A Simple Deterministic Algorithm for Edge Connectivity
aa r X i v : . [ c s . D S ] A ug A Simple Deterministic Algorithm for Edge Connectivity
Thatchaphol SaranurakAugust 20, 2020
Abstract
We show a deterministic algorithm for computing edge connectivity of a simple graph with m edges in m o (1) time. Although the fastest deterministic algorithm by Henzinger, Rao, and Wang[SODA’17] has a faster running time of O ( m log m log log m ), we believe that our algorithm isconceptually simpler. The key tool for this simplication is the expander decomposition . We exploitit in a very straightforward way compared to how it has been previously used in the literature. Edge connectivity is a fundamental measure for robustness of graphs. Given an undirected graph G = ( V, E ) with n vertices and m edges, the edge connectivity λ of G is the minimum number of edgeswhose deletion from G disconnects G . These edges correspond to a (global) minimum cut ( C, V \ C )where the number of edges crossing the cut is | E ( C, V \ C ) | = λ . Numerous algorithms for computingedge connectivity have been discovered and are based on various fascinating techniques, includingexact max flow computation [FF62, HO94, LP20], maximum adjacency ordering [NI92, SW97, Fra09],random contraction [Kar93, KS96], arborescence packing [Gab95, GM98], and greedy tree packingand minimum cuts that 2-respect a tree [Kar00, BLS20, GMW20a, MN20, GMW20b]. All of thesetechniques also extend to weighted graphs where we need to find a cut with minimum total edge weightcrossing the cut.Quite recently, Kawarabayashi and Thorup [KT19] showed a novel technique for computing edgeconnectivity of simple unweighted graphs (i.e. graphs with no parallel edges) in O ( m log n ) timedeterministically. This technique leads to the fastest deterministic algorithm with O ( m log n log log n )time by Henzinger, Rao, and Wang [HRW17], and the fastest randomized algorithm with running timemin { O ( m log n ) , O ( m + n log n ) } with high probability by Ghaffari, Nowicki, and Thorup [GNT20].The state-of-the-art algorithms for non-simple graphs have slower running times.The core idea in this line of work is a new contraction technique that preserves all non-trivial minimum cuts. Recall that trivial cuts ( C, V \ C ) are cuts where min {| C | , | V \ C |} = 1. Althoughthe algorithm by [GNT20] already gave a simple implementation of this idea using randomization, alldeterministic algorithms for finding such a contraction are still quite complicated. For example, theyrequire intricate analysis of personalized PageRank [KT19] and local flow technique [HRW17] and anon-trivial way for combining all algorithmic tools together.In this paper, we observe that such a contraction follows almost immediately from the expanderdecomposition introduced in [KVV04]. Although the best-known implementation of expander decom-position itself is not yet very simple [SW19, CGL + conceptual simplification of this contractiontechnique. Our result is as follows: Theorem 1.1.
There is a deterministic algorithm that, given a simple graph with m edges, computesits edge connectivity in m o (1) time. It is easy to extend the algorithm to compute the corresponding minimum cut but we omit it here. +
19] are at the core of almost-linear time algorithms for many fundamental problems including(directed) Laplacian solvers [ST14, CKP + + +
18, CPS20, CDL +
20] and sketchings [ACK +
16, JS18]. Morerecently, it has been used to break many long-standing barriers in the areas of dynamic algorithms[NS17, Wul17, NSW17, CK19, CGL +
19, BvdBG +
20, BGS20, GRST20, JS20, CS20b] and distributedalgorithms [ER18, CPZ19, DHNS19, CS19, CS20a, CGL20].Unfortunately, how the expander decomposition has been applied is usually highly non-trivial; it iseither a step in a much bigger algorithm containing other complicated components, or the guaranteeof the decomposition is exploited via involved analysis.Both our algorithm and analysis are straightforward. The only key step of the algorithm simplyapplies the expander decomposition followed by the simple trimming and shaving procedures definedin [KT19]. We note that the idea of using expander decomposition for edge connectivity actuallyappeared previously in the distributed algorithm by [DHNS19]. However, that work requires manyother distributed algorithmic components and inevitably played down the simplicity of this approach.In fact, since of the original work by [KT19], their discussion in Sections 1.4 and 1.5 strongly suggestedthat expander decomposition should be useful. We hope that this paper can highlight this simple ideaand serve as a gentle introduction on how to apply expander decomposition in general.The m o (1) factor in Theorem 1.1 solely depends on quality and efficiency of expander decompositionalgorithms. It is believable that this factor can be improved to polylog( n ), which would immediatelyimprove the running time of our algorithm to O ( m polylog( n )). For any graph G = ( V, E ) and a vertex set S , the volume of S is denoted by vol G ( S ) = P v ∈ S deg( v ).For any A, B ⊆ V , let E ( A, B ) denote the set of edges with one endpoint in A and another in B . Let δ denote the minimum vertex degree of G . Now, we state the key tool, the expander decomposition. Lemma 2.1 (Corollary 7.7 of [CGL + . There is an algorithm denoted by expander ( G, φ ) that,given an m -edge graph G = ( V, E ) and a parameter φ ≥ , in O ( mγ ) time where γ = m o (1) , returns apartition X = { X , . . . , X k } of V such that • P i | E ( X i , V \ X i ) | = O ( φmγ ) , and • For each i and each ∅ 6 = S ⊂ X i , | E ( S, X i \ S ) | ≥ φ min { vol G ( S ) , vol G ( X i \ S ) } . Note that if φ ≥ /γ , then the trivial partition X = { v | v ∈ V } satisfies the above guarantees.The next tool is a deterministic algorithm by Gabow for computing edge connectivity. Gabow’salgorithm, in fact, can return the corresponding minimum cut and also works for directed graphs, butwe don’t need these guarantees in this paper. Lemma 2.2 ([Gab95]) . There is an algorithm that, given an m -edge graph G = ( V, E ) and a parameter k , in time O ( m · min { λ, k } ) returns min { λ, k } where λ is the edge connectivity of G . Lastly, we describe the trim and shave procedures from [KT19].
Definition 2.3.
For any vertex set S of a graph G = ( V, E ) , let trim ( S ) ⊆ S be obtained from S as follows: while there exists a vertex v ∈ S where | E ( v, S ) | < v ) / , removes v from S . Let shave ( S ) = { v ∈ S | | E ( v, S ) | > deg( v ) / } . In [CGL + lgorithm 1 Computing edge connectivity λ of a simple graph G
1. Compute X = expander ( G, /δ ), X ′ = { trim ( X ) | X ∈ X } , X ′′ = { shave ( X ′ ) | X ′ ∈ X ′ } .2. Let G ′ be the graph obtained from G by contracting every set X ′′ ∈ X ′′ .3. Using Gabow’s algorithm (Lemma 2.2), return min { λ ′ , δ } where λ ′ is the edge connectivity of G ′ and δ is the minimum vertex degree of G .Note that, for every v ∈ trim ( S ), | E ( v, trim ( S )) | ≥ v ) /
5. Intuitively, the main differencebetween the two procedures is that trim keeps removing a vertex with low “inside-degree” as long asit exists, while shave removes all low “inside-degree” vertices once.
Our algorithm is summarized in Algorithm 1. Step 1 is the step that simplifies the previous algorithmsby [KT19, HRW17]. This step gives us a contracted graph G ′ of G that preserves all non-trivialminimum cuts, as will be proved in Lemma 3.1 below. Previous algorithms for computing suchcontraction are much more involved. For example, they require an intricate analysis of PageRank[KT19] or local flow [HRW17]. Moreover, both algorithms [KT19, HRW17] sequentially contract apart of G into a supervertex and need to distinguish supervertices and regular vertices thereafter. Forus, G ′ is simply obtained by contracting each set X ′′ ∈ X ′′ simultaneously.Besides Step 1 of Algorithm 1 and the key lemma below (Lemma 3.1), other algorithmic stepsand analysis follow the same template in [KT19]. We only show an alternative presentation forcompleteness. Lemma 3.1.
