Dynamic data structures with finite population: A combinatorial analysis

Abstract

This paper analyzes the average behaviour of algorithms that operate on dynamically varying data structures subject to insertions I, deletions D, positive (resp. negative) queries Q⁺ (resp.Q⁻) under the following assumptions:

• i) the universe of keys is finite: U [N]={1, 2, 3,..., N}

• ii) if the size of the data structure is k (k≤N), then the number of possibilities for the operations D and Q⁺ is k, whereas the number of possibilities for the i-th insertion or negative query is equal to N-i+1 for i≤N.

Integrated costs for these dynamic structures are defined as averages of costs taken over the set of all their possible histories (i.e. evolutions considered up to order isomorphism) of length n. We show that the costs can be explicitely calculated for the data structures serving as implementations of linear lists, priority queues and dictionaries. Letting N→∞ we recover the results proved in [9], [10] for an infinite universe of keys (this is not obvious) and we prove also that Knuth's model can be defined as limit model of the model considered here. The method uses continued fractions and orthogonal polynomials techniques like in [10].