Wolfgang W. Bein

Optimal reduction of two-terminal directed acyclic graphs

Algorithms for series-parallel graphs can be extended to arbitrary two-terminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit in-degree (out-degree) into its sole incoming (outgoing) neighbor. This paper gives an

Minimum cost flow algorithms for series-parallel networks

O(n^{2.5} )

A Monge property for the d -dimensional transportation problem

algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NP-hard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly series-parallel.

The 3-server problem in the plane

It is shown that an acyclic multigraph with a single source and a single sink is series-parallel if and only if for arbitrary linear cost functions and arbitrary capacities the corresponding minimum cost flow problem can be solved by a greedy algorithm. Furthermore, for networks of this type with m edges and n vertices, two O(mn + m log m)-algorithms are presented. One of them is based on the greedy scheme.

Cloud storage and online bin packing

In 1963, Hoffman gave necessary and sufficient conditions under which a family of O(mn)-time greedy algorithms solves the classical two-dimensional transportation problem with m sources and n sinks. One member of this family, an algorithm based on the «northwest corner rule», is of particular interest, as its running time is easily reduced to O(m+n). When restricted to this algorithm, Hoffmans result can be expressed as follows: the northwest-corner-rule greedy algorithm solves the two-dimensional transportation problem for all source and supply vectors if and only if the problems cost array C={c[i,j]} possesses what is known as the (two-dimensional) Monge property, which requires c[i 1 ,j 1 ]+c[i 2 ,j 2 ] ≤ c[i 1 ,j 2 ]+c[i 2 ,j 1 ] for i 1 <i 2 and j 1 <j 2 . This paper generalizes this last result to a higher dimensional variant of the transportation problem. We show that the natural extension of the northwest-corner-rule greedy algorithm solves an instance of the d-dimensional transportation problem if and only if the problems cost array possesses a d-dimensional Monge property recently proposed by Aggarwal and Park in the context of their study of monotone arrays. We also give several new examples of cost arrays with this d-dimensional Monge property

The Knuth-Yao quadrangle-inequality speedup is a consequence of total-monotonicity

In the k-server problem we wish to minimize, in an online fashion, the movement cost of k servers in response to a sequence of requests (we assume that k ≥ 2). The request issued at each step is specified by a point r in a given metric space M. To serve this request, one of the k servers must move to r. It is known that if M has at least k + 1 points then no online algorithm for the k-server problem in M has competitive ratio smaller than k. The best known upper bound on the competitive ratio in arbitrary metric spaces, by Koutsoupias and Papadimitriou (J. ACM 42 (1995) 971), is 2k - 1. There are only a few special cases for which k-competitive algorithms are known: for k = 2, when M is a tree, or when M has at most k + 2 points. We prove that the Work Function Algorithm is 3-competitive for the 3-server problem in the Manhattan plane. As a corollary, we obtain a 4.243-competitive algorithm for 3 servers in the Euclidean plane. The best previously known competitive ratio for 3 servers in these metric spaces was 5.

Reliability and fault tolerance of coverage models for sensor networks

We study the problem of allocating memory of servers in a data center based on online requests for storage. Given an online sequence of storage requests and a cost associated with serving the request by allocating space on a certain server one seeks to select the minimum number of servers as to minimize total cost. We use two different algorithms and propose a third algorithm. We show that our proposed algorithm performs better for large number of random requests in terms of the variance in the average number of servers.

Trackless online algorithms for the server problem

There exist several general techniques in the literature for speeding up naive implementations of dynamic programming. Two of the best known are the Knuth-Yao quadrangle inequality speedup and the SMAWK algorithm for finding the row-minima of totally monotone matrices. Although both of these techniques use a quadrangle inequality and seem similar they are actually quite different and have been used differently in the literature.In this paper we show that the Knuth-Yao technique is actually a direct consequence of total monotonicity. As well as providing new derivations of the Knuth-Yao result, this also permits showing how to solve the Knuth-Yao problem directly using the SMAWK algorithm. Another consequence of this approach is a method for solving online versions of problems with the Knuth-Yao property. The online algorithms given here are asymptotically as fast as the best previously known static ones. For example the Knuth-Yao technique speeds up the standard dynamic program for finding the optimal binary search tree of n elements from Θ(n3) down to O(n2), and the results in this paper allow construction of an optimal binary search tree in an online fashion (adding a node to the left or right of the current nodes at each step) in O(n) time per step.We conclude by discussing how the general technique described here is also applicable to later extensions of the Knuth-Yao result, such as those developed by Borchers and Gupta.

Knowledge State Algorithms

We study the coverage problem for sensor networks from the fault tolerance and reliability point of view. Fault tolerance is a critical issue for sensors deployed in places where they are not easily replaceable, repairable and rechargeable. Failure of one node should not incapacitate the entire network. We propose three 1-fault tolerant topologies, namely square, hexagonal and improved 8-node. We show how to extend these to k-fault tolerant schemes and calculate reliabilities using Markov models. The proposed models are compared to one another, as well as with the minimal coverage model of Zhang and Hou. The minimum coverage model is the most unreliable among the models, whereas the improved 8-node model is the most reliable except at the very beginning of the system, where the square model is more reliable. To our knowledge, this is the first paper which studies a pattern from the perspective of reliability.

Limited bookmark randomized online algorithms for the paging problem

Abstract A class of “simple” online algorithms for the k -server problem is identified. This class, for which the term trackless is introduced, includes many known server algorithms. The k -server conjecture fails for trackless algorithms. A lower bound of 23/11 on the competitiveness of any deterministic trackless 2 -server algorithm and a lower bound of 1+ 2 /2 on the competitiveness of any randomized trackless 2 -server problem are given.