Post on 05-Jan-2016
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)
810
911
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
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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
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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
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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