Matthias F. M. Stallmann

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

An augmenting path algorithm for linear matroid parity

O(n^{2.5} )

On Embedding Binary Trees into Hypercubes

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.

Exact and inexact methods for selecting views and indexes for OLAP performance improvement

Linear matroid parity generalizes matroid intersection and graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm was given by Lovász. This paper presents an algorithm that uses timeO(mn3), wherem is the number of elements andn is the rank. (The time isO(mn2.5) using fast matrix multiplication; both bounds assume the uniform cost model). For graphic matroids the time isO(mn2). The algorithm is based on the method of augmenting paths used in the algorithms for all subcases of the problem.

Hypercube embedding heuristics: an evaluation

Hypercubes are known to be able to simulate other structures such as grids and binary trees. It has been shown that an arbitrary binary tree can be embedded into a hypercube with constant expansion and constant dilation. This paper presents a simple linear-time heuristic which embeds an arbitrary binary tree into a hypercube with expansion 1 and average dilation no more than 2. We also give some results extending good embeddings for parity-balanced binary trees to arbitrary binary trees. In particular, we show that a conjecture of I. Havel (?asopis P?est. Mat.109 (1984), 135-152) implies embeddings of binary trees into hypercubes with expansion 1 and either dilation 2 or average dilation approaching 1, and embeddings with expansion 2 and dilation 1.

Heuristics, Experimental Subjects, and Treatment Evaluation in Bigraph Crossing Minimization

In on-line analytical processing (OLAP), precomputing (materializing as views) and indexing auxiliary data aggregations is a common way of reducing query-evaluation time costs for important data-analysis queries. We consider an OLAP view- and index-selection problem stated as an optimization problem, where (i) the inputs include the data-warehouse schema, a set of data-analysis queries of interest, and a storage-limit constraint, and (ii) the output is a set of views and indexes that minimizes the costs of the input queries, subject to the storage limit. While greedy and other heuristic strategies for choosing views or indexes might help to some extent in improving the costs, it is highly nontrivial to arrive at a globally optimum solution, one that reduces the processing costs of typical OLAP queries as much as is theoretically possible. In fact, as observed in [17] and to the best of our knowledge, there is no known approximation algorithm for OLAP view or index selection with nontrivial performance guarantees. In this paper we propose a systematic study of the OLAP view- and index-selection problem. Our specific contributions are as follows: (1) We develop an algorithm that effectively and efficiently prunes the space of potentially beneficial views and indexes when given realistic-size instances of the problem. (2) We provide formal proofs that our pruning algorithm keeps at least one globally optimum solution in the search space, thus the resulting integer-programming model is guaranteed to find an optimal solution. (3) We develop a family of algorithms to further reduce the size of the search space, so that we are able to solve larger problem instances, although we no longer guarantee the global optimality of the resulting solution. (4) Finally, we present an experimental comparison of our proposed approaches with the state-of-the-art approaches of [2, 12]. Our experiments show that our approaches to view and index selection result in high-quality solutions --- in fact, in globally optimum solutions for many realistic-size problem instances. Thus, they compare favorably with the well-known OLAP-centered approach of [12] and provide for a winning combination with the end-to-end framework of [2] for generic view and index selection.

The directional p-median problem: Definition, complexity, and algorithms

The hypercube embedding problem, a restricted version of the general mapping problem, is the problem of mapping a set of communicating processes to a hypercube multiprocessor. The goal is to find a mapping that minimizes the length of the paths between communicating processes. Unfortunately the hypercube embedding problem has been shown to be NP-hard. Thus many heuristics have been proposed for hypercube embedding. This paper evaluates several hypercube embedding heuristics, including simulated annealing, local search, greedy, and recursive mincut bipartitioning. In addition to known heuristics, we propose a new greedy heuristic, a new Kernighan-Lin style heuristic, and some new features to enhance local search. We then assess variations of these strategies (e.g., different neighborhood structures) and combinations of them (e.g., greedy as a front end of iterative improvement heuristics). The asymptotic running times of the heuristics are given, based on efficient implementations using a priority-queue data structure.

Unconstrained via minimization for topological multilayer routing

The bigraph crossing problem, embedding the two node sets of a bipartite graph along two parallel lines so that edge crossings are minimized, has applications to circuit layout and graph drawing. Experimental results for several previously known and two new heuristics suggest continued exploration of the problem, particularly sparse instances. We emphasize careful design of experimental subject classes and present novel views of the results. All source code, data, and scripts are available on-line

Effective bounding techniques for solving unate and binate covering problems

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.

Heuristics and Experimental Design for Bigraph Crossing Number Minimization

A theoretical study which allows determination of the minimum number of vias for realizable multilayer channel routing under a topological model is presented. The theory is sufficiently general to solve a variety of problems under different technological constraints, e.g. VLSI multilayer switchbox and channel routing, through-hole printed circuit board (PCB) channels, and single-layer routing. Topological routing is concerned with wire intersection but not area, zero-width wires and zero-area vias being assumed. Unconstrained via minimization (UVM) is not constrained to a prerouted topology. This paper presents restrictive cases of UVM, finding most to be NP-hard, but the case resulting from constraints of traditional switchbox or channel routing to be solvable in O(kn/sup 2/), where k is the maximum number of pins of a net layer and n is the number of pins. The minimum number of vias for various switchbox and channel routing benchmarks are reported. >