8,868 research outputs found
Fast Robust Shortest Path Computations
We develop a fast method to compute an optimal robust shortest path in large networks like road networks, a fundamental problem in traffic and logistics under uncertainty.
In the robust shortest path problem we are given an s-t-graph D(V,A) and for each arc a nominal length c(a) and a maximal increase d(a) of its length. We consider all scenarios in which for the increased lengths c(a) + bar{d}(a) we have bar{d}(a) <= d(a) and sum_{a in A} (bar{d}(a)/d(a)) <= Gamma. Each path is measured by the length in its worst-case scenario. A classic result [Bertsimas and Sim, 2003] minimizes this path length by solving (|A| + 1)-many shortest path problems. Easily, (|A| + 1) can be replaced by |Theta|, where Theta is the set of all different values d(a) and 0. Still, the approach remains impractical for large graphs.
Using the monotonicity of a part of the objective we devise a Divide and Conquer method to evaluate significantly fewer values of Theta. This methods generalizes to binary linear robust problems. Specifically for shortest paths we derive a lower bound to speed-up the Divide and Conquer of Theta. The bound is based on carefully using previous shortest path computations. We combine the approach with non-preprocessing based acceleration techniques for Dijkstra adapted to the robust case.
In a computational study we document the value of different accelerations tried in the algorithm engineering process. We also give an approximation scheme for the robust shortest path problem which computes a (1 + epsilon)-approximate solution requiring O(log(d^ / (1 + epsilon))) computations of the nominal problem where d^ := max d(A) / min (d(A){0})
A Divide-and-Conquer Algorithm for Two-Point L_1 Shortest Path Queries in Polygonal Domains
Let P be a polygonal domain of h holes and n vertices. We study the problem of constructing a data structure that can compute a shortest path between s and t in P under the L_1 metric for any two query points s and t. To do so, a standard approach is to first find a set of n_s "gateways" for s and a set of n_t "gateways" for t such that there exist a shortest s-t path containing a gateway of s and a gateway of t, and then compute a shortest s-t path using these gateways. Previous algorithms all take quadratic O(n_s * n_t) time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in O(n_s + n_t log n_s) time. As a consequence, we construct a data structure of O(n+(h^2 log^3 h/log log h)) size in O(n+(h^2 log^4 h/log log h)) time such that each query can be answered in O(log n) time
Log-space Algorithms for Paths and Matchings in k-trees
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width 2 [JT07]. However, for
graphs of tree-width larger than 2, no bound better than NL is known. In this
paper, we improve these bounds for k-trees, where k is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed k-trees, and for computation of shortest and longest
paths in directed acyclic k-trees.
Besides the path problems mentioned above, we also consider the problem of
deciding whether a k-tree has a perfect macthing (decision version), and if so,
finding a perfect match- ing (search version), and prove that these two
problems are L-complete. These problems are known to be in P and in RNC for
general graphs, and in SPL for planar bipartite graphs [DKR08].
Our results settle the complexity of these problems for the class of k-trees.
The results are also applicable for bounded tree-width graphs, when a
tree-decomposition is given as input. The technique central to our algorithms
is a careful implementation of divide-and-conquer approach in log-space, along
with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201
Optimal Path Queries in Very Large Spatial Databases
Researchers have been investigating the optimal route query problem for a long time. Optimal route queries are categorized as either unconstrained or constrained queries. Many main memory based algorithms have been developed to deal with the optimal route query problem. Among these, Dijkstra's shortest path algorithm is one of the most popular algorithms for the unconstrained route query problem. The constrained route query problem is more complicated than the unconstrained one, and some constrained route query problems such as the Traveling Salesman Problem and Hamiltonian Path Problem are NP-hard. There are many algorithms dealing with the constrained route query problem, but most of them only solve a specific case. In addition, all of them require that the entire graph resides in the main memory. Recently, due to the need of applications in very large graphs, such as the digital maps managed by Geographic Information Systems (GIS), several disk-based algorithms have been derived by using divide-and-conquer techniques to solve the shortest path problem in a very large graph. However, until now little research has been conducted on the disk-based constrained problem. This thesis presents two algorithms: 1) a new disk-based shortest path algorithm (DiskSPNN), and 2) a new disk-based optimal path algorithm (DiskOP) that answers an optimal route query without passing a set of forbidden edges in a very large graph. Both algorithms fit within the same divide-and-conquer framework as the existing disk-based shortest path algorithms proposed by Ning Zhang and Heechul Lim. Several techniques, including query super graph, successor fragment and open boundary node pruning are proposed to improve the performance of the previous disk-based shortest path algorithms. Furthermore, these techniques are applied to the DiskOP algorithm with minor changes. The proposed DiskOP algorithm depends on the concept of collecting a set of boundary vertices and simultaneously relaxing their adjacent super edges. Even if the forbidden edges are distributed in all the fragments of a graph, the DiskOP algorithm requires little memory. Our experimental results indicate that the DiskSPNN algorithm performs better than the original ones with respect to the I/O cost as well as the running time, and the DiskOP algorithm successfully solves a specific constrained route query problem in a very large graph
Reverse k Nearest Neighbor Search over Trajectories
GPS enables mobile devices to continuously provide new opportunities to
improve our daily lives. For example, the data collected in applications
created by Uber or Public Transport Authorities can be used to plan
transportation routes, estimate capacities, and proactively identify low
coverage areas. In this paper, we study a new kind of query-Reverse k Nearest
Neighbor Search over Trajectories (RkNNT), which can be used for route planning
and capacity estimation. Given a set of existing routes DR, a set of passenger
transitions DT, and a query route Q, a RkNNT query returns all transitions that
take Q as one of its k nearest travel routes. To solve the problem, we first
develop an index to handle dynamic trajectory updates, so that the most
up-to-date transition data are available for answering a RkNNT query. Then we
introduce a filter refinement framework for processing RkNNT queries using the
proposed indexes. Next, we show how to use RkNNT to solve the optimal route
planning problem MaxRkNNT (MinRkNNT), which is to search for the optimal route
from a start location to an end location that could attract the maximum (or
minimum) number of passengers based on a pre-defined travel distance threshold.
Experiments on real datasets demonstrate the efficiency and scalability of our
approaches. To the best of our best knowledge, this is the first work to study
the RkNNT problem for route planning.Comment: 12 page
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