arXiv subjects

Hsin-Hao Su

Publications and source records attributed to Hsin-Hao Su.

At least 19 recordsLinked to original sources

arXiv2026

How Hard Is It for Message-Passing GNNs to Simulate One Weisfeiler-Lehman Color-Refinement Step?

Message-passing graph neural networks (MPGNNs) are commonly compared with the Weisfeiler-Lehman (WL) color-refinement procedure, but this comparison does not quantify the resource parameters a network needs to realize color refinement with bounded-size messages and finite numerical precision. We study the cost of simulating a single color-refinement step on unattributed graphs. We distinguish input-independent, or oblivious, simulation from instance-dependent simulation. In the former, the parameters, or their distributions in randomized models, are fixed before the input instance is known. Our results show that the local form of WL color refinement hides a global relabeling problem. In the oblivious setting, deterministic and zero-error randomized MPGNNs cannot solve this problem in the worst case using only shallow networks with small messages. We complement this lower bound with a nearly matching construction in a stronger rooted, port-aware model. By contrast, when the color set is large, bounded-error randomness can greatly reduce the cost, and a one-layer MPGNN with messages of logarithmic size and a logarithmic number of random bits suffices. We show that this logarithmic number of random bits is essentially necessary for shallow, small-message simulations. When the color set is small, we still obtain a rooted, port-aware simulation, but this construction requires more layers or larger messages. We also prove that this extra cost is partly unavoidable, as small color sets force a nontrivial trade-off between the number of layers and the message size. Finally, instance-dependent simulation can be much shallower, but the required instance-specific parameters are not necessarily easy to find. Together, these results reveal quantitative structure hidden behind the statement that MPGNNs match WL color refinement.

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arXiv2025

Narrowing the LOCAL$\unicode{x2013}$CONGEST Gaps in Sparse Networks via Expander Decompositions

Many combinatorial optimization problems can be approximated within $(1 \pm ε)$ factors in $\text{poly}(\log n, 1/ε)$ rounds in the LOCAL model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches require sending messages of unlimited size, so they do not extend to the CONGEST model, which restricts the message size to be $O(\log n)$ bits. In this paper, we develop a generic framework for obtaining $\text{poly}(\log n, 1/ε)$-round $(1\pm ε)$-approximation algorithms for many combinatorial optimization problems, including maximum weighted matching, maximum independent set, and correlation clustering, in graphs excluding a fixed minor in the CONGEST model. This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs. Furthermore, we show that our framework can be applied to give an efficient distributed property testing algorithm for an arbitrary minor-closed graph property that is closed under taking disjoint union, significantly generalizing the previous distributed property testing algorithm for planarity in [Levi, Medina, and Ron, PODC 2018 & Distributed Computing 2021]. Our framework uses distributed expander decomposition algorithms [Chang and Saranurak, FOCS 2020] to decompose the graph into clusters of high conductance. We show that any graph excluding a fixed minor admits small edge separators. Using this result, we show the existence of a high-degree vertex in each cluster in an expander decomposition, which allows the entire graph topology of the cluster to be routed to a vertex. Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster.

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arXiv2025

Min-Max Correlation Clustering via Neighborhood Similarity

We present an efficient algorithm for the min-max correlation clustering problem. The input is a complete graph where edges are labeled as either positive $(+)$ or negative $(-)$, and the objective is to find a clustering that minimizes the $\ell_{\infty}$-norm of the disagreement vector over all vertices. We resolve this problem with an efficient $(3 + ε)$-approximation algorithm that runs in nearly linear time, $\tilde{O}(|E^+|)$, where $|E^+|$ denotes the number of positive edges. This improves upon the previous best-known approximation guarantee of 4 by Heidrich, Irmai, and Andres, whose algorithm runs in $O(|V|^2 + |V| D^2)$ time, where $|V|$ is the number of nodes and $D$ is the maximum degree in the graph. Furthermore, we extend our algorithm to the massively parallel computation (MPC) model and the semi-streaming model. In the MPC model, our algorithm runs on machines with memory sublinear in the number of nodes and takes $O(1)$ rounds. In the streaming model, our algorithm requires only $\tilde{O}(|V|)$ space, where $|V|$ is the number of vertices in the graph. Our algorithms are purely combinatorial. They are based on a novel structural observation about the optimal min-max instance, which enables the construction of a $(3 + ε)$-approximation algorithm using $O(|E^+|)$ neighborhood similarity queries. By leveraging random projection, we further show these queries can be computed in nearly linear time.

