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

• Martin Dietzfelbinger
• Michael Rink
Conference paper
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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## References

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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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