Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

Quantum algorithms for shortest paths problems in structured instances

The final answer is computed by taking the minimum out of all of the above answers. Ingredient (1) above requires a fast algorithm to compute the special matrix product. Typically, one computes these matrix products by some sort of partitioning of the input, then using fast integer matrix multiplication to compute some of the answers, typically between di↵erent partitions, and then using brute ...

Graph Embedding through Random Walk for Shortest Paths Problems

We present a new probabilistic technique of embedding graphs in Z, the d-dimensional integer lattice, in order to find the shortest paths and shortest distances between pairs of nodes. In our method the nodes of a breath first search (BFS) tree, starting at a particular node, are labeled as the sites found by a branching random walk on Z. After describing a greedy algorithm for routing (distanc...

Approximate Euclidean shortest paths amid convex obstacles

We develop algorithms and data structures for the approximate Euclidean shortest path problem amid a set P of k convex obstacles in R and R, with a total of n faces. The running time of our algorithms is linear in n, and the size and query time of our data structure are independent of n. We follow a “core-set” based approach, i.e., we quickly compute a small sketch Q of P whose size is independ...

Euclidean Shortest Paths - Exact or Approximate Algorithms

Euclidean Shortest Paths in a Simple Polygon

Let p and q be two points in a simple polygon Π. This chapter provides two rubberband algorithms for computing a shortest path between p and q that is contained in Π. The two algorithms use previously known results on triangular or trapezoidal decompositions of simple polygons, and have either O (n) or O (n log n) time complexity (where the super-linear time complexity is only due to preprocess...