Thresholds for Extreme Orientability

Multiple-choice load balancing has been a topic of intense study since the seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be phrased in terms of orientations of a graph, or more generally a k-uniform random hypergraph. A (d,b)-orientation is an assignment of each edge to d of its vertices, such that no vertex has more than b edges assigned to it. Conditions for the existence of such orientations have been completely documented except for the “extreme” case of (k−1,1)-orientations. We consider this remaining case, and establish: The density threshold below which an orientation exists with high probability, and above which it does not exist with high probability.An algorithm for finding an orientation that runs in linear time with high probability, with explicit polynomial bounds on the failure probability. Previously, no closed-form expression for the threshold was known. The only known algorithms for constructing (k−1,1)-orientations worked for k≤3, and were only shown to have expected linear running time.

[1]  Marc Lelarge,et al.  A new approach to the orientation of random hypergraphs , 2012, SODA.

[2]  Friedhelm Meyer auf der Heide,et al.  Simple, efficient shared memory simulations , 1993, SPAA '93.

[3]  Eli Upfal,et al.  How to share memory in a distributed system , 1984, JACM.

[4]  Konstantinos Panagiotou,et al.  The multiple-orientability thresholds for random hypergraphs , 2011, SODA '11.

[5]  Rasmus Pagh,et al.  Cuckoo Hashing , 2001, Encyclopedia of Algorithms.

[6]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[7]  Béla Bollobás,et al.  Random Graphs , 1985 .

[8]  Peter Sanders,et al.  The random graph threshold for k-orientiability and a fast algorithm for optimal multiple-choice allocation , 2007, SODA '07.

[9]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[10]  Konstantinos Panagiotou,et al.  Orientability of Random Hypergraphs and the Power of Multiple Choices , 2010, ICALP.

[11]  Eli Upfal,et al.  Balanced Allocations , 1999, SIAM J. Comput..

[12]  Pu Gao,et al.  Load balancing and orientability thresholds for random hypergraphs , 2010, STOC '10.

[13]  S. Ross.  A random graph , 1981 .

[14]  Alan M. Frieze,et al.  Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables , 2009, Random Struct. Algorithms.

[15]  B. Bollobás.  The evolution of random graphs , 1984 .

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  Rasmus Pagh,et al.  A New Data Layout for Set Intersection on GPUs , 2011, 2011 IEEE International Parallel & Distributed Processing Symposium.

[18]  Tomasz Łuczak,et al.  The phase transition in a random hypergraph , 2002 .

[19]  Alan M. Frieze,et al.  Ramsey games with giants , 2009, Random Struct. Algorithms.

[20]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[21]  Vijaya Ramachandran,et al.  The k-orientability thresholds for Gn, p , 2007, SODA '07.

[22]  Uzi Vishkin,et al.  Simulation of Parallel Random Access Machines by Circuits , 1984, SIAM J. Comput..

[23]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[24]  Berthold Vöcking,et al.  Balanced allocations: the heavily loaded case , 2000, STOC '00.

[25]  Andrea Montanari,et al.  Tight Thresholds for Cuckoo Hashing via XORSAT , 2009, ICALP.

[26]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.