Shortest Path Problems: Multiple Paths in a Stochastic Graph

Shortest path problems arise in a variety of applications ranging from transportation planning to network routing among others. One group of these problems involves finding shortest paths in graphs where the edge weights are defined by probability distributions. While some research has addressed the problem of finding a single shortest path, no research has been done on finding multiple paths in such graphs. This thesis addresses the problem of finding paths for multiple robots through a graph in which the edge weights represent the probability that each edge will fail. The objective is to find paths for n robots that maximize the probability that at least k of them will arrive at the destination. If we make certain restrictions on the edge weights and topology of the graph, this problem can be solved in O(n log n)time. If we restrict only the topology, we can find approximate solutions which are still guaranteed to be better than the single most reliable path

Finding kk Simple Shortest Paths and Cycles

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than $n^3$ (where $n=|V|$) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find $k$ simple shortest paths for all pairs of vertices. For $k=2$, our algorithm runs in $O(mn + n^2 \log n)$ time (where $m=|E|$), which is almost the same bound as for the single pair case, and for $k=3$ we improve earlier bounds. Our approach is based on forming suitable path extensions to find simple shortest paths; this method is different from the `detour finding' technique used in most of the prior work on simple shortest paths, replacement paths, and distance sensitivity oracles. Enumerating simple cycles is a well-studied classical problem. We present new algorithms for generating simple cycles and simple paths in $G$ in non-decreasing order of their weights; the algorithm for generating simple paths is much faster, and uses another variant of path extensions. We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding $k$ simple paths and cycles.Comment: The current version includes new results for undirected graphs. In Section 4, the notion of an (m,n) reduction is generalized to an f(m,n) reductio

Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs

In the replacement paths (RP) problem we are given a graph G and a shortest path P between two nodes s and t . The goal is to find for every edge e ? P, a shortest path from s to t that avoids e. The first result of this paper is a simple reduction from the RP problem to the problem of computing shortest cycles for all nodes on a shortest path. Using this simple reduction we unify and extremely simplify two state of the art solutions for two different well-studied variants of the RP problem. In the first variant (algebraic) we show that by using at most n queries to the Yuster-Zwick distance oracle [FOCS 2005], one can solve the the RP problem for a given directed graph with integer edge weights in the range [-M,M] in O?(M n^?) time . This improves the running time of the state of the art algorithm of Vassilevska Williams [SODA 2011] by a factor of log?n. In the second variant (planar) we show that by using the algorithm of Klein for the multiple-source shortest paths problem (MSSP) [SODA 2005] one can solve the RP problem for directed planar graph with non negative edge weights in O (n log n) time. This matches the state of the art algorithm of Wulff-Nilsen [SODA 2010], but with arguably much simpler algorithm and analysis

Parallel Voronoi Computation for Physics-Based Simulations

International audienceVoronoi diagrams are fundamental data structures in computational geometry, with applications in such areas as physics-based simulations. For non-Euclidean distances, the Voronoi diagram must be performed over a grid-graph, where the edges encode the required distance information. Th e major bottleneck in this case is a shortest path algorithm that must be computed multiple times during the simulation. We present a GPU algorithm for solving the shortest path problem from multiple sources using a generalized distance function. Our algorithm was designed to leverage the grid-based nature of the underlying graph that represents the deformable objects. Experimental results report speed-ups up to 65× over a current reference sequential method

Cost allocation in connection and conflict problems on networks: a cooperative game theoretic approach

This thesis examines settings where multiple decision makers with conflicting interests benefit from cooperation in joint combinatorial optimisation problems. It draws on cooperative game theory, polyhedral theory and graph theory to address cost sharing in joint single-source shortest path problems and joint weighted minimum colouring problems. The primary focus of the thesis are problems where each agent corresponds to a vertex of an undirected complete graph, in which a special vertex represents the common supplier. The joint combinatorial optimisation problem consists of determining the shortest paths from the supplier to all other vertices in the graph. The optimal solution is a shortest path tree of the graph and the aim is to allocate the cost of this shortest path tree amongst the agents. The thesis defines shortest path tree problems, proposes allocation rules and analyses the properties of these allocation rules. It furthermore introduces shortest path tree games and studies the properties of these games. Various core allocations for shortest path tree games are introduced and polyhedral properties of the core are studied. Moreover, computational results on finding the core and the nucleolus of shortest path tree games for the application of cost allocation in Wireless Multihop Networks are presented. The secondary focus of the thesis are problems where each agent is interested in having access to a number of facilities but can be in conflict with other agents. If two agents are in conflict, then they should have access to disjoint sets of facilities. The aim is to allocate the cost of the minimum number of facilities required by the agents amongst them. The thesis models these cost allocation problems as a class of cooperative games called weighted minimum colouring games, and characterises total balancedness and submodularity of this class of games using the properties of the underlying graph

Quantization as histogram segmentation: globally optimal scalar quantizer design in network systems

We propose a polynomial-time algorithm for optimal scalar quantizer design on discrete-alphabet sources. Special cases of the proposed approach yield optimal design algorithms for fixed-rate and entropy-constrained scalar quantizers, multi-resolution scalar quantizers, multiple description scalar quantizers, and Wyner-Ziv scalar quantizers. The algorithm guarantees globally optimal solutions for fixed-rate and entropy-constrained scalar quantizers and constrained optima for the other coding scenarios. We derive the algorithm by demonstrating the connection between scalar quantization, histogram segmentation, and the shortest path problem in a certain directed acyclic graph

Evaluation of Shortest Path Query Algorithm in Spatial Databases

Many variations of algorithms for finding the shortest path in a large graph have been introduced recently due to the needs of applications like the Geographic Information System (GIS) or Intelligent Transportation System (ITS). The primary subjects of those algorithms are materialization and hierarchical path views. Some studies focus on the materialization and sacrifice the pre-computational costs and storage costs for faster computation of a query. Other studies focus on the shortest-path algorithm, which has less pre-computation and storage but takes more time to compute the shortest path. The main objective of this thesis is to accelerate the computation time for the shortest-path queries while keeping the degree of materialization as low as possible. This thesis explores two different categories: 1) the reduction of the I/O-costs for multiple queries, and 2) the reduction of search spaces in a graph. The thesis proposes two simple algorithms to reduce the I/O-costs, especially for multiple queries. To tackle the problem of reducing search spaces, we give two different levels of materializations, namely, the boundary set distance matrix and x-Hop sketch graph, both of which materialize the shortest-path view of the boundary nodes in a partitioned graph. Our experiments show that a combination of the suggested solutions for 1) and 2) performs better than the original Disk-based SP algorithm [7], on which our work is based, and requires much less storage than HEPV [3]