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arXiv2024

Deterministic Expander Routing: Faster and More Versatile

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 $°(v)$ tokens. They developed $\textit{randomized algorithms}$ that solve this problem in $\text{poly}(ϕ^{-1}) \cdot 2^{O(\sqrt{\log n \log \log n})}$ rounds in the $\textsf{CONGEST}$ model, where $ϕ$ is the conductance of the graph. Later, Ghaffari and Li (DISC 2018) gave an improved algorithm. However, both algorithms are randomized, which means that all the resulting applications are also randomized. Recently, Chang and Saranurak (FOCS 2020) gave a deterministic algorithm that solves an expander routing instance in $2^{O(\log^{2/3} n \cdot \log^{1/3} \log n)}$ rounds. The deterministic algorithm is less efficient and does not allow preprocessing/query tradeoffs, which precludes the de-randomization of algorithms that require this feature, such as the $k$-clique enumeration algorithm in general graphs. The main contribution of our work is a new deterministic expander routing algorithm that not only matches the randomized bound of [GKS 2017] but also allows preprocessing/query tradeoffs. Our algorithm solves a single instance of routing query in $2^{{O}(\sqrt{\log n \cdot \log \log n})}$ rounds. Our algorithm achieves the following preprocessing and query tradeoffs: For $0 < ε< 1$, we can answer every routing query in $\log^{O(1/ε)} n$ rounds at the cost of a $(n^{O(ε)} + \log^{O(1/ε)} n)$-round preprocessing procedure. Combining this with the approach of Censor-Hillel, Leitersdorf, and Vulakh (PODC 2022), we obtain a near-optimal $\tilde{O}(n^{1-2/k})$-round deterministic algorithm for $k$-clique enumeration in general graphs, improving the previous state-of-the-art $n^{1-2/k+o(1)}$.

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arXiv2023

Breaking 3-Factor Approximation for Correlation Clustering in Polylogarithmic Rounds

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 $1.994+ε$ ratio achieved by polynomial-time sequential algorithms [CLN22], a significant gap exists. We propose the first poly-logarithmic depth parallel algorithm that achieves a better approximation ratio than 3. Specifically, our algorithm computes a $(2.4+ε)$-approximate solution and uses $\tilde{O}(m^{1.5})$ work. Additionally, it can be translated into a $\tilde{O}(m^{1.5})$-time sequential algorithm and a poly-logarithmic rounds sublinear-memory MPC algorithm with $\tilde{O}(m^{1.5})$ total memory. Our approach is inspired by Awerbuch, Khandekar, and Rao's [AKR12] length-constrained multi-commodity flow algorithm, where we develop an efficient parallel algorithm to solve a truncated correlation clustering linear program of Charikar, Guruswami, and Wirth [CGW05]. Then we show the solution of the truncated linear program can be rounded with a factor of at most 2.4 loss by using the framework of [CMSY15]. Such a rounding framework can then be implemented using parallel pivot-based approaches.

