Single-Source Shortest Path & Minimum Spanning Trees Dijkstra’s, Prim’s, and Kruskal’s Algorithms

Single-Source Shortest Path Defined Given G = (V, E) Source vertex s  V Compute {SP} = set of shortest paths from s to all vertices in V

Path Weight Defined Weight of path p = {v1, v2, … vk} Shortest-path weight from u to v

Determining Path Lengths 1 3 s b 5 2 4 c

Determining Path Lengths 3 1 3 s b 5 5 2 4 c 2

Determining Path Lengths 3 1 3 s b 5 5 2 4 c d[b] <= d[c] + w(c,b) 5 <= 2 + 4 2

Determining Path Lengths d[b] > d[a] + w(a,b) 5 > 3 + 1 3 1 3 s b 5 5 2 4 c 2

Determining Path Lengths d[b] > d[a] + w(a,b) 5 > 3 + 1 3 1 3 s b 5 4 2 4 c 2 Reduce

Path Lengths - FIN a All vertices are in S 3 1 3 s b 5 4 2 4 c 2

Cycles in Paths a b -4 3 -1 3 4 s 6 c d g 5 8 5 11 - -3 2 3 7 e f - 5 11 - -3 2 3 7 e f - - -6

Properties of Shortest Paths Cannot contain a negative-weight cycle No need to contain 0-weight cycle No need to contain positive-weight cycle Thus contains no cycles Will contain at most V-1 edges Assumes no cycles (as stated above) Maintain predecessor information E & V

Shortest Paths Not Unique x 6 3 9 3 s 4 2 1 2 7 3 5 5 11 6 y z t x t x 3 6 9 3 6 9 3 3 s 4 s 4 2 1 2 7 2 1 2 7 3 3 5 11 5 11 5 5 6 6 y y z z

Shortest Path Distance Is Unique x s u z v y [z] must be either x or y, but not both This is true even if d[x] + w(x,z) = d[y] + w(y,z) establishing that if s is source, d[z] is well-defined and unique

Setting up the Solution For each vertex v in V[G] Predecessor of v ([v]) = NIL Distance to v from source (d[v]) =  RELAX: as we process the graph, we’ll “Improve” [v] & d[v] by “relaxing” if (d[v] > d[u] + w(u,v)) d[v] = d[u] + w(u,v) [v] = u

Init & Relax Initialize(G, s) for each v in V[G] d[v] = INFINITY [v] = NULL d[s] = 0 Relax(u, v, w) if (d[v] > d[u] + w(u,v)) d[v] = d[u] + w(u,v) [v] = u

Dijkstra’s Algorithm (1959) Assume G = (V, E), source s, & weight function w Also assume all edges are non-negative Dijkstra(G, w, s) Initialize(G, s) S = {} Q = V[G] while Q != {} u = Extract_Min(Q) S.Add(u) for each vertex v in u.Adjacency() Relax(u, v, w)

Example: Dijkstra’s t x 1   10 s 9 2 3 4 6 7 5   y 2 z

Example: Dijkstra’s t x 1 10  10 s 9 2 3 4 6 7 5 5  y 2 z

Example: Dijkstra’s t x 1 8 14 10 s 9 2 3 4 6 7 5 5 7 y 2 z

Example: Dijkstra’s t x 1 8 13 10 s 9 2 3 4 6 7 5 5 7 y 2 z

Example: Dijkstra’s t x 1 8 9 10 s 9 2 3 4 6 7 5 5 7 y 2 z

Dijkstra’s - FIN t x 1 8 9 10 s 9 2 3 4 6 7 5 5 7 y 2 z

Runtime of Dijkstra’s O(V lgV + E lgV) O(E lgV) O(V lgV + E) If G is connected O(V lgV + E) If we use a Fibonacci heap Better if V << E (i.e. sparsely connected)

MST Defined

Minimum Spanning Tree Connect all nodes at minimum cost. Can start anywhere? Why? One solution. i.e. There is just one minimum Two algorithms Prim’s Kruskal’s

A Graph 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's start 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Prim's 5 4 4 2 5 2 5 6 6 44 5

The Same Graph 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

Kruskal's 5 4 4 2 5 2 5 6 6 44 5