Papers by Martin Zachariasen
On the Approximation of the Rectilinear Steiner Arborescence Problem in the Plane
Algorithms for plane Steiner tree problems
Algorithmic aspects of divisor-based biproportional rounding
Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative... more Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we ®rst give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secondly, we present a new general local search approach which is based on a list of full Steiner trees constructed in a preprocessing phase. This list de®nes a solution space on which three neighbourhood structures are proposed and evaluated. Computational results show that this new approach is very competitive from a cost±bene®t point of view. Furthermore, it has the advantage of being easy to apply to the Steiner tree problem in other metric spaces and to obstacle avoiding variants. Ó
European Symposium on Algorithms, 2007
We present an experimental evaluation of an approximate distance oracle recently suggested by Tho... more We present an experimental evaluation of an approximate distance oracle recently suggested by Thorup [1] for undirected planar graphs. The oracle uses the existence of graph separators for planar graphs, discovered by Lipton and Tarjan [2], in order to divide the graph into smaller subgraphs. For a planar graph with n nodes, the algorithmic variant considered uses O(n(log n)3/ε) preprocessing
Delay-related secondary objectives for rectilinear Steiner minimum trees
The rectilinear Steiner tree problem in the plane is to construct a minimum-length tree interconn... more The rectilinear Steiner tree problem in the plane is to construct a minimum-length tree interconnecting a set of points (called terminals) consisting of horizontal and vertical line segments only. Rectilinear Steiner minimum trees (RSMTs) can today be computed quickly for realistic instances occurring in VLSI design. However, interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the
Heuristic methods for single link shared backup path protection
Journal of Heuristics, 2014
We present an experimental evaluation of an approximate distance oracle recently suggested by Tho... more We present an experimental evaluation of an approximate distance oracle recently suggested by Thorup [1] for undirected planar graphs. The oracle uses the existence of graph separators for planar graphs, discovered by Lipton and Tarjan [2], in order to divide the graph into smaller subgraphs. For a planar graph with n nodes, the algorithmic variant considered uses O(n(log n) 3 / ) preprocessing time and O(n(log n) 2 / ) space to answer factor (1 + ) distance queries in O((log n) 2 / ) time. By performing experiments on randomly generated planar graphs and on planar graphs derived from real world road networks, we investigate some key characteristics of the oracle, such as preprocessing time, query time, precision, and characteristics related to the underlying data structure, including space consumption. For graphs with one million nodes, the average query time is less than 20μs.
A catalog of Hanan grid problems
... by Chiang et al. [4]. They divided the feasible planar region into rectangular regions with d... more ... by Chiang et al. [4]. They divided the feasible planar region into rectangular regions with different weights and the length of a tree was computed accordingly. Also, they proved that a shortest path be-tween two points with regard to this length function can be found in the Hanan ...
International Journal of Geographical Information Science, 2010
Analysis of potential spatial behavior in transport infrastructures is usually carried out by mea... more Analysis of potential spatial behavior in transport infrastructures is usually carried out by means of a digital network. A basic condition for such a network analysis has traditionally been the desire to find solutions to optimization problems and to achieve greater efficiency in industry. Geographic information system (GIS) tools for network analysis are overwhelmingly targeted at finding solutions to optimization problems, which include the shortest path problem and the traveling salesman problem. This article addresses the problem of the lack of tools for finding solutions to a class of constraint satisfaction problems that are of potential interest to behavioral geographers. Constraint satisfaction problems differ from optimization problems in that they lack an expression to be maximized or minimized. We describe how a constraint-based approach to network analysis can be applied to search for 'excess routes' that are longer or in other ways exceed single, optimal routes. Our analysis considers both round-trips and travel from A to B and defines a set of constraints that can characterize such paths. We present a labeling algorithm that can generate solutions to such excess route problems.
Planning of Shared Backup Path Protection
Frá atkvøðum til tingsessir: Samanbering av hættum til útrokning av tingmannabýti
Page 1. Tøknifrágreiðing Frá atkvøðum til tingsessir Samanbering av hættum til útrokning av tingm... more Page 1. Tøknifrágreiðing Frá atkvøðum til tingsessir Samanbering av hættum til útrokning av tingmannabýti Petur Zachariassen Martin Zachariassen NVDRit 2005:1 Page 2. Heiti / Title Frá atkvøðum til tingsessir Samanbering ...