Let G = ( V, E ) be a simple graph. Let ( C, V \ C ) be a non-trivial minimum cut in G .Let X ∈ expander ( G, /δ ) , X ′ = trim ( X ) , and X ′′ = shave ( X ′ ) . We have that1. min {| X ∩ C | , | X \ C |} ≤ λ/ ,2. min {| X ′ ∩ C | , | X ′ \ C |} ≤ , and3. min {| X ′′ ∩ C | , | X ′′ \ C |} = 0 .In particular, G ′ preserves all non-trivial minimum cuts of G .Proof. (1): We have min {| X ∩ C | , | X \ C |} ≤ λ/
40 because of the following: λ ≥ | E ( X ∩ C, X \ C ) | as C is a minimum cut ≥ (40 /δ ) · min { vol G ( X ∩ C ) , vol G ( X \ C ) } by Lemma 2.1 ≥ · min {| X ∩ C | , | X \ C |} . (2): Assume w.l.o.g. that | X ′ ∩ C | ≤ | X ′ \ C | . So, | X ′ ∩ C | ≤ min {| X ∩ C | , | X \ C |} ≤ λ/
40 by (1).Observe that δ ≥ λ ≥ | E ( X ′ ∩ C, X ′ \ C ) | as C is a minimum cut= vol G [ X ′ ] ( X ′ ∩ C ) − | E ( X ′ ∩ C, X ′ ∩ C ) |≥ δ | X ′ ∩ C | − | X ′ ∩ C | as X ′ = trim ( X ) and G is simple . | X ′ ∩ C | ≤
2. Otherwise, | X ′ ∩ C | ≥ δ ≥ (6 / δ − | X ′ ∩ C | ,which implies that | X ′ ∩ C | ≥ δ/
30. But we have | X ′ ∩ C | ≤ λ/ < δ/
30, which is a contradiction.(3): Again, assume w.l.o.g. that | X ′ ∩ C | ≤ | X ′ \ C | . Suppose for contradiction that min {| X ′′ ∩ C | , | X ′′ \ C |} >
0. So there is a vertex v ∈ X ′′ ∩ C ⊆ X ′ ∩ C . As G is simple and | X ′ ∩ C | ≤ | E ( v, X ′ ∩ C ) | ≤
1. Also, we have | E ( v, X ′ ) | > deg( v ) / X ′′ = shave ( X ′ ).Therefore, | E ( v, X ′ \ C ) | = | E ( v, X ′ ) | − | E ( v, X ′ ∩ C ) | > deg( v ) / − v ) / . As (
C, V \ C )is non-trivial, we can switch v from C to V \ C and obtain a smaller cut, contradicting the fact that C is a minimum cut. Corollary 3.2.
Algorithm 1 correctly computes the edge connectivity λ of G .Proof. Note that λ ′ ≥ λ because G ′ is obtained from G by contraction. If λ = δ (i.e. there is a trivialminimum cut), then min { λ ′ , δ } = λ . If λ < δ (i.e. all minimum cuts are non-trivial), then we have λ ′ = λ by Lemma 3.1 and so min { λ ′ , δ } = λ . Lemma 3.3.
