Lecture 26 CSE 331 Nov 1, 2010

Blog posts/Group Scribe leader Please sign up if you have not There are many lectures even in November with <3 students If I have a pick a blogger/leader I will only pick THREE/lecture Will lose out on 5%-10% of your grade

Shortest Path problem 15 5 s u v 100 Input: Directed graph G=(V,E) Edge lengths, le for e in E “start” vertex s in V 15 5 s u v Output: All shortest paths from s to v in V 5 s u

Naïve Algorithm Ω(n!) time

Dijkstra’s shortest path algorithm E. W. Dijkstra (1930-2002)

Dijkstra’s shortest path algorithm 1 d’(v) = min e=(u,v) in E, u in R d(u)+le 1 2 4 3 y 4 3 u d(s) = 0 d(u) = 1 s x 2 4 d(v) = 2 d(x) = 2 d(y) = 3 d(z) = 4 v z 5 4 2 s v Input: Directed G=(V,E), le ≥ 0, s in V u R = {s}, d(s) =0 Shortest paths x While there is a v not in R with (u,v) in E, u in R z y Pick w that minimizes d’(w) Add w to R d(w) = d’(w)

Couple of remarks The Dijkstra’s algo does not explicitly compute the shortest paths Can maintain “shortest path tree” separately Dijkstra’s algorithm does not work with negative weights Left as an exercise

Rest of today’s agenda Prove correctness of Dijkstra’s algorithm Efficient implementation of Dijkstra’s algorithm