Heuristic methods for shared backup path protection planning
2012 IV International Congress on Ultra Modern Telecommunications and Control Systems, 2012
ABSTRACT Protecting communication networks against failures is becoming increasingly important as... more ABSTRACT Protecting communication networks against failures is becoming increasingly important as they have become an integrated part of our society. Cable failures are fairly common, but it is unacceptable for a single cable failure to disconnect communication for more than a few seconds - hence protection schemes are employed. In contrast to manual intervention, automatic protection schemes such as Shared Backup Path Protection (SBPP) can recover from failure quickly and efficiently. SBPP is a simple but efficient protection scheme that can be implemented in backbone networks with technology available today. In SBPP backup paths are planned in advance for every failure scenario in order to recover from failures quickly and efficiently. The SBPP problem is an NP-hard optimization problem, and previous work confirms that it is time-consuming to solve the problem in practice using exact methods. We present heuristic algorithms and lower bound methods for the SBPP planning problem. Experimental results show that the heuristic algorithms are able to find good quality solutions in minutes. A solution gap of less than 3.5% was achieved for more than half of the benchmark instances (and a gap of less than 12% for the remaining instances.)
Fixed orientation interconnection problems: Theory, algorithms and applications
... M. Zachariasen and A. Rohe ... In particular I thank my oldest colleague and Ph.D. supervisor... more ... M. Zachariasen and A. Rohe ... In particular I thank my oldest colleague and Ph.D. supervisor, Pawel Winter (coauthor on 5 papers), my Steiner tree friend and colleague in Melbourne, Marcus Brazil (coauthor on 4 papers) and Benny K. Nielsen, former Ph.D. student (coauthor on 3 ...
Delay-related secondary objectives for rectilinear Steiner minimum trees
Discrete Applied Mathematics, 2004
The rectilinear Steiner tree problem in the plane is to construct a minimum-length tree interconn... more The rectilinear Steiner tree problem in the plane is to construct a minimum-length tree interconnecting a set of points (called terminals) consisting of horizontal and vertical line segments only. Rectilinear Steiner minimum trees (RSMTs) can today be computed quickly for realistic instances occurring in VLSI design. However, interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the
Divisor-Based Biproportional Apportionment in Electoral Systems: A Real-Life Benchmark Study
Management Science, 2010
Biproportional apportionment methods provide two-way proportionality in electoral systems where t... more Biproportional apportionment methods provide two-way proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row- and column-sum requirements are fulfilled. In a divisor-based method for biproportional apportionment, the problem
Lecture Notes in Computer Science, 2004
Spanners are sparse subgraphs that preserve distances up to a given factor in the underlying grap... more Spanners are sparse subgraphs that preserve distances up to a given factor in the underlying graph. Recently spanners have found important practical applications in metric space searching and message distribution in networks. These applications use some variant of the socalled greedy algorithm for constructing the spanner -an algorithm that mimics Kruskal's minimum spanning tree algorithm. Greedy spanners have nice theoretical properties, but their practical performance with respect to total weight is unknown. In this paper we give an exact algorithm for constructing minimum-weight spanners in arbitrary graphs. By using the solutions (and lower bounds) from this algorithm, we experimentally evaluate the performance of the greedy algorithm for a set of realistic problem instances.
A given nonnegative n × n matrix A = (a ij ) is to be scaled, by multiplying its rows and columns... more A given nonnegative n × n matrix A = (a ij ) is to be scaled, by multiplying its rows and columns by unknown positive multipliers λ i and µ j , such that the resulting matrix (a ij λ i µ j ) has specified row and column sums r i and s j .
The Rectilinear Steiner Tree Problem: A Tutorial
Combinatorial Optimization, 2001
ABSTRACT
European Journal of Operational Research, 1999
Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative... more Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we ®rst give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secondly, we present a new general local search approach which is based on a list of full Steiner trees constructed in a preprocessing phase. This list de®nes a solution space on which three neighbourhood structures are proposed and evaluated. Computational results show that this new approach is very competitive from a cost±bene®t point of view. Furthermore, it has the advantage of being easy to apply to the Steiner tree problem in other metric spaces and to obstacle avoiding variants. Ó
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Papers by Martin Zachariasen