Topic
Karger's algorithm
About: Karger's algorithm is a research topic. Over the lifetime, 19 publications have been published within this topic receiving 539 citations.
Papers
6 Jun 2011
TL;DR: A key ingredient of the proofs is a natural generalization of Karger's bound on the number of small cuts in an undirected graph, which is likely to be of independent interest.
Abstract: We present a general framework for constructing cut sparsifiers in undirected graphs --- weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 e). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczur and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczur and Karger.While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)-time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + ~O(n/e2) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczur and Karger) of O(m log2 n) (for the unweighted case) and O(m log3 n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/e2) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n3 m / e2) time.A key ingredient of our proofs is a natural generalization of Karger's bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger's bound, we suspect that our generalization will also be of independent interest.
101 citations
1 Jun 1993
TL;DR: This paper presents the first algorithm that breaks the tl(mn) “max-flow barrier” for finding minimum cuts in weighted undirected graphs by giving a strongly polynomial randomized algorithm which finds a minimum cut with high probability in 0(n2 log3 n) time.
Abstract: A minimum cut is a set of edges of minimum weight whose removal disconnects a given graph. Minimum cut algorithms historically applied duality with maximum flows and thus had the same 0 (inn) running time as maximum flow algorithms. More recent algorithms which are not based on maximum flows also require fl (inn) time. In this paper, we present the first algorithm that breaks the tl(mn) “max-flow barrier” for finding minimum cuts in weighted undirected graphs. We give a strongly polynomial randomized algorithm which finds a minimum cut with high probability in 0(n2 log3 n) time. This suggests that the rein-cut problem might be fundamentally easier to solve than the maximum flow problem. Our algorithm can be implemented in 72JUC using only nz processors—this is the first efficient 7UfC algorithm for the rein-cut problem. Our algorithm is simple and uses no complicated data structures.
90 citations
23 May 1994
TL;DR: The algorithm is a modification of one proposed by Karger and uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree in a connected graph with edge weights.
Abstract: We present a randomized linear-time algorithm for finding a minimum spanning tree in a connected graph with edge weights. The algorithm is a modification of one proposed by Karger and uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.
79 citations
14 Jun 2015
TL;DR: In this paper, the authors presented a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges.
Abstract: We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n).At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.
59 citations
TL;DR: This paper presents a O(mn^3) time exact algorithm for finding a minimum 3-cut in an edge-weighted graph that compares very favorably with the best-known algorithm which takes O (mn^5) time in the worst case.
39 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2018 | 1 |
| 2016 | 1 |
| 2015 | 1 |
| 2014 | 1 |
| 2013 | 2 |