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arXiv2023

$(1-ε)$-Approximate Maximum Weighted Matching in Distributed, Parallel, and Semi-Streaming Settings

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 [FMU22] who gave a $\text{poly}(1/ε, \log n)$-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 $\text{poly}(1/ε, \log n)$ rounds in the CONGEST model. Algorithms with such running times were only known for special graph classes such as bipartite graphs [AKO18] and minor-free graphs [CS22]. For general graphs, the previously known algorithms require exponential in $(1/ε)$ rounds for obtaining a $(1-ε)$-approximate solution [FFK21] or achieve an approximation factor of at most 2/3 [AKO18]. In this work, we settle this open problem by giving a deterministic $\text{poly}(1/ε, \log n)$-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 [FMU22], blends in the sequential algorithm from Duan and Pettie [DP14] and the work of Faour, Fuchs, and Kuhn [FFK21]. Interestingly, this solution also implies a CREW PRAM algorithm with $\text{poly}(1/ε, \log n)$ span using only $O(m)$ processors. In addition, with the reduction from Gupta and Peng [GP13], we further obtain a semi-streaming algorithm with $\text{poly}(1/ε)$ passes. When $ε$ is smaller than a constant $o(1)$ but at least $1/\log^{o(1)} n$, our algorithm is more efficient than both Ahn and Guha's $\text{poly}(1/ε, \log n)$-passes algorithm [AG13] and Gamlath, Kale, Mitrovic, and Svensson's $(1/ε)^{O(1/ε^2)}$-passes algorithm [GKMS19].

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arXiv2022

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Given a graph $G=(V,E)$ with arboricity $α$, we study the problem of decomposing the edges of $G$ into $(1+ε)α$ disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a $(2+ε)α$-forest decomposition using $O(\frac{\log n}ε)$ rounds. Ghaffari and Su [SODA `17] made further progress by computing a $(1+ε) α$-forest decomposition in $O(\frac{\log^3 n}{ε^4})$ rounds when $εα= Ω(\sqrt{α\log n})$, i.e. the limit of their algorithm is an $(α+ Ω(\sqrt{α\log n}))$-forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid \& Reed [Combinatorica `92], in fact provides a decomposition of the graph into \emph{star-forests}, i.e. each forest is a collection of stars. Our main result in this paper is to reduce the threshold of $εα$ in $(1+ε)α$-forest decomposition and star-forest decomposition. This further answers the $10^{\text{th}}$ open question from Barenboim and Elkin's "Distributed Graph Algorithms" book. Moreover, it gives the first $(1+ε)α$-orientation algorithms with {\it linear dependencies} on $ε^{-1}$. At a high level, our results for forest-decomposition are based on a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. Our result for star-forest decomposition uses a more careful probabilistic analysis for the construction of Alon, McDiarmid, \& Reed; the bounds on star-arboricity here were not previously known, even non-constructively.

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arXiv2022

Distributed Dense Subgraph Detection and Low Outdegree Orientation

The densest subgraph problem, introduced in the 80s by Picard and Queyranne as well as Goldberg, 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 $\tilde{D} \leq D$ and $0 < ε< 1$, we show that a subgraph with density at least $(1-ε)\tilde{D}$ 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 $O((\log^3 n) / ε^3)$ rounds with high probability. Our techniques also lead to an $O(diameter + (\log^4 n)/ε^4)$-round algorithm that yields a $1-ε$ approximation to the densest subgraph. This improves upon the previous $O(diameter /ε\cdot \log n)$-round algorithm by Das Sarma et al. [DISC 2012] that only yields a $1/2-ε$ approximation. Given an integer $\tilde{D} \geq D$ and $Ω(1/\tilde{D}) < ε< 1/4$, we give a deterministic, $\tilde{O}((\log^2 n) /ε^2)$-round algorithm in the CONGEST model that computes an orientation where the outdegree of every vertex is upper bounded by $(1+ε)\tilde{D}$. Previously, the best deterministic algorithm and randomized algorithm by Harris [FOCS 2019] run in $\tilde{O}((\log^6 n)/ ε^4)$ rounds and $\tilde{O}((\log^3 n) /ε^3)$ rounds respectively and only work in the LOCAL model.

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arXiv2021

Adaptive Massively Parallel Constant-round Tree Contraction

Miller and Reif's FOCS'85 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 $O(1/ε^3)$ 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. 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.

