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The determination of collision-free shortest paths among growing discs has previously been studied for discs with fixed growing rates. Here, we study a more general case of this problem, where: (1) the speeds at which the discs are growing are polynomial functions of degree $dd$, and (2) the source and destination points are given as query points. We show how to preprocess the $n$ growing discs so that, for two given query points $s$ and $d$, a shortest path from $s$ to $d$ can be found in $O(n^2 log (dd n))$ time. The preprocessing time of our algorithm is $O(n^2 log n + k log k)$ where $k$ is the number of intersections between the growing discs and the tangent paths (straight line paths which touch the boundaries of two growing discs). We also prove that $k in O(n^3dd)$.
The determination of time-dependent collision-free shortest paths has received a fair amount of attention. Here, we study the problem of computing a time-dependent shortest path among growing discs which has been previously studied for the instance w
Physarum Polycephalum is a slime mold that is apparently able to solve shortest path problems. A mathematical model has been proposed by biologists to describe the feedback mechanism used by the slime mold to adapt its tubular channels while foragi
We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shor
In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Delta$-stepping algorithm proposed by Meyer and Sanders has been widely
In the decremental $(1+epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s in V$, and we are asked to process edge deletion