Fortgeschrittene Datenstrukturen — Vorlesung: Distance Oracles (ctd.), String B-Trees

In a graph G = (V,E) with non-negative edge weights ω(e), e ∈ E, Dijkstra’s single source shortest path (SSSP) algorithm finds for a given source vertex s ∈ V the shortests path between that vertex s and every other vertex. We denote the length of the shortest path from v to w by δ(v, w), i.e. Dijktra’s algorithm computes δ(s, v) for all v ∈ V . Algorithm 1 shows the full pseudo-code of Dijkstra’s algorithm. In short, Dijkstra’s algorithm declares all nodes unscanned and initializes an array d with ∞ which holds an upper bound on the shortest path: δ(s, v) ≤ d[v]. First all edges out of s are relaxed. While there are unscanned nodes with d[v] < ∞, take such a node v with minimal d[v] and relax all its outgoing edges and declare v scanned. The relax procedure for node v takes all edges (v, w) and sets d[w] to d[v] + ω(v, w) if this new distance is smaller then the current estimate d[w]. In a graph with non-negative edge weights ω(v, w), Dijkstra’s algorithm ensures that for each scanned node d[v] = δ(s, v) and thus at termination for all reachable nodes v : d[v] = δ(s, v). The time complexity of Dijkstra’s algorithm depends upon the time complexity of the insert, deleteMin and decreaseKey operations of the used priority queue as follows: