We design efficient deterministic algorithms for finding short edge-disjoint paths in expanders. Specifically, given an \(n\)-vertex \(m\)-edge expander \(G\) of conductance \(\phi\) and minimum degree \(\delta\), and a set of pairs \(\{(s_i,t_i)\}_i\) such that each vertex appears in at most \(k\) pairs, our algorithm deterministically computes a set of edge-disjoint paths from \(s_i\) to \(t_i\), one for every \(i\): (1) each of length at most \(18 \log(n)/\phi\) and in \(mn^{1+o(1)} \min\{k,\phi^{-1}\}\) total time, assuming \(\phi^3 \delta \ge (35 \log n)^3 k\), or (2) each of length at most \(n^{o(1)}/\phi\) and in total \(m^{1+o(1)}\) time, assuming \(\phi^3 \delta \ge n^{o(1)} k\). Before our work, deterministic polynomial-time algorithms were known only for expanders with constant conductance and were significantly slower. To obtain our result, we give an almost-linear time algorithm for hypergraph perfect matching under generalizations of Hall-type conditions (Haxell 1995), a powerful framework with applications in various settings, which until now has only admitted large polynomial-time algorithms (Annamalai 2018).

We consider the expander routing problem formulated by Ghaffari, Kuhn, and Su (PODC 2017), where the goal is to route all the tokens to their destinations given that each vertex is the source and the destination of at most deg(υ) tokens. They developed randomized algorithms that solve this problem in poly [EQUATION] rounds in the CONGEST model, where ϕ is the conductance of the graph. In addition, as noted by Chang, Pettie, Saranurak, and Zhang (JACM 2021), it is possible to obtain a preprocessing/query tradeoff so that the routing queries can be answered faster at the cost of more preprocessing time. The efficiency and flexibility of the processing/query tradeoff of expander routing have led to many other distributed algorithms in the CONGEST model, such as subpolynomial-round minimum spanning tree algorithms in expander graphs and near-optimal algorithms for k-clique enumeration in general graphs.

In this paper, we study parallel algorithms for the correlation clustering problem, where every pair of two different entities is labeled with similar or dissimilar. The goal is to partition the entities into clusters to minimize the number of disagreements with the labels. Currently, all efficient parallel algorithms have an approximation ratio of at least 3. In comparison with the 1.994 + ɛ ratio achieved by polynomial-time sequential algorithms [25], a significant gap exists.

A cactus representation of a graph, introduced by Dinitz et al. in 1976, is an edge sparsifier of O(n) size that exactly captures all global minimum cuts of the graph. It is a central combinatorial object that has been a key ingredient in almost all algorithms for the connectivity augmentation problems and for maintaining minimum cuts under edge insertions (e.g. [Naor et al. SICOMP’97], [Cen et al. SODA’22], [Henzinger ICALP’95]). This sparsifier was generalized to Steiner cactus for a vertex set T, which can be seen as a vertex sparsifier of O(|T|) size that captures all partitions of T corresponding to a T-Steiner minimum cut, and also hypercactus, an analogous concept in hypergraphs. These generalizations further extend the applications of cactus to the Steiner and hypergraph settings.

Given an undirected weighted graph with n vertices and m edges, we give the first deterministic m1+o(1)-time algorithm for constructing the cactus representation of all global minimum cuts. This improves the current n2+o(1)-time state-of-the-art deterministic algorithm, which can be obtained by combining ideas implicitly from three papers [22, 27, 12]. The known explicitly stated deterministic algorithm has a runtime of Õ(mn) [9, 34]. Using our technique, we can even speed up the fastest randomized algorithm of [23] whose running time is at least Ω(m log4 n) to O(m log3 n).

Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha’s [ 13 ] 1984 protocol (see also Ben-Or [ 8 ]) achieved f < n/3 resilience at the cost of exponential expected latency 2 Θ ( n ) , a bound that has never been improved in this model with f = ⌊ (n-1)/3 ⌋ corruptions. In this article, we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt f < n/3 parties, while incurring only polynomial latency with high probability . Our protocol follows an earlier polynomial latency protocol of King and Saia [ 33 , 34 ], which had suboptimal resilience, namely f ≈ n /10 9 [ 33 , 34 ]. Resilience f = (n-1)/3 is uniquely difficult, as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventually lets us flip a coin with an unambiguous outcome. In the beginning, the influence of the Byzantine players is too powerful to overcome, and they can essentially fix the coin’s behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can “blacklist” players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection .

The maximum weighted matching (mwm) problem is one of the most well-studied combinatorial optimization problems in distributed graph algorithms. Despite a long development on the problem, and the recent progress of Fischer, Mitrovic, and Uitto [16] who gave a poly(1/ϵ, log n)-round algorithm for obtaining a (1 − ϵ)-approximate solution for unweighted maximum matching, it had been an open problem whether a (1 − ϵ)-approximate mwm can be obtained in poly(1/ϵ, log n) rounds in the CONGEST model. Algorithms with such running times were only known for special graph classes such as bipartite graphs [1] and minor-free graphs [8]. For general graphs, the previously known algorithms require exponential in (1/ϵ) rounds for obtaining a (1 − ϵ)-approximate solution [13] or achieve an approximation factor of at most 2/3 [1]. In this work, we settle this open problem by giving a deterministic poly(1/ϵ, log n)-round algorithm for computing a (1 − ϵ)-approximate mwm for general graphs in the CONGEST model. Our proposed solution extends the algorithm of Fischer, Mitrovic, and Uitto [16], blends in the sequential algorithm from Duan and Pettie [11] and the work of Faour, Fuchs, and Kuhn [13]. Interestingly, this solution also implies a CREW PRAM algorithm with poly(1/ϵ, log n) span using only O(m) processors, and a poly(1/ϵ)-passes algorithm in the semi-streaming model.

Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha's [Bra87] 1984 protocol (see also Ben-Or [Ben83]) achieved f < n/3 resilience at the cost of exponential expected latency 2θ(n), a bound that has never been improved in this model with f = ⌊(n- 1)/3⌋ corruptions.

The paper presents parallel algorithms for multiplying implicit simple unit-Monge matrices (Krusche and Tiskin, PPAM 2009) of size n x n in the EREW PRAM model. We show implicit simple unit-Monge matrices multiplication of size n x n can be achieved by a deterministic EREW PRAM algorithm with O(n log n log log n) total work and O(log3 n) span. This implies that there is a deterministic EREW PRAM algorithm solving the longest increasing subsequence (LIS) problem in O(n log2 n log log n) work and O(log 4 n) span. Furthermore, with randomization and bitwise operations, implicitly multiplying two simple unit-Monge matrices can be improved to O(n log n) work and O(log3n) span, which leads to a randomized EREW PRAM algorithm obtaining LIS in O(nlog2n) work and O(log4n) span with high probability. In the regime where the LIS has length k = Ψ(log3n), our results improve the span from Õ(n2/3) (Krusche and Tiskin, SPAA 2010) and O(klog n) (Gu, Men, Shen, Sun, and Wan, SPAA 2023) to O(log4 n) while the total work remains near optimal Õ (n).

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for $c$-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for $c$-edge st-connectivity [Jin and Sun FOCS'21(arXiv:2004.07650)]. We study a natural extension called vertex sparsifiers for $c$-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph $G=(V,E)$ with $n$ vertices and $m$ hyperedges with $k$ terminal vertices and a parameter $c$, there exists a hypergraph $H$ containing only $O(kc^{3})$ hyperedges that preserves all minimum cuts (up to value $c$) between all subset of terminals. This matches the best bound of $O(kc^{3})$ edges for normal graphs by [Liu'20(arXiv:2011.15101)]. Moreover, $H$ can be constructed in almost-linear $O(p^{1+o(1)} + n(rc\log n)^{O(rc)}\log m)$ time where $r=\max_{e\in E}|e|$ is the rank of $G$ and $p=\sum_{e\in E}|e|$ is the total size of $G$, or in $\text{poly}(m, n)$ time if we slightly relax the size to $O(kc^{3}\log^{1.5}(kc))$ hyperedges.

It has been known since the early 1980s that Byzantine Agreement in the full information, asynchronous model is impossible to solve deterministically against even one crash fault [FLP 1985], but that it can be solved with probability 1 [Ben-Or 1983], even against an adversary that controls the scheduling of all messages and corrupts up to f

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We prove that any $O(n)$-size shortcut set cannot bring the diameter below $\Omega(n^{1/6})$ and that any $O(m)$-size shortcut set cannot bring it below $\Omega(n^{1/11})$. These improve Hesse's [ Proceedings of the 14 th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Baltimore, MD, 2003] lower bound of $\Omega(n^{1/17})$. By combining these constructions with Abboud and Bodwin's [ J. ACM, 64 (2017), 28] edge-splitting technique, we get additive stretch lower bounds of $+\Omega(n^{1/11})$ for $O(n)$-size spanners and $+\Omega(n^{1/18})$ for $O(n)$-size emulators. These improve Abboud and Bodwin's $+\Omega(n^{1/22})$ lower bounds for both spanners and emulators.

Computing the strongly connected Components (SCCs) in a graph $G=(V,E)$ is known to take only $O(m + n)$ time using an algorithm by Tarjan [SIAM J. Comput., 1 (1972), pp. 146--160] where $m = |E|$, $n=|V|$. For fully dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for decremental graphs, i.e., graphs that undergo edge deletions. In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time $\tilde{O}(m)$, thus only a polylogarithmic factor from the optimal running time. (We use $\tilde{O}(f(n))$ notation to suppress logarithmic factors, i.e., $g(n) = \tilde{O}(f(n))$ if $g(n) = O(f(n) {polylog}(n)).$) Our result also yields the fastest algorithm for the decremental single-source reachability (SSR) problem which can be reduced to decrementally maintaining SCCs. Using a well-known reduction, we use our decremental result to achieve new update/query-time trade-offs in the fully dynamic setting. We can maintain the reachability of pairs $S \times V$, $S \subseteq V$ in fully dynamic graphs with update time $\tilde{O}(\frac{|S|m}{t})$ and query time $O(t)$ for all $t \in [1,|S|]$; this matches to polylogarithmic factors the best all-pairs reachability algorithm for $S = V$.