CSCI2100 Data Structures Tutorial Graph Revisited Hongyi Zhang CSCI2100 Data Structures Tutorial 1 1

Outline Graph Adjacency Representation Topological Sort Minimum Spanning Tree Kruskal’s Algorithm Prim’s Algorithm Shortest Path Dijkstra’ Algorithm 2 2

Graph - adjacency representation Adjacency matrix G = (V, E) V = { A, B, C, D, E } E = { (A, C), (A, E), (B, D), (C, B), (D, C), (E, C), (E, D) } B A B C D E A 1 1 C D B 1 C 1 D 1 E 1 1 A E 3

Graph - adjacency representation Degree of a vertex v number of edges incident on that vertex For directed graph, Degree = InDegree + OutDegree InDegree of a vertex v sum of column v OutDegree of a vertex v sum of row v A B C D E A B C D E A 1 1 B 1 C 1 D 1 E 1 1 For example, InDegree for C is 3, OutDegree for C is 1 4

Graph - adjacency representation Adjacency matrix G = (V, E) V = { A, B, C, D, E } E = { (A, C), (A, E), (B, D), (C, B), (D, C), (E, C), (E, D) } B A B C D E A 1 1 C D B 1 1 C 1 1 1 1 D 1 1 1 E 1 1 1 A E 5

Graph - adjacency representation Adjacency list B A C E B D C D C B D C E C D A E 6

Graph - Complexity

Topological Sort A topological sort of a DAG (Directed Acyclic Graph) G is a linear ordering of all its vertices such that if G contains an edge (u, v), then u appears before v in the ordering. 8 8

Topological Sort V1 V2 V3 V4 V5 V6 V7 9 9

How to find the topological ordering? Define the indegree of a vertex v as the # of edges (u,v). We compute the indegrees of all vertices in the graph. Find any vertex with no incoming edges (or the indegree is 0). We print this vertex, and remove it, along with its edges, from the graph. Then we apply this same strategy to the rest of the graph. 10 10

Topological Order V1 V2 V5 V4 V3 V7 V6 V1 V2 V2 V3 V4 V5 V3 V4 V5 V6 11 11

Real Life Application Course prerequisite in university studies A directed edge (u,v) indicates that course u must be completed before course v may be attempted A topological ordering of these courses is any course sequence that does not violate the prerequisite requirement. 12 12

Spanning Tree 13 13

Minimum Spanning Tree 14 14

Real Life Application One example would be a cable TV company laying cable to a new neighborhood. The graph represents which houses are connected by those cables. A spanning tree for this graph would be a subset of those paths that has no cycles but still connects to every house. There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost. 15 15

MST Algorithm 16 16

Kruskal’s Algorithm a b c h d g f i e a b c h d g f i e a b c h d g f 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 a b c h d g f i e 4 8 7 14 9 1 10 11 2 6 Edge Weight <h, g> 1 <c, i> 2 <g, f> <a, b> 4 <c, f> <g, i> 6 <c, d> 7 <h, i> <a, h> 8 <b, c> <d, e> 9 <e, f> 10 <b, h> 11 <d, f> 14 a b c h d g f i e 4 8 7 14 9 1 10 11 2 6 a b c h d g f i e 4 8 7 14 9 1 10 11 2 6 a b c h d g f i e 4 8 7 14 9 1 10 11 2 6 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 17 17

Algorithm will stop here since there are already (n-1) edges found. Kruskal’s Algorithm a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 Edge Weight <h, g> 1 <c, i> 2 <g, f> <a, b> 4 <c, f> <g, i> 6 <c, d> 7 <h, i> <a, h> 8 <b, c> <d, e> 9 <e, f> 10 <b, h> 11 <d, f> 14 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 a b c h d g f i e 4 8 14 9 1 10 11 2 7 6 Algorithm will stop here since there are already (n-1) edges found. 18 18

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Prim’s Algorithm a b c h d g f i e a b c h d g f i e a b c h d g f i e 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 20 20

Prim’s Algorithm a b c h d g f i e a b c h d g f i e a b c h d g f i e 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 7 8 a b c h d g f i e 4 14 9 1 10 11 2 6 21 21

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Shortest Path Problem 23 23

The shortest path from V0 to other vertices Dijkstra’s Algorithm V5 60 100 V0 V4 30 10 20 10 V1 V3 50 5 V2 Dest The shortest path from V0 to other vertices Step 1 Step 2 Step 3 Step 4 Step 5 V1 Inf Inf Inf Inf Inf V2 10 (V0, V2) V3 Inf 60 (V0, V2, V3) 50 (V0, V4, V3) V4 30 (V0, V4) 30 (V0, V4) V5 100 (V0, V5) 100 (V0, V5) 90 (V0, V4, V5) 60 (V0, V4, V3, V5) Vj V2 V4 V3 V5 S {V0, V2} {V0, V2, V4} {V0, V2, V3, V4} {V0, V2, V3, V4, V5} 24 24

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