Linear probing and graphs

TL;DR

This paper explores the mathematical connections between trees, graphs, and the parking problem to analyze the behavior and cost of hashing with linear probing, providing new insights into its probabilistic properties.

Contribution

It combines classical results on trees, graphs, and parking problems to offer a novel analysis of linear probing hashing, including higher moments of search cost.

Findings

Derived a simple analysis of hashing by linear probing

Connected inversions in trees to the parking problem

Provided insights into higher moments of search cost

Abstract

Mallows and Riordan showed in 1968 that labeled trees with a small number of inversions are related to labeled graphs that are connected and sparse. Wright enumerated sparse connected graphs in 1977, and Kreweras related the inversions of trees to the so-called ``parking problem'' in 1980. A~combination of these three results leads to a surprisingly simple analysis of the behavior of hashing by linear probing, including higher moments of the cost of successful search.