Week -7-8 Topic - Graph Algorithms CSE

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Week -7-8Topic - Graph AlgorithmsCSE
5311

• Prepared by-
• Sushruth Puttaswamy
• Lekhendro Lisham

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Contents

• Different Parts of Vertices used in Graph
  Algorithms
• Analysis of DFS
• Analysis of BFS
• Minimum Spanning Tree
• Kruskals Algorithm
• Shortest Paths
• Dijkshitras Algorithm

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Different Parts of Vertices (used in Graph
Algorithms)

• Vertices of the Graph are kept in 3 parts as
  below
• Visited Vertices already selected.
• Fringe Vertices to be considered for
• next selection.
• Unvisited Vertices yet to consider as a
  possible candidate.

Visited
Fringe
Unvisited

• For DFS, the Fringe Part is kept in a STACK
  while for BFS
• QUEUE is used.

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Brief Description of DFS

• Explores a graph using greedy method.
• Vertices are divided into 3 parts as it proceeds
  i.e. into Visited, Fringe Unvisited
• Fringe part is maintained in a Stack.
• The path is traced from last visited node.
• Element pointed by TOS is the next vertex to be
  considered by DFS algorithm to select for Visited
  Part.

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Data Structures use in DFS
a b e
d c Graph
with 5 Vertices
b
V F U F U
a b c d e

d
a
c
e

b
e
d

e
a
c


TOS
b
b
d
c
d


(U/V/F) 1-D Array
Adjacency list
Fringe (Stack)
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Analysis of DFS

• For a Graph G(V, E) and n V mE
• When Adjacency List is used
• Complexity is O(mn)
• When Adjacency Matrix is used
• This again requires a stack to maintain the
  fringe an array for maintaining the state of a
  node.
• Scanning each row for checking the connectivity
  of a Vertex is in order O(n).
• So, Complexity is O(n2).

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Applications of DFS

• To check if a graph is connected. The algorithm
  ends when the Stack becomes empty.
• Graph is CONNECTED if all the vertices are
  visited.
• Graph is DISCONNECTED if one or more vertices
• remained unvisited.
• Finding the connected components of a
  disconnected graph. Repeat 1 until all the
  vertices become visited. Next vertex can also be
  taken randomly.
• Check if a graph is bi-connected (i.e. if 2
  distinct paths exist between any 2 nodes.).
• Detecting if a given directed graph is strongly
  connected( path exists between a-b b-a).
• Hence DFS is the champion algorithm for all
  connectivity problems

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Analysis of BFS

• Fringe part is maintained in a Queue.
• The trees developed are more branched wider.
• When Adjacency List is used
• Complexity is O(mn)
• When Adjacency Matrix is used
• This requires a Queue to maintain the fringe an
  array for maintaining the state of a node.
• Scanning each row for checking the connectivity
  of a Vertex is in order O(n).
• So, Complexity is O(n2).

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Minimum Spanning Tree (MST)

• A spanning tree of a connected Graph G is a
  sub-graph of G which covers all the vertices of
  G.
• A minimum-cost spanning tree is one whose edge
  weights add up to the least among all the
  spanning trees.
• A given graph may have more than one spanning
  tree.
• DFS BFS give rise to spanning trees, but they
  dont consider weights.

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Properties MST

• The least weight edge of the graph is always
  present in the MST.
• Proof by contradiction

w3
w2
c
h
d
Let this be a MST. Also let us assume that the
shortest edge is not part of this tree. Consider
node g h. All edges on the path between these
nodes are longer.(w1,w2,w3) Let us take any edge
(ex-w1) delete it. We will then connect g h
directly. But this edge has a lighter weight then
already existing edges which is a contradiction.
w1
g
CONTRADICTION
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. Properties of MST

• For any pair of vertices a b, the edges along
  the path from a to b have to be greater than the
  weight of (a,b).

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Kruskals Algorithm

• An algorithm to find the minimum spanning tree of
  connected graph.
• It makes use of the previously mentioned
  properties.
• Step 1 Initially all the edges of the graph are
  sorted based on their weights.
• Step2 Select the edge with minimum weight from
  the sorted list in step 1.Selected edge shouldnt
  form a cycle. Selected edge is added into the
  tree or forest.
• Step 3 Repeat step 2 till the tree contains all
  nodes of the graph.

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. Kruskals Algorithm

• This algorithm works because when any edge is
  rejected it will be longer than the already
  existing edge(s).
• The Union-Find data structure is tailor made to
  implement this algorithm.
• Cycle formation between any 2 vertices a b is
  checked if Find(a)Find(b).
• Applet Demo Link.

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Analysis of Kruskals Alg.

• If an adjacency list is used.
• We have n vertices m edges the following steps
  are followed.
• Sort the edges mlogm
• Find operation 2m O(m)
• Union - n-1 O(n)
• O(mlogm) O(mlogn ) O(mlog n).
• If we use path compression then find union
  become m times inverse Ackermans function.
• But sorting always takes mlogm time.

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Shortest Paths

• The problem is to find shortest paths among
  nodes.
• There are 3 different versions of this problem as
  shown below
• Single source single destination shortest path
  (SSSP)
• Single source all destinations shortest path
  (SSAP).
• All pairs shortest path.
• For un-weighted graphs
• BFS can be used for 1.
• BFS can used to for 2.
• Running BFS from all nodes is presently the best
  algorithm for 3.

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Dijkshitras Algorithm

• This is an algorithm for finding the shortest
  path in a weighted graph.
• It finds the shortest path from a single source
  node to all other nodes.
• The shortest path from a single node to all
  destinations is a tree.
• The following applet gives a very good
  demonstration of the Dijkshitras Algorithm. It
  finds the shortest distances from 1 city to all
  the remaining cities to which it has a path.
• Applet

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. Dijkshitras Algorithm

• DIJKSTRA (G, w, s)
• INITIALIZE-SINGLE-SOURCE(G, s)
• S ? ø
• Q ? VG
• while Q ? ø
• do u ? EXTRACT-MIN(Q)
• S ? S U u
• for each vertex v ? Adju do
• if dv gt du w (u, v)
• then dv ? du w (u, v)
• p v ? u
• The algorithm repeatedly selects the
  vertex u with the minimum shortest-path estimate,
  adds u to S, and relaxes all edges leaving u.

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. Analysis of Dijkshitras Alg.

• For a Graph G(V, E) and n V mE
• When Binary Heap is used
• Complexity is O((mn) log n)
• When Fibonacci Heap is used
• Complexity is O( m n log n)

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• Thank you.