The contracted graph G ′ has at most O ( mγ/δ ) edges.Proof. Assume that δ ≥ G/ X denote the graph obtainedfrom G by contracting each X ∈ X into a single vertex. Let G/ X ′ and G/ X ′′ be similarly defined.Note that G ′ = G/ X ′′ . We would like to bound | E ( G/ X ′′ ) | = | E ( G/ X ) | + | E ( G/ X ′ ) \ E ( G/ X ) | + | E ( G/ X ′′ ) \ E ( G/ X ′ ) | . We will show that each term is bounded by O ( mγ/δ ) where γ is the factorfrom Lemma 2.1.First, the set E ( G/ X ) contains exactly the edges crossing the partition X of V . So | E ( G/ X ) | = P X ∈X | E ( X, V \ X ) | = O ( mγ/δ ) by Lemma 2.1.Second, the set E ( G/ X ′ ) \ E ( G/ X ) contains all edges that are “trimmed from” each X ∈ X .Consider the trim procedure executing on X until X becomes X ′ . Whenever a vertex v is removedfrom X , | E ( X, V \ X ) | is decreased by at least deg( v ) /
5, because at that point of time | E ( v, X ) | ≤ v ) / | E ( v, V \ X ) | ≥ v ) /
5. On the other hand, the number of trimmed edges, E ( G/ X ′ ) \ E ( G/ X ), is increased by at most | E ( v, X ) | ≤ v ) /
5. Initially, we have P X ∈X | E ( X, V \ X ) | =2 | E ( G/ X ) | . As we argued, every two units in | E ( G/ X ′ ) \ E ( G/ X ) | can be charged to one unit in P X ∈X | E ( X, V \ X ) | . So | E ( G/ X ′ ) \ E ( G/ X ) | ≤ | E ( G/ X ) | = O ( mγ/δ ).Last, the set E ( G/ X ′′ ) \ E ( G/ X ′ ) contains all edges that are “shaved from” each X ′ ∈ X ′ . Thenumber of shaved edges from X ′ is bounded by P v ∈ X ′ \ shave ( X ′ ) | E ( v, X ′ ) | . By definition of shave , foreach vertex v ∈ X ′ \ shave ( X ′ ), we have | E ( v, X ′ ) | < deg( v ) / | E ( v, V \ X ′ ) | > deg( v ) / − δ ≥
4, we have | E ( v, X ′ ) | < | E ( v, V \ X ′ ) | and so P v ∈ X ′ \ shave ( X ′ ) | E ( v, X ′ ) | ≤ | E ( X ′ , V \ X ′ ) | . Summing over all X ′ ∈ X ′ , we have | E ( G/ X ′′ ) \ E ( G/ X ′ ) | ≤ P X ′ ∈X ′ | E ( X ′ , V \ X ′ ) | ≤ P X ∈X | E ( X, V \ X ) | = O ( mγ/δ ). The last inequality is because the trim procedure only decreases | E ( X, V \ X ) | and so | E ( X ′ , V \ X ′ ) | ≤ | E ( X, V \ X ) | for each X ′ = trim ( X ). Corollary 3.4.
Algorithm 1 takes O ( mγ ) = m o (1) time.Proof. In Step 1, X can be computed in O ( mγ ) time by Lemma 2.1. X ′ and X ′′ can be computed in O ( m ) by using straightforward implementations for trim and shave . Contracting G into G ′ can bedone in O ( m ) time in Step 2. Finally, in Step 3, the minimum degree δ can be computed in O ( m )time, and Gabow’s algorithm takes O ( | E ( G ′ ) | δ ) = O ( mγ ) time by Lemma 3.3.To conclude, Theorem 1.1 follows immediately from Corollaries 3.2 and 3.4.4 cknowledgements I thank Aaron Bernstein for encouragement for writing this note up. Also, thanks to Sayan Bhat-tacharya, Maximilian Probst Gutenberg, Jason Li, Danupon Nanongkai, and Di Wang for helpfulcomments on the write-up.
References [ACK +
16] Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P. Woodruff, andQin Zhang. On sketching quadratic forms. In
Proceedings of the 2016 ACM Conferenceon Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16,2016 , pages 311–319, 2016. 2[BGS20] Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deter-ministic decremental reachability, scc, and shortest paths via directed expanders andcongestion balancing. 2020. To appear at FOCS’20. 2[BLS20] Nalin Bhardwaj, Antonio Molina Lovett, and Bryce Sandlund. A simple algorithm forminimum cuts in near-linear time. In , pages12:1–12:18, 2020. 1[BvdBG +
20] Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, DanuponNanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-dynamic graphsparsifiers against an adaptive adversary.
