su, Hsin-Hao and Vu, Hoa T.}, title = {{Distributed Dense Subgraph Detection and Low Outdegree Orientation}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {15:1--15:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.15}, URN = {urn:nbn:de:0030-drops-130938}, doi = {10.4230/LIPIcs.DISC.2020.15}, annote = {Keywords: Distributed Algorithms, Network Algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the problem of distributed task allocation in multi-agent systems. Suppose there is a collection of agents, a collection of tasks, and a demand vector, which specifies the number of agents required to perform each task. The goal of the agents is to cooperatively allocate themselves to the tasks to satisfy the demand vector. We study the dynamic version of the problem where the demand vector changes over time. Here, the goal is to minimize the switching cost, which is the number of agents that change tasks in response to a change in the demand vector. The switching cost is an important metric since changing tasks may incur significant overhead.
We study a mathematical formalization of the above problem introduced by Su, Su, Dornhaus, and Lynch [Su et al., 2017], which can be reformulated as a question of finding a low distortion embedding from symmetric difference to Hamming distance. In this model it is trivial to prove that the switching cost is at least 2. We present the first non-trivial lower bounds for the switching cost, by giving lower bounds of 3 and 4 for different ranges of the parameters.

Hsin-Hao Su and Nicole Wein. Lower Bounds for Dynamic Distributed Task Allocation. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 99:1-99:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{su_et_al:LIPIcs.ICALP.2020.99, author = {Su, Hsin-Hao and Wein, Nicole}, title = {{Lower Bounds for Dynamic Distributed Task Allocation}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {99:1--99:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.99}, URN = {urn:nbn:de:0030-drops-125063}, doi = {10.4230/LIPIcs.ICALP.2020.99}, annote = {Keywords: distributed task allocation, combinatorics, lower bounds, multi-agent systems, low-distortion embedding, dynamic algorithms, biological distributed algorithms} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

We study distributed algorithms for some fundamental problems in data summarization. Given a communication graph G of n nodes each of which may hold a value initially, we focus on computing sum_{i=1}^N g(f_i), where f_i is the number of occurrences of value i and g is some fixed function. This includes important statistics such as the number of distinct elements, frequency moments, and the empirical entropy of the data.
In the CONGEST~ model, a simple adaptation from streaming lower bounds shows that it requires Omega~(D+ n) rounds, where D is the diameter of the graph, to compute some of these statistics exactly. However, these lower bounds do not hold for graphs that are well-connected. We give an algorithm that computes sum_{i=1}^{N} g(f_i) exactly in {tau_{G}} * 2^{O(sqrt{log n})} rounds where {tau_{G}} is the mixing time of G. This also has applications in computing the top k most frequent elements.
We demonstrate that there is a high similarity between the GOSSIP~ model and the CONGEST~ model in well-connected graphs. In particular, we show that each round of the GOSSIP~ model can be simulated almost perfectly in O~({tau_{G}}) rounds of the CONGEST~ model. To this end, we develop a new algorithm for the GOSSIP~ model that 1 +/- epsilon approximates the p-th frequency moment F_p = sum_{i=1}^N f_i^p in O~(epsilon^{-2} n^{1-k/p}) rounds , for p >= 2, when the number of distinct elements F_0 is at most O(n^{1/(k-1)}). This result can be translated back to the CONGEST~ model with a factor O~({tau_{G}}) blow-up in the number of rounds.

Hsin-Hao Su and Hoa T. Vu. Distributed Data Summarization in Well-Connected Networks. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{su_et_al:LIPIcs.DISC.2019.33, author = {Su, Hsin-Hao and Vu, Hoa T.}, title = {{Distributed Data Summarization in Well-Connected Networks}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {33:1--33:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.33}, URN = {urn:nbn:de:0030-drops-113400}, doi = {10.4230/LIPIcs.DISC.2019.33}, annote = {Keywords: Distributed Algorithms, Network Algorithms, Data Summarization} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

(Delta+1)-vertex coloring is one of the most fundamental symmetry breaking graph problems, receiving tremendous amount of attention over the last decades. We consider the congested clique model where in each round, every pair of vertices can exchange O(log n) bits of information.
In a recent breakthrough, Yi-Jun Chang, Wenzheng Li, and Seth Pettie [CLP-STOC'18] presented a randomized (Delta+1)-list coloring algorithm in the LOCAL model that works in O(log^*n+Det_{deg}(log log n)) rounds, where Det_{deg}(n') is the deterministic LOCAL complexity of (deg+1)-list coloring algorithm on n'-vertex graphs. Unfortunately, the CLP algorithm uses large messages and hence cannot be efficiently implemented in the congested clique model when the maximum degree Delta is large (in particular, when Delta=omega(sqrt{n})).
Merav Parter [P-ICALP'18] recently provided a randomized (Delta+1)-coloring algorithm in O(log log Delta * log^* Delta) congested clique rounds based on a careful partitioning of the input graph into almost-independent subgraphs with maximum degree sqrt{n}. In this work, we significantly improve upon this result and present a randomized (Delta+1)-coloring algorithm with O(log^* Delta) rounds, with high probability. At the heart of our algorithm is an adaptation of the CLP algorithm for coloring a subgraph with o(n) vertices and maximum degree Omega(n^{5/8}) in O(log^* Delta) rounds. The approach is built upon a combination of techniques, this includes: the graph sparsification of [Parter-ICALP'18], and a palette sampling technique adopted to the CLP framework.

Merav Parter and Hsin-Hao Su. Randomized (Delta+1)-Coloring in O(log* Delta) Congested Clique Rounds. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{parter_et_al:LIPIcs.DISC.2018.39, author = {Parter, Merav and Su, Hsin-Hao}, title = {{Randomized (Delta+1)-Coloring in O(log* Delta) Congested Clique Rounds}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {39:1--39:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.39}, URN = {urn:nbn:de:0030-drops-98286}, doi = {10.4230/LIPIcs.DISC.2018.39}, annote = {Keywords: Distributed Graph Algorithms, Coloring, congested clique} }

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