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Towards Optimal Degree-Distributions for Left-Perfect Matchings in Random Bipartite Graphs

  • Martin Dietzfelbinger
  • Michael Rink
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

Consider a random bipartite multigraph G with n left nodes and m ≥ n ≥ 2 right nodes. Each left node x has d x  ≥ 1 random right neighbors. The average left degree \(\bar{{\mathrm{\scriptstyle\Delta}}}\) is fixed, \(\bar{{\mathrm{\scriptstyle\Delta}}}\geq2\). We ask whether for the probability that G has a left-perfect matching it is advantageous not to fix d x for each left node x but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If \(\bar{{\mathrm{\scriptstyle\Delta}}}\) is an integer then it is optimal to use a fixed degree of \(\bar{{\mathrm{\scriptstyle\Delta}}}\) for all left nodes. If \(\bar{{\mathrm{\scriptstyle\Delta}}}\) is non-integral then an optimal degree-distribution has the property that each left node x has two possible degrees, \(\ensuremath{\lfloor \bar{{\mathrm{\scriptstyle\Delta}}}\rfloor}\) and \(\ensuremath{\lceil \bar{{\mathrm{\scriptstyle\Delta}}}\rceil}\), with probability p x and 1 − p x , respectively, where p x is from the closed interval [0,1] and the average over all p x equals \(\ensuremath{\lceil \bar{{\mathrm{\scriptstyle\Delta}}}\rceil}-\bar{{\mathrm{\scriptstyle\Delta}}}\). Furthermore, if n = c·m and \(\bar{{\mathrm{\scriptstyle\Delta}}}>2\) is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold \(c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})\) that has the following property as n goes to infinity: If \(c<c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})\) then there exists a left-perfect matching with high probability. If \(c>c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})\) then there exists no left-perfect matching with high probability. The threshold \(c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})\) is the same as the known threshold for offline k-ary cuckoo hashing for integral or non-integral \(k=\bar{{\mathrm{\scriptstyle\Delta}}}\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Martin Dietzfelbinger
    • 1
  • Michael Rink
    • 1
  1. 1.Fakultät für Informatik und AutomatisierungTechnische Universität IlmenauGermany

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