A New Approach to Analyzing Robin Hood Hashing

Robin Hood hashing is a variation on open addressing hashing designed to reduce the maximum search time as well as the variance in the search time for elements in the hash table. While the case of insertions only using Robin Hood hashing is well understood, the behavior with deletions has remained open. Here we show that Robin Hood hashing can be analyzed under the framework of finite-level finite-dimensional jump Markov chains. This framework allows us to re-derive some past results for the insertion-only case with some new insight, as well as provide a new analysis for a standard deletion model, where we alternate between deleting a random old key and inserting a new one. In particular, we show that a simple but apparently unstudied approach for handling deletions with Robin Hood hashing offers good performance even under high loads.Comment: 19 pages, draft version. Updated from the previous version with some new proofs, in particular a full proof of the log log n + O(1) maximum age bound in the static setting, by converting the fluid limit argument into a layered inductio

Hashing with Linear Probing and Referential Integrity

We describe a variant of linear probing hash tables that never moves elements and thus supports referential integrity, i.e., pointers to elements remain valid while this element is in the hash table. This is achieved by the folklore method of marking some table entries as formerly occupied (tombstones). The innovation is that the number of tombstones is minimized. Experiments indicate that this allows an unbounded number of operations with bounded overhead compared to linear probing without tombstones (and without referential integrity).Comment: 5 pages, 3 figure

Robin Hood Hashing really has constant average search cost and variance in full tables

Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of $1.883$ when the table was full. Furthermore, by using a non-standard mean-centered search algorithm, this would imply that searches could be performed in expected constant time even in a full table. In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centered search algorithm runs in expected constant time. We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by $1/(1-\alpha)+O(1)$, where $\alpha$ is the load factor. To model the behavior of these hash tables, we use a unified approach that can be applied also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions