Data Structures & Algorithms

Graphs

Richard Newmanbased on book by R. Sedgewick

and slides by S. Sahni

Definitions

• G = (V,E)• V is the vertex set• Vertices are also called nodes• E is the edge set• Each edge connects two different

vertices. • Edges are also called arcs

Definitions

• Directed edge has an orientation (u,v)

• Undirected edge has no orientation {u,v}

• Undirected graph => no oriented edge

• Directed graph (a.k.a. digraph) => every edge has an orientation

u v

u v

(Undirected) Graph

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Directed Graph

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Application: Communication NW

Vertex = city, edge = communication link

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Driving Distance/Time Map

Vertex = cityedge weight = driving distance/time

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Street Map2

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Some streets are one way.

Complete Undirected Graph

Has all possible edges.

n = 1 n = 2 n = 3 n = 4

Also known as clique

Number Of Edges—Undirected Graph

• Each edge is of the form (u,v), u != v• Number of such pairs in an n = |V|

vertex graph is n(n-1)• Since edge (u,v) is the same as edge

(v,u), the number of edges in a complete undirected graph is n(n-1)/2

• Number of edges in an undirected graph is |E| <= n(n-1)/2

Number Of Edges—Directed Graph

• Each edge is of the form (u,v), u != v• Number of such pairs in an n = |V|

vertex graph is n(n-1)• Since edge (u,v) is NOT the same as

edge (v,u), the number of edges in a complete directed graph is n(n-1)

• Number of edges in a directed graph is |E| <= n(n-1)

Vertex Degree

Number of edges incident to vertex.

degree(2) = 2, degree(5) = 3, degree(3) = 1

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Sum Of Vertex Degrees

Sum of degrees = 2e (e is number of edges)

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In-Degree Of A Vertex

in-degree is number of incoming edges

indegree(2) = 1, indegree(8) = 0

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Out-Degree Of A Vertex

out-degree is number of outbound edges

outdegree(2) = 1, outdegree(8) = 2

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Sum Of In- And Out-Degrees

• each edge contributes 1 to the in-degree of some vertex and 1 to the out-degree of some other vertex

• sum of in-degrees = sum of out-degrees = e,

• where e = |E| is the number of edges in the digraph

Sample Graph Problems

• Path problems• Connectedness problems• Spanning tree problems• Flow problems• Coloring problems

Path FindingPath between 1 and 8

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Path length is 20

Path FindingAnother path between 1 and 8

Path length is 28

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Path FindingNo path between 1 and 10

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Connected Graph

• Undirected graph• There is a path between every pair of

vertices

Example of Not Connected

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Example of Connected

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Connected Component

• A maximal subgraph that is connected

• Cannot add vertices and edges from original graph and retain connectedness

• A connected graph has exactly 1 component

Connected Components

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Communication Network

Each edge is a link that can be constructed (i.e., a feasible link)

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Communication Network Problems

• Is the network connected?• Can we communicate between every

pair of cities?• Find the components• Want to construct smallest number of

feasible links so that resulting network is connected

Cycles And Connectedness2

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Removal of an edge that is on a cycle does not affect connectedness.

Cycles And Connectedness2

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Connected subgraph with all vertices and minimum number of edges has no cycles

Tree

• Connected graph that has no cycles• n vertex connected graph with n-1

edges

Spanning Tree

• Subgraph that includes all vertices of the original graph

• Subgraph is a tree• If original graph has n vertices, the

spanning tree has n vertices and n-1 edges

Minimum Cost Spanning Tree

Tree cost is sum of edge weights/costs

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A Spanning Tree

Spanning tree cost = 51

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Minimum Cost Spanning Tree

Spanning tree cost = 41.

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A Wireless Broadcast Tree

Source = 1, weights = needed power.Cost = 4 + 8 + 5 + 6 + 7 + 8 + 3 = 41

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Graph Representation

• Adjacency Matrix• Adjacency Lists

• Linked Adjacency Lists• Array Adjacency Lists

Adjacency Matrix

• Binary (0/1) n x n matrix, • where n = # of vertices• A(i,j) = 1 iff (i,j) is an edge

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1 2 3 4 512345

0 1 0 1 01 0 0 0 10 0 0 0 11 0 0 0 10 1 1 1 0

Adjacency Matrix Properties

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1 2 3 4 512345

0 1 0 1 01 0 0 0 10 0 0 0 11 0 0 0 10 1 1 1 0

• Diagonal entries are zero• Adjacency matrix of an undirected

graph is symmetric• A(i,j) = A(j,i) for all i and j

Adjacency Matrix (Digraph)

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1 2 3 4 5

1

2

3

4

5

0 0 0 1 0

1 0 0 0 1

0 0 0 0 0

0 0 0 0 1

0 1 1 0 0

•Diagonal entries are zero

•Adjacency matrix of a digraph need not be symmetric.

Adjacency Matrix

• n2 bits of space• For an undirected graph, may store

only lower or upper triangle (exclude diagonal).

• Space? (n-1)n/2 bits• Time to find vertex degree and/or

vertices adjacent to a given vertex? O(n)

Adjacency Lists

Adjacency list for vertex i is a linear list of vertices adjacent from vertex i

Graph is an array of n adjacency lists

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aList[1] = (2,4)

aList[2] = (1,5)

aList[3] = (5)

aList[4] = (5,1)

aList[5] = (2,4,3)

Linked Adjacency Lists

Each adjacency list is a chain.

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aList[1]

aList[5]

[2]

[3]

[4]

2 4

1 5

5

5 1

2 4 3

Array Length = n# of chain nodes = 2e (undirected graph)# of chain nodes = e (digraph)

Array Adjacency Lists

Each adjacency list is an array list

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aList[1]

aList[5]

[2]

[3]

[4]

2 4

1 5

5

5 1

2 4 3

Array Length = n# of list elements = 2e (undirected graph)# of list elements = e (digraph)

Weighted Graph Representations

• Cost adjacency matrix• C(i,j) = cost of edge (i,j)

• Adjacency lists • Each list element is a pair • (adjacent vertex, edge weight)

Paths

• Simple Path

• List of distinct vertices v0, v1, … , vn

• Each pair of successive vertices is an edge in E (vi, vi+1)

• Hamilton Path • Visit each node exactly once

• Euler Path• Visit each edge exactly once

Bipartite Graph

• Vertex set V can be partitioned into two disjoint subsets, V0 and V1

• No two vertices in Vi have an edge between them

Complete bipartite graphs

Hamilton PathDoes this graph have a Hamilton path?

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Yes! 2 or 7 to 10 Very hard in general

Euler PathDoes this graph have a Euler path?

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Yes! 9 to 10 Very easy in general

Euler Path• Bridges of Königsberg• People wanted to walk over all bridges

without crossing one twice

So they asked Euler …

Euler Path• Bridges of Königsberg• People wanted to walk over all bridges

without crossing one twice

Does this graph have a Euler path?

Euler Path

• Easy to determine existence• Graph must be connected• Degree of all nodes…• … must be even, except for two

• A little work to find the path• But also efficient…• Find path between odd nodes• Add loops as encountered

Graph Problems

• These two problems• Seem very similar• One is easy• The other is really hard!

• Classify graph problems• Easy• Tractable• Intractable• Unknown

Easy Graph Problems

• Simple, efficient algorithms exist• Linear or small polynomial time

• Simple connectivity• Strong connectivity in digraphs• Transitive closure• Minimum spanning tree• Single-source shortest path (SSSP)

Tractable Graph Problems

• Polynomial time algorithm is known… but

… hard to make into a practical program• Planarity

•Can graph be drawn without any lines representing edges intersecting?

• Matching•Largest subset of edges where no two connect to same vertex

• Even cycles in digraphs

Graph Planarity

• Kuratowski’s theorem:

For a graph to be non-planar

It must contain (after removal of degree-2 nodes) a subgraph isomorphic to either

6-node complete bipartite graph, or

5-clique

More Tractable Graph Problems

• Often tractable problems can be solved with general purpose algorithm through graph transformation

• Assignment (network-flow)• Bipartite weighted matching – minimum

weight perfect matching in bipartite graph• Edge-connectivity (network-flow)

• What is minimum number of edges whose removal will partition graph?

• Node-connectivity (network-flow)

More Tractable Graph Problems

• Mail carrier problem• Tour with minimum number of edges that

uses every edge at least once• Harder than Euler tour• Easier than Hamilton tour

Intractable Graph Problems

• No known polynomial time algorithm• NP-hard complexity class• However, may be polynomial time to

verify a solution (class NP)• Longest path (version of Hamilton Path)

• What is the longest simple path in G?• Independent Set

• Largest subset of vertices where no two have an edge between them

Intractable Graph Problems

• Colorability• Assign k colors to vertices such that no

edge connects two vertices of same color• Easy for k=2 (bipartite graph)

• Only even length cycles• Hard for k=3!

• Clique• What is largest complete subgraph?

(What is relationship to Independent Set?)

Graph Coloring

Can this graph be 2-colored?

No! Why not?

Odd cycle!

Can it be 3-colored?

Yes!

Graph Edge Coloring

Color each edge so that no two edges of same color are adjacent to same vertex

Chromatic Index = min # colors needed

Vizing’s Theorem: CI is either max degree or +1

Can it be 4-edge colored? Yes!

Graph Edge Coloring

• Despite the narrow range of possibilities• This problem is still intractable!!!!!

• There are polynomial time algorithms for bipartite graphs

• Brings up larger issue:• Can we solve exactly hard problem for

special case?• Can we “get close” for all cases?• How close is “close”?

Unknown Graph Problems

• Graph Isomorphism• Are two graphs identical other than the

names of their vertices?• Note that subgraph isomorphism is hard!

• How do you know this already?• Clique problem!

Summary

• Graph definitions, properties

• Graph representations

• Graph problems and classifications

• Next: Graph Search, Digraphs, DAGs