A Queueing Analysis of Hashing With Lazy Deletion

Hashing with lazy deletion is a simple method for maintaining a dynamic dictionary: items are inserted and sought as usual in a separate-chaining hash table; however, items that no longer need to be in the data structure remain until a later insertion operation stumbles on them and removes them from the table. Because hashing with lazy deletion does not delete items as soon as possible, it keeps more items in the dictionary than methods that use more careful deletion strategies. On the other hand, its space overhead is much smaller than those more careful methods, so if the number of extra items is not too large, hashing with lazy deletion can be a practical algorithm when space is scarce. In this paper, we analyze the expected amount of excess space used by hashing with lazy deletion

The Maximum Size of Dynamic Data Structures

This paper develops two probabilistic methods that allow the analysis of the maximum data structure size encountered during a sequence of insertions and deletions in data structures such as priority queues, dictionaries, linear lists, and symbol tables, and in sweepline structures for geometry and Very-Large-Scale-Integration (VLSI) applications. The notion of the "maximum" is basic to issues of resource preallocation. The methods here are applied to combinatorial models of file histories and probabilistic models, as well as to a non-Markovian process (algorithm) for processing sweepline information in an efficient way, called "hashing with lazy deletion" (HwLD). Expressions are derived for the expected maximum data structure size that are asymptotically exact, that is, correct up to lower-order terms; in several cases of interest the expected value of the maximum size is asymptotically equal to the maximum expected size. This solves several open problems, including longstanding questions in queueing theory. Both of these approaches are robust and rely upon novel applications of techniques from the analysis of algorithms. At a high level, the first method isolates the primary contribution to the maximum and bounds the lesser effects. In the second technique the continuous-time probabilistic model is related to its discrete analog--the maximum slot occupancy in hashing