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S Sameen Fatima 1
GRAPH THEORY
Prof S Sameen FatimaDept of Computer Science and EngineeringOsmania University College of Engineering
Hyderabad [email protected]
Graph Theory
S Sameen Fatima 2
OVERVIEW
• BASICS• REPRESENTATION OF GRAPHS• MINIMUM SPANNING TREE• SEARCH ALGORITHMS• EXAMPLES
Graph Theory
BASICS
1. What Is a Graph?
2. Kinds of Graphs
3. Vertex Degree
4. Paths and Cycles
Graph Theory S Sameen Fatima 3
The KÖnigsberg Bridge Problem • Königsber is a city on the Pregel river in
Prussia
• The city occupied two islands plus areas on both banks
• Problem: Whether they could leave home, cross every bridge exactly once, and return home.
Graph Theory S Sameen Fatima 4
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A Model
• A vertex : an island
• An edge : a path(bridge) between two islands
Graph Theory S Sameen Fatima 5
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General Model
• A vertex : an object
• An edge : a relation between two objects
Graph Theory S Sameen Fatima 6
common
member
Committee 1 Committee 2
What Is a Graph?
• A graph G is an ordered pair (V, E) consisting of:– A vertex set V = {W, X, Y, Z}– An edge set E = {e1, e2, e3, e4, e5, e6, e7}
Graph Theory S Sameen Fatima 7
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What Is a Graph?
• A graph, G is an ordered triple (V, E, f)
consisting of– V is a set of nodes, points, or vertices. – E is a set, whose elements are known as
edges or lines. – f is a function that maps each element
of E to an unordered pair of vertices in V.
Graph Theory
Loop, Multiple edges
• Loop : An edge whose endpoints are equal
• Multiple edges : Edges have the same pair of endpoints
Graph Theory S Sameen Fatima 9
loop
Multiple edges
Graph Theory S Sameen Fatima 10
Simple Graph
Simple graph : A graph has no loops or multiple
edges
loopMultiple edges
It is not simple. It is a simple graph.
Adjacent, neighbors• Two vertices are adjacent and are
neighbors if they are the endpoints of an edge
• Example:– A and B are adjacent– A and D are not adjacent
Graph Theory S Sameen Fatima 11
A B
C D
Finite Graph, Null Graph
• Finite graph : an graph whose vertex set and edge set are finite
• Null graph : the graph whose vertex set and edges are empty
Graph Theory S Sameen Fatima 12
Connected and Disconnected
• Connected : There exists at least one path between two vertices
• Disconnected : Otherwise
• Example: – H1 and H2 are connected
– H3 is disconnected
Graph Theory S Sameen Fatima 13
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Complete Graph
• Complete Graph: A simple graph in which every pair of vertices are adjacent
• If no of vertices = n, then there are n(n-1) edges
Graph Theory
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Sparse/Dense Graph
• A graph is sparse if | E | | V |• A graph is dense if | E | | V |2.
Graph Theory
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Directed Graph (digraph)
In a digraph edges have directions
Graph Theory
S Sameen Fatima 17
Weighted Graph
Weighted graph is a graph for which each edge has an associated weight, usually given by a weight function w: E R.
Graph Theory
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Planar Graph
• Can be drawn on a plane such that no two edges intersect
Graph Theory
Complement
Complement of G: The complement G’ of a simple graph G :– A simple graph– V(G’) = V(G) – E(G’) = { uv | uv E(G) }
Graph Theory S Sameen Fatima 19
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Subgraphs
• A subgraph of a graph G is a graph H such that:– V(H) V(G) and E(H) E(G) and– The assignment of endpoints to edges
in H is the same as in G.
