Graph implementation Cinda Heeren / Andy Roth / Will Evans / March - PowerPoint PPT Presentation

Graph implementation Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 1 Geoffrey Tien

Graph vocabulary Quiz yourself! 𝐻 List the edges incident to vertex 𝑐 1. 𝑐 What is the degree of vertex 𝑒 ? 2. 𝑏 𝑑 List the vertices adjacent to vertex 𝑗 3. 8 9 𝑓 𝑒 Give a path from 0 to 7 4. Give a path from 𝑙 to ℎ 5. 6 7 𝑕 𝑔 Vertices in the largest complete subgraph in 𝐻 6. How many connected components are in 𝐻 ? 7. 𝑝 4 5 8. How many edges in a spanning forest? 𝑜 𝑛 How many simple paths connect 0 and 9 ? 9. 2 3 𝑚 10. Can you draw 𝐻 with no edge crossings (as a 𝑙 𝑘 planar graph)? 0 1 𝑗 ℎ 𝑞 𝑟 Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 2 Geoffrey Tien

Weighted graphs • In a weighted graph each edge is assigned a weight – Edges are labeled with their 3 1 weights • Each edge’s weight represents 4 1 2 the cost to travel along that edge – The cost could be distance, time, 2 5 3 2 money or some other measure – The cost depends on the underlying problem 3 Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 3 Geoffrey Tien

Graph density • A sparse graph has 𝑃 𝑊 edges • A dense graph has Θ 𝑊 2 edges – Anything in between is either on the sparse side or on the dense side, depending critically on context! • Analysis of graph operations typically must be expressed in terms of both 𝑊 and 𝐹 Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 4 Geoffrey Tien

Connectivity • Undirected graphs are connected if there is a path between any two vertices • Directed graphs are strongly connected if there is a directed path from any vertex to any other • Digraphs are weakly connected if there is a path between any two vertices, ignoring direction • A complete graph has an edge between every pair of vertices Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 5 Geoffrey Tien

Isomorphism and subgraphs • We often care only about the structure of a graph, not the names of its vertices. Then, we can ask: – "Are two graphs isomorphic ?" i.e. do the graphs have identical structure / can you "line up" their vertices so that their edges match? – "Is one graph a subgraph of another?" i.e. is one graph isomorphic to a part of the other graph (a subset of vertices and a subset of edges connecting those vertices)? • 𝐻 ′ = 𝑊 ′ , 𝐹′ 𝑊′ ⊆ 𝑊 , 𝐹′ ⊆ 𝐹 , if 𝑣, 𝑤 ∈ 𝐹 ′ then 𝑣, 𝑤 ∈ 𝑊 ′ Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 6 Geoffrey Tien

Degree • The degree of a vertex 𝑤 ∈ 𝑊 is denoted deg 𝑤 and represents the number of edges incident on 𝑤 • Handshaking theorem: ( )  = – If 𝐻 = 𝑊, 𝐹 is an undirected graph, then deg v 2 E  v V 𝐹 = 7 Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 7 Geoffrey Tien

Degree for directed graphs • The in-degree of a vertex 𝑤 ∈ 𝑊 is denoted deg − 𝑤 and is the number of edges entering 𝑤 • The out-degree of a vertex 𝑤 ∈ 𝑊 is denoted deg + 𝑤 and is the number of edges leaving 𝑤 • We let deg 𝑤 = deg + 𝑤 + deg − 𝑤 • Then: ( ) ( ) 1 ( )    − + = = = deg v deg v deg v E 2    v V v V v V Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 8 Geoffrey Tien

Graph implementation Adjacency matrix Adjacency list Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 9 Geoffrey Tien

Adjacency matrix Data structure for set 𝐹 • A 𝑊 × 𝑊 array in which an element 𝑣, 𝑤 is true if and only if there is an edge from 𝑣 to 𝑤 – Note that 𝑊 is typically a set of integers used as array indices Tom Tom Shelly hamster popcorn Tom Shelly hamster Shelly popcorn hamster popcorn Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 10 Geoffrey Tien

Adjacency matrix examples With weights and/or directed edges 5 A B C A B C 1 1 3 1 2 2 5 D E D E 2 4 3 F G F G 8 A B C D E F G A B C D E F G A A B B C C D D E E F F G G Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 11 Geoffrey Tien

Adjacency matrix performance • What would be the complexity of these operations? – insertVertex(Vertex v) – removeVertex(Vertex v) – areAdjacent(Vertex v, Vertex u) – incidentEdges(Vertex v) – insertEdge(Vertex v, Vertex u) 𝑣 𝑤 𝑥 𝑨 – removeEdge(Vertex v, Vertex u) 𝑣 0 1 1 0 𝑣 𝑤 1 0 1 0 𝑥 1 1 0 1 𝑤 𝑥 𝑨 𝑨 0 0 1 0 • What is the required space usage of this representation? Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 12 Geoffrey Tien

Adjacency list • A 𝑊 -ary list (array) in which each entry stores a list (linked list) of all adjacent vertices … Tom Tom … Shelly Shelly … hamster hamster … popcorn popcorn Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 13 Geoffrey Tien

Adjacency list example 5 A B C A B C 1 1 3 1 2 2 5 D E D E 2 4 3 F G F G 8 A A B B C C D D E E F F G G Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 14 Geoffrey Tien

Adjacency list performance • What would be the complexity of these operations? – insertVertex(Vertex v) – removeVertex(Vertex v) – areAdjacent(Vertex v, Vertex u) – incidentEdges(Vertex v) – insertEdge(Vertex v, Vertex u) – removeEdge(Vertex v, Vertex u) 𝑣 𝑤 𝑥 𝑣 𝑤 𝑣 𝑥 𝑥 𝑣 𝑤 𝑥 𝑤 𝑥 𝑨 𝑨 𝑥 • What is the required space usage of this representation? Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 15 Geoffrey Tien

Graph performance Implementation affects performance! 𝑜 vertices 𝑛 edges No self-edges Adjacency list Adjacency matrix Space incidentEdges( 𝑤 ) areAdjacent( 𝑤 , 𝑥 ) insertVertex( 𝑦 ) removeVertex( 𝑤 ) insertEdge( 𝑤 , 𝑥 ) removeEdge( 𝑤 , 𝑥 ) Cinda Heeren / Andy Roth / Will Evans / March 25, 2020 16 Geoffrey Tien