Graph Algorithms

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Graph Algorithms
By Shruti Aggrawal, Preeti M. Palkar
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Table Of Contents

• Graphs
• Eulers Circuit Problem
• Graph Terminologies (Paths, Cycles, Subgraph,
  Connectivity, Spanning Tree, Forest)
• Depth First Search
• Breadth First Search
• Minimum Cost Spanning Tree
• Kruskals Algorithm

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Graphs

• Used to model relationships between objects.
• Graphs consist of nodes and edges.
• The edges may or may not have direction.
• If they dont have a direction then they are
  called as undirected graphs.
• If they have a direction then they are called as
  directed graphs.

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Examples of modeling with Graphs
(Directed Graph)
Graph of a road network (Undirected Graph)
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Eulers Circuit problem
Problem Whether it was possible to start walking
from anywhere in town and return to the starting
point by crossing all bridges exactly once. This
is an undirected graph so one can walk in any
direction. In this case edges cant repeat but
vertices can. Degree of node Number of Edges
connected to it. Example Degree of C 5. In the
Eulers circuit problem, if you enter a node, you
have to exit it too, in order to cover all paths
(edges). Hence all the nodes should have an even
degree. In the above case, the nodes do not have
even degree hence it is not possible to start
walking from anywhere in town and return to the
starting point by crossing all bridges exactly
once. For a graph to be Eulers circuit all
nodes degrees should be even.
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Graph Terminologies

• A Graph consists of a set 'V' of vertices (or
  nodes) and a set 'E' of edges (or links).
• A graph can be directed or undirected.
• Edges in a directed graph are ordered pairs. The
  order between the two vertices is important.
• Example (S,P) is an ordered pair because the
  edge
• starts at S and terminates at P.
• The edge is unidirectional
• Edges of an undirected graph form unordered
  pairs.
• A multigraph is a graph with possibly several
  edges between the same pair of vertices.
• Graphs that are not multigraphs are called simple
  graphs.

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Graphs Terminologies
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Graphs Terminologies

• The degree d(v) of a vertex v is the number of
  edges incident to v.
• d (A) three, d (D) two
• In directed graphs, indegree is the number of
  incoming edges at the vertex and outdegree is the
  number of outgoing edges from the vertex.
• The indegree of P is 2, its outdegree is 1.
• The indegree of Q is 1, its outdegree is 1.

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Paths and Cycles
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Paths and Cycles
It is similar to the traveling salesman problem.
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Subgraphs
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Spanning tree
A spanning tree of a graph G is a subgraph of G
that is a tree and contains all the vertices of G.
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Connectivity

• A graph is said to be connected if there is a
  path from any vertex to any other vertex in the
  graph.
• A forest is a graph that does not contain a
  cycle.
• A tree is a connected forest.
• A spanning forest of an undirected graph G is a
  subgraph of G that is a forest and contains all
  the vertices of G.
• If a graph G(V,E) is not connected, then it can
  be partitioned in a unique way into a set of
  connected subgraphs called connected components.
• A connected component of G is a connected
  subgraph
• of G such that no other connected subgraph of G
• contains it.

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Forest
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Graphs Representations in undirected graph

• Matrix
• 1 indicates that a link is present between the
  nodes.
• In case of an undirected graph we get a
  symmetrical matrix.

• Edges are counted twice.
• For a graph of n nodes, size of matrix is n2.
• Maximum number of edges n(n-1)/2, because n
  nodes are connected to n-1 nodes and counting
  each link once so divide by 2.

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Graphs Representations in directed graph

• List
• The 0s from the matrix representation are
  eliminated.
• It has varying row length.

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Depth First Search (DFS)

• The algorithm starts from a node, selects a
  node connected to it, then selects a node
  connected to this new node and so on, till no new
  nodes remain. The it backtracks to the latest
  node and discovers any new nodes connected to it.
• The data structure suitable for this purpose
  (LIFO) is a stack.
• Once stack is empty the algorithm ends.

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DFS

• DFS is an aggressive algorithm
• It finds a spanning tree
• Produces automatic ordering.
• Ordering not unique
• DFS is written recursively as it uses stack
• Application-connected components

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Algorithm for DFS
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Implementing DFS

• To draw a depth first search spanning tree lets
  start with A. Put A in the stack.
• Next we can pick either of D,E,B. Order does not
  matter. In this example we r picking B. Put B in
  stack.
• After B we can either pick C or E. In this case
  we put C in the stack. D is selected next and put
  in stack.
• After D there is just one edge to A. As A is
  already visited, we start back track. We pop out
  D and look at C.
• From C there is one node left which is E. Add E
  to stack.
• From E again there is no other node so we pop E
  out and look at C again. As there is no other
  node left which needs to be explored so we pop C
  out and similarly pop B and A out.
• Finally as the stack is empty algorithm comes to
  an end and we have corresponding tree .