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arXiv2020

Lower Bounds for Dynamic Distributed Task Allocation

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, 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.

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arXiv2019

Distributed Data Summarization in Well-Connected Networks

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 $\tildeΩ(D+ n)$ 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 $τ_G \cdot 2^{O(\sqrt{\log n})}$ rounds where $τ_G$ 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 $\tilde{O}(τ_G $ rounds of the CONGEST model. To this end, we develop a new algorithm for the GOSSIP model that $1\pm ε$ approximates the $p$-th frequency moment $F_p = \sum_{i=1}^N f_i^p$ in $\tilde{O}(ε^{-2} n^{1-k/p})$ rounds, for $p \geq2$, when the number of distinct elements $F_0$ is at most $O\left(n^{1/(k-1)}\right)$. This result can be translated back to the CONGEST model with a factor $\tilde{O}(τ_G)$ blow-up in the number of rounds.

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arXiv2019

Towards the Locality of Vizing's Theorem

Vizing showed that it suffices to color the edges of a simple graph using $Δ+ 1$ colors, where $Δ$ is the maximum degree of the graph. However, up to this date, no efficient distributed edge-coloring algorithms are known for obtaining such a coloring, even for constant degree graphs. The current algorithms that get closest to this number of colors are the randomized $(Δ+ \tildeΘ(\sqrtΔ))$-edge-coloring algorithm that runs in $\text{polylog}(n)$ rounds by Chang et al. (SODA '18) and the deterministic $(Δ+ \text{polylog}(n))$-edge-coloring algorithm that runs in $\text{poly}(Δ, \log n)$ rounds by Ghaffari et al. (STOC '18). We present two distributed edge-coloring algorithms that run in $\text{poly}(Δ,\log n)$ rounds. The first algorithm, with randomization, uses only $Δ+2$ colors. The second algorithm is a deterministic algorithm that uses $Δ+ O(\log n/ \log \log n)$ colors. Our approach is to reduce the distributed edge-coloring problem into an online, restricted version of balls-into-bins problem. If $\ell$ is the maximum load of the bins, our algorithm uses $Δ+ 2\ell - 1$ colors. We show how to achieve $\ell = 1$ with randomization and $\ell = O(\log n / \log \log n)$ without randomization.

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arXiv2019

Ant-Inspired Density Estimation via Random Walks

Many ant species employ distributed population density estimation in applications ranging from quorum sensing [Pra05], to task allocation [Gor99], to appraisal of enemy colony strength [Ada90]. It has been shown that ants estimate density by tracking encounter rates -- the higher the population density, the more often the ants bump into each other [Pra05,GPT93]. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides new tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using $local\ mixing\ properties$ of the underlying graph. Our results extend beyond the grid to more general graphs and we discuss applications to size estimation for social networks and density estimation for robot swarms.

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arXiv2018

Distributed $(Δ+1)$-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for $(Δ+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log Δ})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$Δ$. This implies that the $(Δ+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $Ω\left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log Δ}{\log \log Δ} \right) \right)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $Δ+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts.

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arXiv2017

Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

This paper gives drastically faster gossip algorithms to compute exact and approximate quantiles. Gossip algorithms, which allow each node to contact a uniformly random other node in each round, have been intensely studied and been adopted in many applications due to their fast convergence and their robustness to failures. Kempe et al. [FOCS'03] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful $O(\log n + \log \frac{1}ε)$ round algorithm to $ε$-approximate the sum of all values and an $O(\log^2 n)$ round algorithm to compute the exact $ϕ$-quantile, i.e., the the $\lceil ϕn \rceil$ smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact $ϕ$-quantile problem which runs in $O(\log n)$ rounds. We furthermore show that one can achieve an exponential speedup if one allows for an $ε$-approximation. We give an $O(\log \log n + \log \frac{1}ε)$ round gossip algorithm which computes a value of rank between $ϕn$ and $(ϕ+ε)n$ at every node.% for any $0 \leq ϕ\leq 1$ and $0 < ε< 1$. Our algorithms are extremely simple and very robust - they can be operated with the same running times even if every transmission fails with a, potentially different, constant probability. We also give a matching $Ω(\log \log n + \log \frac{1}ε)$ lower bound which shows that our algorithm is optimal for all values of $ε$.

