Cache Oblivious Minimum Cut

We show how to compute the minimum cut of a graph cache-efficiently. Let B be the width of a cache line and M be the size of the cache. On a graph with V vertices and E edges, we give a cache oblivious algorithm that incurs \(O(\lceil \frac{E}{B} (\log ^4 E) \log _{M/B} E\rceil )\) cache misses and a simpler one that incurs \(O(\lceil \frac{V^2}{B} \log ^3 V\rceil )\) cache misses.

[1]  Matteo Frigo,et al.  Cache-oblivious algorithms , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[2]  Toshihide Ibaraki,et al.  Computing Edge-Connectivity in Multigraphs and Capacitated Graphs , 1992, SIAM J. Discret. Math..

[3]  Edward F. Grove,et al.  External-memory graph algorithms , 1995, SODA '95.

[4]  Ulrich Meyer,et al.  Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths , 2004, SWAT.

[5]  Ken-ichi Kawarabayashi,et al.  Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time , 2014, STOC.

[6]  Gerth Stølting Brodal,et al.  On the limits of cache-obliviousness , 2003, STOC '03.

[7]  Michael A. Bender,et al.  An Optimal Cache-Oblivious Priority Queue and Its Application to Graph Algorithms , 2007, SIAM J. Comput..

[8]  Francesco Silvestri,et al.  On the Limits of Cache-Oblivious Matrix Transposition , 2006, TGC.

[9]  Guy E. Blelloch,et al.  Low depth cache-oblivious algorithms , 2010, SPAA '10.

[10]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[11]  Mechthild Stoer,et al.  A simple min-cut algorithm , 1997, JACM.

[12]  Sajith Gopalan,et al.  I/O efficient algorithms for the minimum cut problem on unweighted undirected graphs , 2014, Theor. Comput. Sci..

[13]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[14]  Kamesh Munagala,et al.  I/O-complexity of graph algorithms , 1999, SODA '99.

[15]  Lars Arge,et al.  On external-memory MST, SSSP and multi-way planar graph separation , 2000, J. Algorithms.

[16]  Uzi Vishkin,et al.  On Finding Lowest Common Ancestors: Simplification and Parallelization , 1988, AWOC.

[17]  Alok Aggarwal,et al.  The input/output complexity of sorting and related problems , 1988, CACM.

[18]  David R. Karger,et al.  Minimum cuts in near-linear time , 1998, JACM.