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Running time of DFS

• V vertex set n
• E edges set O(n2)
• Running time in case of Adjacency List
  representation O(VE)
• We need to add V because there can be isolated
  nodes.
• Hence this is a linear time algorithm.
• If graph is very dense we can use matrix
  representation.
• Running time in case of Adjacency Matrix
  representation O(n2)

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Breadth First Search (BFS)

• Compared to DFS breadth first search is a simple
  algorithm
• Timidly tries one edge. Totally exhaust neighbors
  of a vertex then goes to next neighbors. It
  radiates in waves in balanced manner.
• Implemented using queues
• Whatever is in queues tells u what to explore
  next. Once the queue is empty algorithm comes to
  an end

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Algorithm for BFS

• Procedure BFS_Tree G(V,E)
• Input G (V,E) Q is a queue - initially empty
• x ?Q remove the front item of queue and
• denote it by x
• initially mark all vertices new
• Lx refers to the adjacency list of x.
• T ? 0
• Output The BFS tree T
• 1. v ?old v? V
• 2. insert (Q,v)
• 3. while Q is nonempty do
• 4. x ? Q
• 5. for each vertex w in Lx and
  marked new
• 6. T ? T ? x,w
• 7. w ? old
• 8. insert (Q,w)

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BFS
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BFS
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Implementing BFS using Queues

• Start from A. Add it to queue

• When visited A is removed from the queue, Add
  D,E,B in queue (order does not matter)

A
D
E
B
C
E
B

• Next when D is pulled out vertex C which is
  adjacent to D is added

A
C
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Running time of BFS

• G(V,E) , Vn, Em
• Running time of BFSO (n m)
• There is no cost to insert or delete
• When a vertex is inserted or deleted it is always
  at the back of the queue

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Application of BFS Shortest Path

• Using BFS Shortest path to all the vertices
• To find shortest path from A to vertices
  B,C,D,E,F we first create a spanning tree and
  from that we can find shortest path
• Figure 1 shows the graph and figure 2 shows the
  corresponding spanning tree.
• From A we can see the shortest path to reach E is
  through A to C to E

Figure 1
Figure 2
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DFS Versus BFS
Graph
DFS
BFS

• DFS trees usually have thin and tall structure.
  In case of complete graph (dense) it has one
  straight line all the way. BFS trees usually are
  bushy like a star.
• In both trees, number of vertices is same. Both
  are following the edges and hence can be used to
  find connected components.
• DFS can be recursively written. It is difficult
  to do so in case of BFS.

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• DFS and BFS break graph in 3 parts
• visited nodes that have been visited.
• Fringe nodes that it will consider to visit
  next.
• Remaining vertices
• For DFS, the fringe nodes are in the stack while
  for BFS they are in queue.
• DFS and BFS each create a different shape.
• BFS
• DFS

Visited
Fringed
Remaining
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MCST

• MCST is minimum cost spanning tree . The
  minimum-cost spanning tree (MCST) is one whose
  edge weights add up to the least among all the
  spanning trees
• Minimum cost Spanning tree is a fixed connected
  subgraph, containing all the vertices such that
  the sum of the costs of the edges in the subgraph
  is minimum also it does not contain any cycles.
• A given graph may have more than one spanning
  tree

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MCST
Consider a network of computers connected through
bidirectional links. Each link is associated with
a positive cost the cost of sending a message on
each link. The network is represented as an
undirected graph with positive costs on each
edge. In bidirectional networks we can assume
that the cost of sending a message on link does
not depend on the direction. Suppose we want to
broadcast a message to all the computers from an
arbitrary computer. The cost of the broadcast is
the sum of the costs of links used to forward the
message.
A
3
2
3
1
B
2
3
1
C
D
5
5
4
7
E
F
An undirected graph with positive costs on each
edge.
Graph with corresponding sorted order written
beside the edges
Minimum cost spanning tree
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MCST

• Why does the least cost edge have to be there in
  MCST?
• Proof using contradiction
• Assume minimum cost edge is not in the MCST.
• Assume it is between u and v(the dotted line
  shown in the graph).
• Since this edge is not in the MCST, there exist
  another path between u and v.
• Remove one edge (between x and y) and use the
  minimum cost edge.
• Now you get a cheaper tree.
• Hence least cost edge has to be there in MCST.

x
y
U
v
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Kruskals Algorithm
Graph is given, find MCST
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Kruskals Algorithm
A
3
3
3
2
1
2
B
1
2
1
6
4
C
D
6
4
3
4
5
4
7
5
7
5
E
2
8
F
8

• Figure on the left shows a graph with sorted
  order (inside text boxes) ,cost written beside
  the edges. Initially graph has only
  vertices.Starting from A we can see A to C has
  order one so this edge will be added first to
  spanning tree

• Edge AD and AB are next as their order is 2 and
  3 respectively

• Edge between C and D has order 4 but it creates
  a cycle between ACD therefore it is not added to
  the tree

• Next edge between D and F is added and edge
  between B and C is dropped as it creates cycle
  between A,B,C. Lastly edge between C,E is added
  and edge between E,F is dropped

• Edges that do not belong to the tree are the
  heavier ones than those that are in the tree.

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Kruskals Algorithm

• Running time
• Vn , Em
• Running time of sorting O (m log n)

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Example Why two already visited vertices are not
sufficient to check for a cycle
A
A
4
3
4
3
3
3
1
B
2
1
2
B
1
1
4
7
C
D
C
D
3
5
6
8
5
6
4
4
E
E
F
2
1
F
2
1

• Figure on the left shows a graph with sorted
  order (inside text boxes) written beside the
  edges plus the cost corresponding to each edge.
  Starting from A we can see A to C has order one
  so this edge will be added first to spanning tree

• Second in order is an edge between E and F so we
  add it.

• Next we add edge between A and D and one edge
  between A and B. Edge between C and D is
  discarded as it creates a cycle

• Next an edge between D and F is added even if D
  and F have been explored. Edge between B and C is
  dropped and edge between C and E is dropped

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Example When an edge is added when its not.