Graph Theory S Sameen Fatima 21
Subgraphs
• Example: H1, H2, and H3 are subgraphs of G
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Bipartite Graphs• A graph G is bipartite if V(G) is the union
of two disjoint independent sets called partite sets of G
• Also: The vertices can be partitioned into two sets such that each set is independent
• Matching Problem
• Job Assignment Problem
Graph Theory S Sameen Fatima 23
Workers
Jobs
Boys
Girls
Chromatic Number
• The chromatic number of a graph G, written x(G), is the minimum number of colors needed to label the vertices so that adjacent vertices receive different colors
Graph Theory S Sameen Fatima 24
Red
Green
Blue
Bluex(G) = 3
Maps and coloring
• A map is a partition of the plane into connected regions
• Can we color the regions of every map using at most four colors so that neighboring regions have different colors?
• Map Coloring graph coloring– A region A vertex– Adjacency An edge
Graph Theory S Sameen Fatima 25
Scheduling and Graph Coloring
• Two committees can not hold meetings at the same time if two committees have common member
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common
member
Committee 1 Committee 2
Scheduling and Graph Coloring• Model:– One committee being represented by a
vertex– An edge between two vertices if two
corresponding committees have common member
– Two adjacent vertices can not receive the same color
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common
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Scheduling and Graph Coloring
• Scheduling problem is equivalent to graph coloring problem
Graph Theory S Sameen Fatima 28
Common
MemberCommittee 1
Committee 2
Committee 3
Common Member
Different Color
No Common Member
Same Color OK
Same time slot OK
S Sameen Fatima 29
Degree
Degree: Number of edges incident on a node
Graph Theory
A
D E F
B C
The degree of B is 2.
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Degree (Directed Graphs)• In degree: Number of edges entering a node• Out degree: Number of edges leaving a node• Degree = Indegree + Outdegree
Graph Theory
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The in degree of 2 is 2 andthe out degree of 2 is 3.
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Degree: Simple Facts
• If G is a digraph with m edges, then
indeg(v) = outdeg(v) = m = |E |
• If G is a graph with m edges, then
deg(v) = 2m = 2 |E |
– Number of Odd degree Nodes is even
Graph Theory
S Sameen Fatima 32
Path• A path is a sequence of vertices such that there
is an edge from each vertex to its successor. • A path is simple if each vertex is distinct.• A circuit is a path in which the terminal vertex
coincides with the initial vertex
Graph Theory
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Simple path: [ 1, 2, 4, 5 ]Path: [ 1, 2, 4, 5, 4]Circuit: [ 1, 2, 4, 5, 4, 1]
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Cycle
• A path from a vertex to itself is called a cycle. • A graph is called cyclic if it contains a cycle; – otherwise it is called acyclic
Graph Theory
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Cycle
Other Paths• Geodesic path: shortest path
– Geodesic paths are not necessarily unique: It is quite possible to have more than one path of equal length between a given pair of vertices
– Diameter of a graph: the length of the longest geodesic path between any pair of vertices in the network for which a path actually exists
• Eulerian path: a path that traverses each edge in a network exactly once
The Königsberg bridge problem
• Hamilton path: a path that visits each vertex in a network exactly once
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Euclerian Path
• An undirected graph possesses an Euclerian Path if and only if it is connected and has either zero or two vertices of odd degree
OR• An undirected graph possesses an Euclerian Path if
and only if it is connected and its vertices are all of even degree
There is no Euclerian Path for the Konigsberg Bridge Problem
Graph Theory
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GRAPH REPRESENTATION
• Adjacency Matrix• Incidence Matrix• Adjacency List
Graph Theory
Adjacency, Incidence, and Degree
• Assume ei is an edge whose endpoints are (vj,vk)
• The vertices vj and vk are said to be adjacent
• The edge ei is said to be incident upon vj
• Degree of a vertex vk is the number of edges incident upon vk . It is denoted as d(vk)
Graph Theory S Sameen Fatima 37
eivj vk
Adjacency Matrix
• Let G = (V, E), |V| = n and |E|=m• The adjacency matrix of G written A(G),
is the |V| x |V| matrix in which entry ai,j is the number of edges in G with endpoints {vi, vj}.
