Computing Lengths of Shortest Non-Crossing Paths in Planar Graphs

Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.Comment: 17 pages, 11 figure

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

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

Long induced paths in graphs

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on $n$ vertices, also contains an induced path on $\Omega(\sqrt{\log n})$ vertices. We conjecture that for any $k$, there is a contant $c(k)$ such that any $k$-degenerate graph with a path on $n$ vertices also contains an induced path on $\Omega((\log n)^{c(k)})$ vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio

Approximating shortest paths in large networks

In the classroom students are introduced to shortest route calculation using small datasets (those that can be hand-drawn.) For demonstrating the application of an algorithm a small dataset is typically sufficient. However, real-world applications of shortest path calculations seem to be useful only when applied to large datasets. This paper presents research on a computer based implementation of a modified Dijkstra algorithm as applied to large datasets including tens of thousands of arcs. In an attempt to improve the performance of calculating paths two heuristics are also examined. The intuition behind the heuristics is to remove the arcs that will likely not be traversed by the optimal path from the set of arcs that can possibly be traversed by the optimal path. By reducing this number less labeling is required, resulting in fewer CPU cycles being used to generate a route. This paper compares the results of the optimal against those of the two heuristics