Efficiently listing bounded length st-paths

The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s. For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971 and 1972], and an $O(K(m+n \log n))$ algorithm for the undirected version by Katoh et al. [Networks, 1982], both using $O(Kn + m)$ space. In this work, we consider a different parameterization for this problem: instead of bounding the number of $st$-paths output, we bound their length. For the bounded length parameterization, we propose new non-trivial algorithms matching the time complexity of the classic algorithms but using only $O(m+n)$ space. Moreover, we provide a unified framework such that the solutions to both parameterizations -- the classic $K$-shortest and the new length-bounded paths -- can be seen as two different traversals of a same tree, a Dijkstra-like and a DFS-like traversal, respectively.Comment: 12 pages, accepted to IWOCA 201

UniALT for regular language contrained shortest paths on a multi-modal transportation network

Shortest paths on road networks can be efficiently calculated using Dijkstra\u27s algorithm (D). In addition to roads, multi-modal transportation networks include public transportation, bicycle lanes, etc. For paths on this type of network, further constraints, e.g., preferences in using certain modes of transportation, may arise. The regular language constrained shortest path problem deals with this kind of problem. It uses a regular language to model the constraints. The problem can be solved efficiently by using a generalization of Dijkstra\u27s algorithm (D_RegLC). In this paper we propose an adaption of the speed-up technique uniALT, in order to accelerate D_RegLC. We call our algorithm SDALT. We provide experimental results on a realistic multi-modal public transportation network including time-dependent cost functions on arcs. The experiments show that our algorithm performs well, with speed-ups of a factor 2 to 20

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

A GIS based multi-modal multiple optimal path transit advanced traveler information system

A method for the design and use of a Transit Advanced Traveler Information System (TATIS) using an off-the-shelf Geographic Information System (GIS) is developed in this thesis. The research design included: 1) representing multi-modal transit networks in a digital form with schedule databases; 2) development of a multiple optimal path algorithm that takes into account walking transfers using published time schedules; 3) incorporating user preferences and penalties in the algorithm; 4) development of a user-interface with suitable output capabilities; 5) using the prototype for sample inquiries giving performance measures. This prototype was developed using the Arc/Info GIS developed by ESRI, Inc. The principal results of the research demonstrated the effectiveness and robustness of the TATIS prototype with respect to the five previously mentioned issues. Areas of future improvement and research focus on performance measures and added functionality