Finding k-Dissimilar Paths with Minimum Collective Length

Shortest path computation is a fundamental problem in road networks. However, in many real-world scenarios, determining solely the shortest path is not enough. In this paper, we study the problem of finding k-Dissimilar Paths with Minimum Collective Length (kDPwML), which aims at computing a set of paths from a source s to a target t such that all paths are pairwise dissimilar by at least \theta and the sum of the path lengths is minimal. We introduce an exact algorithm for the kDPwML problem, which iterates over all possible s-t paths while employing two pruning techniques to reduce the prohibitively expensive computational cost. To achieve scalability, we also define the much smaller set of the simple single-via paths, and we adapt two algorithms for kDPwML queries to iterate over this set. Our experimental analysis on real road networks shows that iterating over all paths is impractical, while iterating over the set of simple single-via paths can lead to scalable solutions with only a small trade-off in the quality of the results.Comment: Extended version of the SIGSPATIAL'18 paper under the same titl

Faster Parametric Shortest Path and Minimum Balance Algorithms

The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents fast algorithms for both problems. The algorithms run in O(nm+n^2 log n) time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding a minimum-mean cycle (an important special case of both problems) is O(n log(n) + m)

Efficient Algorithms for Finding Maximal Matching in Graphs

This paper surveys the techniques used for designing the most efficient algorithms for finding a maximum cardinality or weighted matching in (general or bipartite) graphs. It also lists some open problems concerning possible improvements in existing algorithms and the existence of fast parallel algorithms for these problems

On-line and Dynamic Shortest Paths through Graph Decompositions (Preliminary Version)

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. We give both sequential and parallel algorithms that work on a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. The parallel algorithms presented here are the first known ones for solving this problem. Our results can be extended to hold for digraphs of genus $o(n)$

The Parallel Complexity of Growth Models

This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale as $O(\log^2 N)$, where $N$ is the system size, and the number of processors required scale as a polynomial in $N$. The algorithms are based on fast parallel procedures for finding minimum weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist.Comment: 20 pages, latex, submitted to J. Stat. Phys., UNH-TR94-0