Abstract

We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n and |R| = m. We construct a random graph G by allowing each x∈L to choose d random neighbours in R. The question discussed is as to the size μ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is μ(G) = n whp? We answer this question exactly when d is at least three. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Highlights

  • For a graph G we let μ(G) denote the size of the maximum matching in G

  • In essence this paper provides an analysis of μ(G) in the following model of a random bipartite graph

  • Under the assumption that the hash functions are completely random we see that G has the same distribution as the random graph defined in the previous paragraph

Read more

Summary

Introduction

For a graph G we let μ(G) denote the size of the maximum matching in G. In essence this paper provides an analysis of μ(G) in the following model of a random bipartite graph. Under the assumption that the hash functions are completely random we see that G has the same distribution as the random graph defined in the previous paragraph. We assume each location can hold only one item. When an item x is inserted into the table, it can be placed immediately if one of its d locations is currently empty. One of the items in its d locations must be displaced and moved to another of its d choices to make room for x. Having inserted k items, we have constructed a matching M of size k in G. All n items will be insertable in this way iff G contains a matching of size n. We will revert to the abstract question posed in first paragraph of the paper

Definitions and Results
5: Select a vertex ξ uniformly at random from the set of vertices of degree one
Structure of the paper
Probability Model for Phase 1
Differential Equations
Analysis of Phase 1
Threshold for Phase 1 to be sufficient
Finishing the proof of Theorem 2
Proof of Theorem 3
Useful Lemmas
Case 1
Case 2
Medium Witnesses
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.