CoRR , abs/2004.08432, 2020. 2[CDL +
20] Parinya Chalermsook, Syamantak Das, Bundit Laekhanukit, Yunbum Kook, Yang P.Liu, Richard Peng, Mark Sellke, and Daniel Vaz. Vertex sparsification for edge connec-tivity.
CoRR , abs/2007.07862, 2020. 2[CGL +
19] Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and ThatchapholSaranurak. A deterministic algorithm for balanced cut with applications to dynamicconnectivity, flows, and beyond.
CoRR , abs/1910.08025, 2019. To appear at FOCS’20.1, 2, 9[CGL20] Keren Censor-Hillel, Fran¸cois Le Gall, and Dean Leitersdorf. On distributed listing ofcliques. In
PODC ’20: ACM Symposium on Principles of Distributed Computing, VirtualEvent, Italy, August 3-7, 2020 , pages 474–482, 2020. 2[CGP +
18] Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, and Junx-ing Wang. Graph sparsification, spectral sketches, and faster resistance computation,via short cycle decompositions. In , pages 361–372, 2018.2[CK19] Julia Chuzhoy and Sanjeev Khanna. A new algorithm for decremental single-sourceshortest paths with applications to vertex-capacitated flow and cut problems. In
Pro-ceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC2019, Phoenix, AZ, USA, June 23-26, 2019 , pages 389–400, 2019. 2[CKP +
17] Michael B. Cohen, Jonathan A. Kelner, John Peebles, Richard Peng, Anup B. Rao,Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for markov chains and5ew spectral primitives for directed graphs. In
Proceedings of the 49th Annual ACMSIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada,June 19-23, 2017 , pages 410–419, 2017. 2[CKS05] Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. Multicommodity flow,well-linked terminals, and routing problems. In
Proceedings of the 37th Annual ACMSymposium on Theory of Computing, Baltimore, MD, USA, May 22-24, 2005 , pages183–192, 2005. 2[CKS13] Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. The all-or-nothing multi-commodity flow problem.
SIAM J. Comput. , 42(4):1467–1493, 2013. 2[CPS20] Yu Cheng, Debmalya Panigrahi, and Kevin Sun. Sparsification of balanced directedgraphs. arXiv preprint arXiv:2006.01975 , 2020. 2[CPZ19] Yi-Jun Chang, Seth Pettie, and Hengjie Zhang. Distributed triangle detection via ex-pander decomposition. In
Proceedings of the Thirtieth Annual ACM-SIAM Symposiumon Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019 ,pages 821–840, 2019. 2[CS19] Yi-Jun Chang and Thatchaphol Saranurak. Improved distributed expander decomposi-tion and nearly optimal triangle enumeration. In
Proceedings of the 2019 ACM Sympo-sium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July29 - August 2, 2019 , pages 66–73, 2019. 2[CS20a] Yi-Jun Chang and Thatchaphol Saranurak. Deterministic distributed expander de-composition and routing with applications in distributed derandomization.
CoRR ,abs/2007.14898, 2020. To appear at FOCS’20. 2[CS20b] Julia Chuzhoy and Thatchaphol Saranurak. Deterministic algorithms for decrementalshortest paths via layered core decomposition. 2020. 2, 9[DHNS19] Mohit Daga, Monika Henzinger, Danupon Nanongkai, and Thatchaphol Saranurak. Dis-tributed edge connectivity in sublinear time. In
Proceedings of the 51st Annual ACMSIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June23-26, 2019 , pages 343–354, 2019. 2[ER18] Talya Eden and Will Rosenbaum. On sampling edges almost uniformly. In
Proc. of theSymposium on Simplicity in Algorithms (SOSA) , pages 7:1–7:9, 2018. 2[FF62] LR Ford and DR Fulkerson. Flows in networks. 1962. 1[Fra09] Andr´as Frank. On the edge-connectivity algorithm of nagamochi and ibaraki. 2009. 1[Gab95] Harold N. Gabow. A matroid approach to finding edge connectivity and packing arbores-cences.
