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

Graph implementation

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

Graph vocabulary

1. List the edges incident to vertex 𝑐 2. What is the degree of vertex 𝑒? 3. List the vertices adjacent to vertex 𝑗 4. Give a path from 0 to 7 5. Give a path from 𝑙 to ℎ 6. Vertices in the largest complete subgraph in 𝐻 7. How many connected components are in 𝐻? 8. How many edges in a spanning forest? 9. How many simple paths connect 0 and 9?

• 10. Can you draw 𝐻 with no edge crossings (as a

planar graph)?

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 2

Quiz yourself!

𝑐 𝑏 𝑑 𝑓 𝑒 𝑕 𝑔 𝑜 𝑛 𝑙 𝑘 𝑝 𝑚 𝑗 𝑟 𝑞 8 9 6 7 4 5 2 3 1 ℎ

𝐻

Weighted graphs

• In a weighted graph each edge

is assigned a weight

– Edges are labeled with their weights

• Each edge’s weight represents

the cost to travel along that edge

– The cost could be distance, time, money or some other measure – The cost depends on the underlying problem

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 3

1 2 4 3 1 3 5 2 2 3

Graph density

• A sparse graph has 𝑃 𝑊

edges

• A dense graph has Θ 𝑊2

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

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 4

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

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 5

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′

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 6

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

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 7

( )

E v

V v

2 deg =





𝐹 = 7

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:

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 8

( ) ( ) ( )

E v v v

V v V v V v

= = =

  

  +  −

deg 2 1 deg deg

Graph implementation

Adjacency matrix Adjacency list

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 9

Adjacency matrix

• 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

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 10

Data structure for set 𝐹

Tom Shelly hamster popcorn Tom Tom Shelly Shelly hamster hamster popcorn popcorn

Adjacency matrix examples

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 11

With weights and/or directed edges

A B C D E F G A B C D E F G

3 1 1 1 2 2 2 3 4 5 5 8

A B C D E F G A B C D E F G A B C D E F G A B C D E F G

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) – removeEdge(Vertex v, Vertex u)

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 12

𝑤 𝑥 𝑨 𝑣 𝑣 𝑤 𝑥 𝑨 𝑣 1 1 𝑤 1 1 𝑥 1 1 1 𝑨 1

• What is the required space usage of this representation?

Adjacency list

• A 𝑊 -ary list (array) in which each entry stores a list (linked

list) of all adjacent vertices

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 13

Tom Shelly hamster popcorn Tom Shelly hamster popcorn … … … …

Adjacency list example

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 14

A B C D E F G A B C D E F G

3 1 1 1 2 2 2 3 4 5 5 8

A B C D E F G A B C D E F G

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)

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 15

𝑤 𝑥 𝑨 𝑣

• What is the required space usage of this representation?

𝑣 𝑤 𝑥 𝑨 𝑤 𝑥 𝑣 𝑥 𝑤 𝑥 𝑣 𝑥

Graph performance

March 25, 2020 Cinda Heeren / Andy Roth / Will Evans / Geoffrey Tien 16

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)