CSE 521 Winter 2006 Notes on Hashing

The dictionary data structure is ubiquitous in computer science. A dictionary holds a set of items from an ordered universe and supports the operations of inserting an item, deleting an item, and searching for an item (membership queries). This is the dynamic dictionary problem. If the set of items is fixed and never changes, we only need to efficiently support membership queries (of the form “x ∈ S?”), and this is called the static dictionary problem. The fundamental “time-space” trade-off in any data structures problem is the space needed to store the data structure, and the time required to perform the operations (such as updates and answering queries). Let us consider this trade-off for the dictionary data structure. Let us assume the universe U is the set [N ] = {0, 1, 2, . . . , N − 1}, and we need to store a subset S ⊆ U of n N items. A trivial solution is to store a bit array of size N that holds the characteristic vector of the set. This supports O(1) time operations but has very poor storage. On the other hand, at the other extreme, we can store the set as a linked list, which has optimal O(n) storage, but requires Θ(n) time for operations. A binary search tree improves upon both of these with O(n) storage and O(log n) time operations. For the static problem, a sorted array supports search in O(log n) time. The logarithmic time is in fact optimal in the comparison model (if only allowed operations are comparisons between integers). The objective of hashing is to use other operations on integers and simultaneously achieve optimal O(n) storage as well as O(1) time for the operations.