Journal of Computer and System Sciences , 50(2):259–273, 1995. Announced atSTOC’91. 1, 2[GM98] Harold N. Gabow and K. S. Manu. Packing algorithms for arborescences (and spanningtrees) in capacitated graphs.
Math. Program. , 82:83–109, 1998. 1[GMW20a] Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Minimum cut in o(m log n)time. In , pages 57:1–57:15, 2020. 1 6GMW20b] Pawel Gawrychowski, Shay Mozes, and Oren Weimann. A note on a recent algorithmfor minimum cut. CoRR , abs/2008.02060, 2020. 1[GNT20] Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. Faster algorithms for edgeconnectivity via random 2-out contractions. In
Proceedings of the 2020 ACM-SIAMSymposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January5-8, 2020 , pages 1260–1279, 2020. 1[GR02] Oded Goldreich and Dana Ron. Property testing in bounded degree graphs.
Algorith-mica , 32(2):302–343, 2002. Announced at STOC’97. 2[GRST20] Gramoz Goranci, Harald R¨acke, Thatchaphol Saranurak, and Zihan Tan. The expanderhierarchy and its applications to dynamic graph algorithms.
CoRR , abs/2005.02369,2020. 2[HO94] Jianxiu Hao and James B. Orlin. A faster algorithm for finding the minimum cut in adirected graph.
J. Algorithms , 17(3):424–446, 1994. 1[HRW17] Monika Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edgeconnectivity. In
Proc. of the Symposium on Discrete Algorithms (SODA) , pages 1919–1938, 2017. 1, 3[JS18] Arun Jambulapati and Aaron Sidford. Efficient ˜ o ( n/ǫ ) spectral sketches for the laplacianand its pseudoinverse. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Sympo-sium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018 ,pages 2487–2503, 2018. 2[JS20] Wenyu Jin and Xiaorui Sun. Fully dynamic c-edge connectivity in subpolynomial time.
CoRR , abs/2004.07650, 2020. 2[Kar93] David R. Karger. Global min-cuts in rnc, and other ramifications of a simple min-cutalgorithm. In
Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium onDiscrete Algorithms, 25-27 January 1993, Austin, Texas, USA , pages 21–30, 1993. 1[Kar00] David R. Karger. Minimum cuts in near-linear time.
Journal of the ACM , 47(1):46–76,2000. Announced at STOC’96. 1[KLOS14] Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicom-modity generalizations. In
Proceedings of the Twenty-Fifth Annual ACM-SIAM Sympo-sium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014 ,pages 217–226, 2014. 2[KS96] David R. Karger and Clifford Stein. A new approach to the minimum cut problem.
J.ACM , 43(4):601–640, 1996. 1[KT19] Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time.
Journal of the ACM , 66(1):4:1–4:50, 2019. Announced at STOC’15. 1, 2,3[KVV04] Ravi Kannan, Santosh Vempala, and Adrian Vetta. On clusterings: Good, bad andspectral.
J. ACM , 51(3):497–515, 2004. 1[LP20] Jason Li and Debmalya Panigrahi. Deterministic min-cut in poly-logarithmic max-flows.2020. To appear at FOCS’20. 1 7MN20] Sagnik Mukhopadhyay and Danupon Nanongkai. Weighted min-cut: sequential, cut-query, and streaming algorithms. In
Proccedings of the 52nd Annual ACM SIGACTSymposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020 ,pages 496–509, 2020. 1[NI92] Hiroshi Nagamochi and Toshihide Ibaraki. Computing edge-connectivity in multigraphsand capacitated graphs.
SIAM J. Discret. Math. , 5(1):54–66, 1992. 1[NS17] Danupon Nanongkai and Thatchaphol Saranurak. Dynamic spanning forest with worst-case update time: adaptive, las vegas, and o ( n / − ǫ )-time. In STOC , pages 1122–1129.ACM, 2017. 2[NSW17] Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamicminimum spanning forest with subpolynomial worst-case update time. In
FOCS , pages950–961. IEEE Computer Society, 2017. 2[OSV12] Lorenzo Orecchia, Sushant Sachdeva, and Nisheeth K. Vishnoi. Approximating the expo-nential, the lanczos method and an ˜o( m )-time spectral algorithm for balanced separator.In STOC , pages 1141–1160. ACM, 2012. 2[OV11] Lorenzo Orecchia and Nisheeth K. Vishnoi. Towards an sdp-based approach to spectralmethods: A nearly-linear-time algorithm for graph partitioning and decomposition. In
SODA , pages 532–545. SIAM, 2011. 2[ST11] Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs.
SIAM J.Comput. , 40(4):981–1025, 2011. 2[ST13] Daniel A. Spielman and Shang-Hua Teng. A local clustering algorithm for massivegraphs and its application to nearly linear time graph partitioning.
SIAM J. Comput. ,42(1):1–26, 2013. 2[ST14] Daniel A. Spielman and Shang-Hua Teng. Nearly linear time algorithms for precondi-tioning and solving symmetric, diagonally dominant linear systems.
SIAM J. MatrixAnal. Appl. , 35(3):835–885, 2014. 2[SW97] Mechthild Stoer and Frank Wagner. A simple min-cut algorithm.
J. ACM , 44(4):585–591, 1997. 1[SW19] Thatchaphol Saranurak and Di Wang. Expander decomposition and pruning: Faster,stronger, and simpler. In
Proc. of the Symposium on Discrete Algorithms (SODA) , pages2616–2635, 2019. 1, 2[Tre05] Luca Trevisan. Approximation algorithms for unique games. In , pages 197–205, 2005. 2[vdBLN +
20] Jan van den Brand, Yin-Tat Lee, Danupon Nanongkai, Richard Peng, ThatchapholSaranurak, Aaron Sidford, Zhao Song, and Di Wang. Bipartite matching in nearly-lineartime on moderately dense graphs. To appear at FOCS 2020, 2020. 2[Wul17] Christian Wulff-Nilsen. Fully-dynamic minimum spanning forest with improved worst-case update time. In
STOC , pages 1130–1143. ACM, 2017. 28
Variants of Expander Decomposition
The guarantee for expander decomposition is usually stated in a weaker form: for every ∅ 6 = S ⊂ X i and i , we have | E ( S, X i \ S ) | ≥ φ min { vol G [ X i ] ( S ) , vol G [ X i ] ( X i \ S ) } instead of | E ( S, X i \ S ) | ≥ φ min { vol G ( S ) , vol G ( X i \ S ) } as in Lemma 2.1. In [CGL + G = ( V, E ) be any m -edge graph and let G ′ be obtained from G byadding deg G ( v ) self-loops to each vertex v . So G ′ has m ′ = O ( m ) edges. Suppose we have obtaineda weaker form of expander decomposition X = { X , . . . , X k } of G ′ . That is, P i | E G ′ ( X i , V \ X i ) | = O ( φm ′ γ ) and | E G ′ ( S, X i \ S ) | ≥ φ min { vol G ′ [ X i ] ( S ) , vol G ′ [ X i ] ( X i \ S ) } for every ∅ 6 = S ⊂ X i and i .Observe that E G ( A, B ) = E G ′ ( A, B ) for any two disjoint sets
A, B ⊆ V . So P i | E G ( X i , V \ X i ) | = O ( φm ′ γ ) = O ( φmγ ). Also, we have | E G ( S, X i \ S ) | = | E G ′ ( S, X i \ S ) | ≥ φ min { vol G ′ [ X i ] ( S ) , vol G ′ [ X i ] ( X i \ S ) } ≥ φ min { vol G ( S ) , vol G ( X i \ S ) } where the last inequality is because of the self-loops in G ′ . That is, X is indeed a stronger form ofexpander decomposition of G (modulo losing a constant factor in the bound of P i | E G ′ ( X i , V \ X i ) ||