Lower bounds for computing geometric spanners and approximate shortest paths

We consider the problems of constructing geometric spanners, possibly containing Steiner points, for a set of n input points in d-dimensional space , and constructing spanners and approximate shortest paths among a collection of polygonal obstacles on the plane. The complexities of these problems are shown to be Ω(n log n) in the algebraic computation tree model. Since O(n log n)-time algorithms are known for solving these problems, our lower bounds are tight up to a constant factor.     

Reconstructing shortest paths

This paper develops a polynomial-time algorithm that reconstructs a shortest path between two vertices using only the all pairs shortest path distance matrix. For graphs with positive edge weights, the algorithm requiresO(⦹V|log|V|) time. When the graph contains both positive and negative, but not zero, edge weights, and all cycles have positive length, the algorithm runs inO(|V|²) time. The remarkable feature about the algorithm is that it does not require explicit information about edges in the original graph.     

Reconfiguration graphs of shortest paths

For a graph $G$ and$a,b\in V\left(G\right)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and$b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and$b$ in $G$. Two vertices in $V\left(S(G,a,b)\right)$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice.     

Computing homotopic shortest paths efficiently

We give deterministic and randomized algorithms to find shortest paths homotopic to a given collection Π of disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time , and the randomized algorithm runs in expected time O(k_out+k_inlogn+n(logn)^1+ε). Here k_in is the number of edges in all the paths of Π, and k_out is the number of edges in the output paths.     

Disjoint shortest paths in graphs

It is an interesting problem that how much connectivity ensures the existence ofn disjoint paths joining givenn pairs of vertices, but to get a sharp bound seems to be very difficult. In this paper, we study how muchgeodetic connectivity ensures the existence ofn disjointgeodesics joining givenn pairs of vertices, where a graph is calledk-geodetically connected if the removal of anyk−1 vertices does not change the distance between any remaining vertices.     

Polynomial auction algorithms for shortest paths

In this paper we consider strongly polynomial variations of the auction algorithm for the single origin/many destinations shortest path problem. These variations are based on the idea of graph reduction, that is, deleting unnecessary arcs of the graph by using certain bounds naturally obtained in the course of the algorithm. We study the structure of the reduced graph and we exploit this structure to obtain algorithms withO (n min{m, n logn}) andO(n ²) running time. Our computational experiments show that these algorithms outperform their closest competitors on randomly generated dense all destinations problems, and on a broad variety of few destination problems.Research supported by NSF under Grant No. DDM-8903385, by the ARO under Grant DAAL03-86-K-0171, by a CNR-GNIM grant, and by a Fullbright grant     

Incremental network design with shortest paths

We introduce a class of incremental network design problems focused on investigating the optimal choice and timing of network expansions. We concentrate on an incremental network design problem with shortest paths. We investigate structural properties of optimal solutions, show that the simplest variant is NP-hard, analyze the worst-case performance of natural greedy heuristics, derive a 4-approximation algorithm, and conduct a small computational study.     

Visibility graphs and obstacle-avoiding shortest paths

Summary Two closely related problems in Computational Geometry are determining visibility graphs and shortest paths in a two- or three-dimensional environment containing obstacles. Applications are within Computer Graphics and Robotics. We give a survey on recent research done on efficient algorithms for these problems.

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Zusammenfassung Zwei eng miteinander verwandte Probleme in der algorithmischen Geometrie sind die Bestimmung von Sichtbarkeitsgraphen und kürzesten Wegen in einer zwei- oder dreidimensionalen Umgebung mit Hindernissen. Anwendungen finden sich insbesondere auf den Gebieten Computergrafik und Robotik. Diese Arbeit gibt einen Überblick über kürzlich erschienene Arbeiten zum Entwurf effizienter Algorithmen für diese Probleme.

     

Parallel asynchronous label-correcting methods for shortest paths

We develop parallel asynchronous implementations of some known and some new label-correcting methods for finding a shortest path from a single origin to all the other nodes of a directed graph. We compare these implementations on a shared-memory multiprocessor, the Alliant FX/80, using several types of randomly generated problems. Excellent (sometimes superlinear) speedup is achieved with some of the methods, and it is found that the asynchronous versions of these methods are substantially faster than their synchronous counterparts.The authors acknowledge the director and the staff of CERFACS, Toulouse, France for the use of the Alliant FX/80.This research was supported by the National Science Foundation under Grants 9108058-CCR, 9221293-INT, and 9300494-DMI.     

The undirected two disjoint shortest paths problem

The $k$ disjoint shortest paths problem ($k$-DSPP) on a graph with $k$ source–sink pairs $(s_i,t_i)$ asks if there exist $k$ pairwise edge- or vertex-disjoint shortest $s_i$–$t_i$-paths. It is known to be NP-complete if $k$ is part of the input. Restricting to $2$-DSPP with strictly positive lengths, it becomes solvable in polynomial time. We extend this result by allowing zero edge lengths and give a polynomial-time algorithm based on dynamic programming for $2$-DSPP on undirected graphs with non-negative edge lengths.     

Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log²n) time per output vertex which is an improvement by a factor of O(n/log²n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O(n²⁺) for any positive constant . In the case k<n²⁺, where k is the total size of the input and output, we improve the running to O((n+k+(nk)^2/3)log^O(1)n).     

Finding the shortest paths by node combination

By repeatedly combining the source node’s nearest neighbor, we propose a node combination (NC) method to implement the Dijkstra’s algorithm. The NC algorithm finds the shortest paths with three simple iterative steps: find the nearest neighbor of the source node, combine that node with the source node, and modify the weights on edges that connect to the nearest neighbor. The NC algorithm is more comprehensible and convenient for programming as there is no need to maintain a set with the nodes’ distances. Experimental evaluations on various networks reveal that the NC algorithm is as efficient as Dijkstra’s algorithm. As the whole process of the NC algorithm can be implemented with vectors, we also show how to find the shortest paths on a weight matrix.     

On finding shortest paths in nonnegative networks

A modification of Dantzig's algorithm for the all pairs shortest paths problem is given. The new algorithm applies only to graphs with nonnegative arc lengths. For an N-node complete graph it has a worst case running time of $\text{2}\text{3}\text{N}{}^3$ triple operations of the form D_ij: = min(D_ij, D_ik + D_kj) and N² log N other comparisons. This contrasts with a lower bound of N(N ? 1) (N ? 2) triples in any pure triple operation algorithm, and seems to be the first algorithm in which no operation need be repeated N³ times. Sparsity and some other conditions may also be utilized.     

Estimate the shortest paths on fractal m-gons

The self-similar sets seem to be a class of fractals which is most suitable for mathematical treatment. The study of their structural properties is important. In this paper, we estimate the formula for the mean geodesic distance of self-similar set (denote fractal m-gons). The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractal considered. Out of result, obtained exact solution exhibits that the mean geodesic distance approximately increases as a exponential function of the number of nodes (small copies with the same size) with exponent equal to the reciprocal of the fractal dimension.     

The shortest path problem with forbidden paths

We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.     

On shortest disjoint paths in planar graphs

For a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}, the k disjoint paths problem is to find k vertex-disjoint paths P₁,…,P_k, where P_i is a path from s_i to t_i for each i=1,…,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P_i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3.Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.     

Point-to-point shortest paths on dynamic time-dependent road networks

This a summary of the author’s PhD thesis supervised by Leo Liberti, Philippe Baptiste and Daniel Krob and defended on 18 June 2009 at Ecole Polytechnique, Palaiseau, France. The thesis is written in English and is available at . The computation of point-to-point shortest paths in road networks has many practical applications that require very fast solution times, meaning that Dijkstra’s algorithm is not a viable option. In this work we develop an efficient algorithm to find the shortest route between two nodes of a large-scale, time-dependent graph, where we also allow the time-dependent arc cost functions to be updated at regular intervals. Furthermore, we propose a mathematical programming formulation for the shortest paths problem on time-dependent networks, that gives rise to integer programs. Within the context of solving Mixed-Integer Linear Programs through a Branch-and-Bound algorithm, we propose a new strategy for branching, mixing branching on single variables with branching on general hyperplanes. Finally, we introduce an effective heuristic for nonconvex Mixed-Integer Nonlinear Programss which combines VNS, Local Branching, NLP local search and Branch-and-Bound.     

Bottleneck shortest paths on a partially ordered scale

In bottleneck combinatorial problems, admissible solutions are compared with respect to their maximal elements. In such problems, one may work with an ordinal evaluation scale instead of a numerical scale. We consider here a generalization of this problem in which one only has a partially ordered scale (instead of a completely ordered scale). After the introduction of a mappimax comparison operator between sets of evaluations (which boils down to the classical operator when the order is complete), we establish computational complexity results for this variation of the shortest path problem. Finally, we formulate our problem as an algebraic shortest path problem and suggest adequate algorithms to solve it in the subsequent semiring.Received: June 2002, Revised: March 2003, AMS classification: 05C50, 16Y60, 90B06Olivier Spanjaard: Corresponding author.     

Noncrossing Hamiltonian paths in geometric graphs

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that . We also establish several results related to special classes of geometric graphs. Let h₁(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h₁(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that . Let h₂(n) denote the largest number such that when we remove an arbitrary star with at most h₂(n) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h₂(n)=⌈n/2⌉-1. Further we prove that when we remove any matching from a complete geometric graph the resulting graph will have a noncrossing Hamiltonian path.