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arXiv2017

Scaling Algorithms for Weighted Matching in General Graphs

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in $O(m\sqrt{n}\log(nN))$ time, $O(m\sqrt{n})$ per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs. Here $m,n,$ and $N$ bound the number of edges, vertices, and magnitude of any edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in $O(m\sqrt{n\log nα(m,n)} \log(nN))$ time.

cs.DS↗

arXiv2016

Distributed Degree Splitting, Edge Coloring, and Orientations

We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: -- We present a $poly(\log n)$ round deterministic algorithm for $(2Δ-1)\cdot (1+o(1))$-edge-coloring, where $Δ$ denotes the maximum degree. Modulo the $1+o(1)$ factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of $(2Δ-1)\cdot poly(\log Δ)$-edge-coloring in $poly(\log n)$ rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. -- We show that sinkless orientation---i.e., orienting edges such that each node has at least one outgoing edge---on $Δ$-regular graphs can be solved in $O(\log_Δ \log n)$ rounds randomized and in $O(\log_Δ n)$ rounds deterministically. These prove the corresponding lower bounds by Brandt et al. [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of Chang et al. for $Δ$-coloring $Δ$-regular trees. -- We present a randomized $O(\log^4 n)$ round algorithm for orienting $a$-arboricity graphs with maximum out-degree $a(1+ε)$. This can be also turned into a decomposition into $a (1+ε)$ forests when $a=Ω(\log n)$ and into $a (1+ε)$ pseduo-forests when $a=o(\log n)$. Obtaining an efficient distributed decomposition into less than $2a$ forests was stated as the 10th open problem in the book by Barenboim and Elkin.

cs.DS↗

arXiv2014

Almost-Tight Distributed Minimum Cut Algorithms

We study the problem of computing the minimum cut in a weighted distributed message-passing networks (the CONGEST model). Let $λ$ be the minimum cut, $n$ be the number of nodes in the network, and $D$ be the network diameter. Our algorithm can compute $λ$ exactly in $O((\sqrt{n} \log^{*} n+D)λ^4 \log^2 n)$ time. To the best of our knowledge, this is the first paper that explicitly studies computing the exact minimum cut in the distributed setting. Previously, non-trivial sublinear time algorithms for this problem are known only for unweighted graphs when $λ\leq 3$ due to Pritchard and Thurimella's $O(D)$-time and $O(D+n^{1/2}\log^* n)$-time algorithms for computing $2$-edge-connected and $3$-edge-connected components. By using the edge sampling technique of Karger's, we can convert this algorithm into a $(1+ε)$-approximation $O((\sqrt{n}\log^{*} n+D)ε^{-5}\log^3 n)$-time algorithm for any $ε>0$. This improves over the previous $(2+ε)$-approximation $O((\sqrt{n}\log^{*} n+D)ε^{-5}\log^2 n\log\log n)$-time algorithm and $O(ε^{-1})$-approximation $O(D+n^{\frac{1}{2}+ε} \mathrm{poly}\log n)$-time algorithm of Ghaffari and Kuhn. Due to the lower bound of $Ω(D+n^{1/2}/\log n)$ by Das Sarma et al. which holds for any approximation algorithm, this running time is tight up to a $ \mathrm{poly}\log n$ factor. To get the stated running time, we developed an approximation algorithm which combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It saves an $ε^{-9}\log^{7} n$ factor as compared to applying Thorup's tree packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning algorithm and Karger's dynamic programming to achieve an efficient distributed algorithm that finds the minimum cut when we are given a spanning tree that crosses the minimum cut exactly once.

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