Cascading A*: a Parallel Approach to Approximate Heuristic Search

Definition 1. Given a undirected graph G = (V,E), two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = (v1, v2, . . . , vn) ∈ V n such that vi is adjacent to vi+1 for 1 ≤ i < n. Such a path P is called a path of length n from v1 to vn. Let ei,j be the edge incident to both vi and vj. Given a real-valued weight function f : E → R, and an undirected graph G, the shortest path from v to v′ is the path P = (v1, v2, . . . , vn) that over all possible n minimizes the sum ∑n−1 i=1 f(ei,i+1), v1 = v and vn = v ′.