Probabilistic Analysis of Online Stacking Algorithms

Abstract

Consider the situation where some items arrive to a storage location where they are temporarily stored in bounded capacity LIFO stacks until their departure. We consider the problem of deciding where to put an arriving item with the objective of using as few stacks as possible. The decision has to be made as soon as an item arrives and we assume that we only have information on the departure times for the arriving item and the items currently at the storage area. We are only allowed to put an item on top of another item if the item below departs at a later time. We assume that the numbers defining the storage time intervals are picked independently and uniformly at random from the interval [0, 1]. We present a simple polynomial time online algorithm for the problem and prove the following: For any positive real numbers $\epsilon _1, \epsilon _2 > 0$ there exists an $N > 0$ such that the algorithm uses no more than $(1+\epsilon _1)OPT$ stacks with probability at least $1-\epsilon _2$ if the number of items is at least N where OPT denotes the optimal number of stacks. The result even holds if the stack capacity is $o(\sqrt{n})$ where n is the number of items.