Minimum-weight spanning tree algorithms A survey and empirical study

Abstract

The minimum-weight spanning tree problem is one of the most typical and well-known problems of combinatorial optimisation. Efficient solution techniques had been known for many years. However, in the last two decades asymptotically faster algorithms have been invented. Each new algorithm brought the time bound one step closer to linearity and finally Karger, Klein and Tarjan proposed the only known expected linear-time method. Modern algorithms make use of more advanced data structures and appear to be more complicated to implement. Most authors and practitioners refer to these but still use the classical ones, which are considerably simpler but asymptotically slower. The paper first presents a survey of the classical methods and the more recent algorithmic developments. Modern algorithms are then compared with the classical ones and their relative performance is evaluated through extensive empirical tests, using reasonably large-size problem instances. Randomly generated problem instances used in the tests range from small networks having 512 nodes and 1024 edges to quite large ones with 16 384 nodes and 524 288 edges. The purpose of the comparative study is to investigate the conjecture that modern algorithms are also easy to apply and have constants of proportionality small enough to make them competitive in practice with the older ones. Scope and purpose The minimum-weight spanning tree (MST) problem is a well-known combinatorial optimisation problem concerned with finding a spanning tree of an undirected, connected graph, such that the sum of the weights of the selected edges is minimum. The importance of this problem derives from its direct applications in the design of computer, communication, transportation, power and piping networks; from its appearance as part of solution methods to other problems to which it applies less directly such as network reliability, clustering and classification problems and from its occurrence as a subproblem in the solution of other problems like the travelling salesman problem, the multi-terminal flow problem, the matching problem and the capacitated MST problem. Although efficient solution techniques capable of solving large instances had existed, there has been sustained effort over the last two decades to invent asymptotically faster algorithms. With each new algorithm found the time bound approached linearity. Finally, an expected linear-time method has been proposed. The purpose of this work is to survey the classical and modern solution techniques and empirically compare the performance of the existing methods.