Competitive Algorithms for Layered Graph Traversal

Abstract

A layered graph is a connected graph whose vertices are partitioned into sets L 0 =s, L 1 , L 2 ,..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is {|L i |}. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1.We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9 w )-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2 w-2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor. Abstract. A layered graph is a connected graph whose vertices are partitioned into sets L 0 = {s}, L 1 , L 2 , ..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is max{|L i |}. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i−1 and i are only revealed when the searcher reaches layer i−1.We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9 w )-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2 w−2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.

Combining online algorithms for rejection and acceptance

“…We consider the problem both for the measure of the number of rejected requests and for the measure of the number accepted requests. We observe that using known techniques [7,9,6] we can construct a combined randomized preemptive algorithm which is at most O(log k) worse with respect to the number of rejected requests of the best algorithm among the k. Using known results [2] we can also construct a combined randomized preemptive algorithm which is at most O(log k) worse with respect to the number of accepted requests of the best algorithm among the k. These two combined algorithms can be combined to one master algorithm using our main result to guarantee both rejection and acceptance competitiveness.…”

“…Results of this form already exist in the literature [2,6,7,9] but our main point here is that (a) these known techniques can be applied in our abstract model, and (b) using our main result we can combine the two master algorithms that result into one combined algorithm which guarantees both rejection and acceptance competitiveness.…”

“…Since this problem can be viewed as a variant of the layered graph traversal [7,9,6] one can show the following: Theorem 4. The randomized algorithm REJ rand rejects at most O(log k) times more requests than R * (σ) for any sequence σ of requests.…”

On the two-dimensional cow search problem

“…Lower and upper bounds of 9 were given there and are one of the usual introductory examples at the courses on on-line algorithms. Some of the problems stated in this paper and other, closely related, were afterwards considered by many researchers [6,4,7]. We refer the reader to [3] for a more modern and general reflection on the subject, or [5] for a more classic approach to search problems.…”

The -server problem

“…For randomized algorithms with memory, the two problems are intertwined. Another related problem that is more general than the k-server problem and a special case of metrical task systems is the metrical service systems [50,51] which is equivalent to the layered graph traversal problem [52,53]. Also the CNN problem [54,55] is similar to the k-server problem, especially to the weighted version in which the servers have different speeds (or costs) [56].…”