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Adjacency Matrix• Let G = (V, E), |V| = n and |E|=m
• The adjacency matrix of G written A(G), is the |V| x |V| matrix in which entry ai,j is 1 if an edge exists otherwise it is 0
Graph Theory
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Adjacency Matrix (Weighted Graph)• Let G = (V, E), |V| = n and |E|=m
• The adjacency matrix of G written A(G), is the |V| x |V| matrix in which entry ai,j is weight of the edge if it exists otherwise it is 0
Graph Theory
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Incidence Matrix
• Let G = (V, E), |V| = n and |E|=m• The incidence matrix M(G) is the |V| x
|E| matrix in which entry mi,j is 1 if vi is an endpoint of ei and otherwise is 0.
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Adjacency List Representation
• Adjacency-list representation – an array of |V | elements, one for each vertex in V– For each u V , ADJ [ u ] points to all its adjacent
vertices.
Graph Theory
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Adjacency List Representation for a Digraph
Graph Theory
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Minimum Spanning Tree
Graph Theory
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Minimum Spanning Tree
• What is MST?
• Kruskal's Algorithm
• Prim's Algorithm
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A spanning tree of a graph is just a subgraph that contains all the vertices and is a tree.
A graph may have many spanning trees.
or or or
Some Spanning Trees from Graph AGraph A
Spanning Trees
Graph Theory
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All 16 of its Spanning TreesComplete Graph
Graph Theory
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Minimum Spanning Trees
The Minimum Spanning Tree for a given graph is the Spanning Tree of minimum cost for that graph.
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Kruskal's Algorithm
This algorithm creates a forest of trees. Initially the forest consists of n single node trees (and no edges). At each step, we add one edge (the cheapest one) so that it joins two trees together. If it were to form a cycle, it would simply link two nodes that were already part of a single connected tree, so that this edge would not be needed.
Graph Theory
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The steps are:
1. The forest is constructed - with each node in a separate tree. 2. The edges are placed in a priority queue. 3. Until we've added n-1 edges, 1. Extract the cheapest edge from the queue, 2. If it forms a cycle, reject it, 3. Else add it to the forest. Adding it to the forest will join two trees together.
Every step will have joined two trees in the forest together, so that at the end, there will only be one tree in T.
Graph Theory
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(in reality they are placed in a priority queue - not sorted - but sorting them
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Prim's Algorithm
This algorithm starts with one node. It then, one by one, adds a node that is unconnected to the new graph, each time selecting the node whose connecting edge has the smallest weight out of the available nodes’ connecting edges.
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The steps are:
1. The new graph is constructed - with one node from the old graph. 2. While new graph has fewer than n nodes, 1. Find the node from the old graph with the smallest connecting
edge to the new graph, 2. Add it to the new graph
Every step will have joined one node, so that at the end we will have one graph with all the nodes and it will be a minimum spanning tree of the original graph.
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Old Graph New Graph
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Graph Theory
S Sameen Fatima 91
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Complete Graph Minimum Spanning Tree
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Graph Theory
S Sameen Fatima 101
Examples
• Cost of wiring electronic components• Shortest route between two cities.• Shortest distance between all pairs of cities in
a road atlas.• Matching / Resource Allocation• Task scheduling
Graph Theory
S Sameen Fatima 102
Examples
• Flow of material – liquid flowing through pipes– current through electrical networks– information through communication networks– parts through an assembly line
• In Operating systems to model resource handling (deadlock problems)
• In compilers for parsing and optimizing the code.
Graph Theory
S Sameen Fatima 106
Graph Algorithms
Graph Theory
S Sameen Fatima 107
Search Algorithms
• Breadth First Search• Depth Dirst Search
Graph Theory
S Sameen Fatima 108
Breadth-First Search(BFS)
1. open (initial state).2. If open is empty , report failure , stop.3. s pop ( open )4. If s is a solution , report s, stop.5. succs successors(s).6. Add succs to tail of open.7. go to 2.
