Hash Table - Performance Analysis

Performance Analysis

In the simplest model, the hash function is completely unspecified and the table does not resize. For the best possible choice of hash function, a table of size n with open addressing has no collisions and holds up to n elements, with a single comparison for successful lookup, and a table of size n with chaining and k keys has the minimum max(0, k-n) collisions and O(1 + k/n) comparisons for lookup. For the worst choice of hash function, every insertion causes a collision, and hash tables degenerate to linear search, with Ω(k) amortized comparisons per insertion and up to k comparisons for a successful lookup.

Adding rehashing to this model is straightforward. As in a dynamic array, geometric resizing by a factor of b implies that only k/bi keys are inserted i or more times, so that the total number of insertions is bounded above by bk/(b-1), which is O(k). By using rehashing to maintain k < n, tables using both chaining and open addressing can have unlimited elements and perform successful lookup in a single comparison for the best choice of hash function.

In more realistic models, the hash function is a random variable over a probability distribution of hash functions, and performance is computed on average over the choice of hash function. When this distribution is uniform, the assumption is called "simple uniform hashing" and it can be shown that hashing with chaining requires Θ(1 + k/n) comparisons on average for an unsuccessful lookup, and hashing with open addressing requires Θ(1/(1 - k/n)). Both these bounds are constant, if we maintain k/n < c using table resizing, where c is a fixed constant less than 1.