Graph ADT

1
Graph ADT
2
Introduction

• relationships between people, things
• callgraph - which functions call which others
• dependence graphs - which variables are defined
  and used at which statements
•  Computer graphics
• Computer networks
• Maps and Driving direction applications
• Circuit Layout

3
Graph ADT

• a graph G is represented as G (V, E)
• V is a set of vertices v1, v2, , vn
• E is a set of edges e1, e2, , em where each
  ei connects two vertices (vi1, vi2)
• operations include
• iterating over vertices
• iterating over edges
• iterating over vertices adjacent to a specific
  vertex
• asking whether an edge exists connected two
  vertices

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Graph Representations

• 2-D matrix of vertices (marking edges in the
  cells)
• adjacency matrix
• List of vertices each with a list of adjacent
  vertices
• adjacency list

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Graph Representations

• Consider the following graph, give the following
• Mathematical representation of the graph
• Adjacent matrix representation
• Adjacent list representation

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Adjacency Matrix

• A V x V array in which an element (u, v) is
  true if and only if there is an edge from u to v

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Adjacency List

• A V-ary list (array) in which each entry stores
  a list (linked list) of all adjacent vertices

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Directed Graphs
0
2
1

• In directed graphs, edges have a specific
  direction

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Undirected Graphs

• In undirected graphs, The edges dont (edges are
  two-way) have direction
• Vertices u and v are adjacent if (u, v) ? E

0
2
1
10
Weighted Graphs
Each edge has an associated weight or cost.
100
0
2
500
1
200
The weight may represent distance, driving time,
cost of building the highway, etc
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Paths

• A path is a list of vertices v1, v2, , vn such
  that (vi, vi1) ? E for all 0 ? i lt n.

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Paths

• P1 A, D, E
• P2 B, E, D, C

100
C
30
50
50
200
10
20
150
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Path Length and Cost

• Path length
• the number of edges in the path
• Path cost
• the sum of the costs of each edge
• What are the path length and path cost of P1 and
  P2 on the previous slides?

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Simple Paths

• A simple path repeats no vertices (except that
  the first can be the last)
• P1 A, D, E
• P2 B, E, D, C
• Both P1 and P2 are simple paths.

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Cycles and Simple Cycles

• A cycle is a path that starts and ends at the
  same node
• p3 A, D, E, B, A is a cycle
• A simple cycle is a cycle that repeats no
  vertices except that the first vertex is also the
  last (in undirected graphs, no edge can be
  repeated)

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Connectivity

• Undirected graphs are connected if there is a
  path between any two vertices

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Connectivity

• Directed graphs are strongly connected if there
  is a path from any one vertex to any other
• Di-graphs are weakly connected if there is a path
  between any two vertices, ignoring direction

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Trees as Graphs

• Every tree is a graph with some restrictions
• the tree is directed
• there are no cycles (directed or undirected)
• there is a directed path from the root to every
  node

A
B
C
D
E
F
H
G
J
I
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Directed Acyclic Graphs (DAGs)

• DAGs are directed graphs with no cycles.

main()
Sort()
Swap()
Assign()
Compare()
Trees ? DAGs ? Graphs