Graph Theory
S Sameen Fatima 109
Space for BFS
Space is calculated in terms of open list. In the worst case:The solution may be the rightmost node at the last levelAt the last level (level d) the no. of nodes = bd
Therefore, Total no: of nodes in the open list = bd
Space O (bd)
Graph Theory
S Sameen Fatima 110
Time for BFS
In the worst case, the solution will be the right most node at depth d, that is all the nodes would be expanded upto depth d.
No. of nodes processed at 1st level = 1No. of nodes processed at the 2nd level = bNo: of nodes processed at the 3rd level = b2
…………..No. of nodes processed at the dth level = bd
Therefore,Total no. of nodes processed = 1 + b + b2 +……..bd. = b ( bd – 1 ) (b – 1 ) O ( bd ) (ignoring 1 in comparison to b)
Graph Theory
S Sameen Fatima 111
Depth-First Search(DFS)
1. open ( initial state )2. If open is empty , report failure ,stop.3. s pop ( open )4. If s is a solution, report s, stop.5. succs successors (s).6. add succs to head of open7. go to 2.
Graph Theory
S Sameen Fatima 112
Space for DFS
Space is calculated in terms of open list. In the worst case: At the last level (level d) the no. of nodes = b At each of the preceding (d-1) levels i.e., 1, 2, 3, …., (d-1), the no. of nodes = b-1 Therefore, Total no: of nodes at the preceding (d-1) levels = (d-1)(b-1) Space b+ (d-1) ( b-1) b + db – d – b + 1 d ( b – 1) + 1 bd O ( d ) ( In terms of open list )
Graph Theory
S Sameen Fatima 113
Time for DFS
In the worst case, the solution will be the right most node at depth d, that is all the nodes would be expanded upto depth d.
No. of nodes processed at 1st level = 1No. of nodes processed at the 2nd level = bNo: of nodes processed at the 3rd level = b2
…………..No. of nodes processed at the dth level = bd
Therefore,Total no. of nodes processed = 1 + b + b2 +……..bd. = b ( bd – 1 ) (b – 1 ) O ( bd ) (ignoring 1 in comparison to b)
Graph Theory
S Sameen Fatima 114
Depth-First Iterative Deepening (DFID)
The depth-first iterative deepening algorithm combines the advantage of low space requirement of depth first search (DFS) and advantage of finding an optimal solution of the breadth first search (BFS)
time requirement, which is the same for both BFS and DFS 1. d 12. result depth first (initial state, d)3. (Comment: try to find a solution of length d using depth first search)4. If result ≠ NIL, report it, stop5. d d+ 16. go to 2
Graph Theory
S Sameen Fatima 115
Time requirement for DFID
On a search to depth d =1, b nodes are visited. For d = 2 , b1 + b2 nodes are visited and so on. At d = k, b1 + b2 + …….. + bk nodes are visited Let’s define the cost of DFID as a recurrence relation DFID(1) = b1DFID(k) = + DFID(k-1) This expands to
bk + bk-1 + bk-2 + ………………….. + b1
bk-1 + bk-2 + ………………….. + b1
bk-2 + ………………….. + b1
……
…….
……..
___________________________________________________________________
bk +2bk-1 +3bk-2 + ………………….. +kb1
= bk
For large k the above expression asymptotes to bk (as the expression in the parenthesis asymptotes to 1). Hence time required for DFID is O(bk)
Graph Theory
S Sameen Fatima 116
Examples
• Shortest route between two cities.• Shortest distance between all pairs of cities in
a road atlas.• Matching / Resource Allocation• Task scheduling• Cost of wiring electronic components• Visibility / Coverage
Graph Theory
S Sameen Fatima 117
Examples
• Flow of material – liquid flowing through pipes– current through electrical networks– information through communication networks– parts through an assembly line
• In Operating systems to model resource handling (deadlock problems)
• In compilers for parsing and optimizing the code.
Graph Theory
S Sameen Fatima 118
Thank you
Graph Theory
