APPLICATIONS OF GRAPHS

GRAPHS• Graph theory has turned out to be a vast area with

innumerable applications in the field of social networks , data organization , communication network and so on…

• We have considered here 1.Dijkstra’s algorithm 2. Fingerprint classification using graph theory

DIJKSTRA’S ALGORITHM

DIJKSTRA’S ALGORITHM

• Djikstra's algorithm (named after its discover, E.W. Dijkstra) solves the problem of finding the shortest path from a point in a graph (the source) to a destination.

• It turns out that one can find the shortest paths from a given source to all points in a graph in the same time, hence this problem is sometimes called the single-source shortest paths problem.

• Given a network of cities and the distances between them, the objective of the single-source, shortest-path problem is to find the shortest path from a city to all other cities connected to it.

• The network of cities with their distances is represented as a weighted graph.

• Let V be a set of N cities (vertices) of the digraph. Here the source city is 1.

• The set T is initialized to city 1.• The DISTANCE vector, DISTANCE [2:N] initially records the

distances of cities 2 to N connected to the source by an edge (not path!)

110

3

4

5

2

1

60 2030

20 40

70

SOURCE

• If there is no edge directly connecting the city to source, then we initialize its DISTANCE value to ‘∞’

• Once the algorithm completes iterations, the DISTANCE vector holds the shortest distance of two cities 2 to N from the source city 1

• It is convenient to represent the weighted digraph using its cost matrix COSTN x N

• The cost matrix records the distances between cities connected by an edge.

• Dijkstra’s algorithm has a complexity of O(N2) where N is the number of vertices (cities) in the weighted digraph

DIJKSTRA’S ALGORITHMProcedure DIJKSTRA_SSSP(N, COST)

/* N is the number of vertices labeled {1,2,3,….,N} of the weighted digraph. If there is no edge then COST [I, j] = ∞ */

/* The procedure computes the cost of the shortest path from vertex 1 the source, to every other vertex of the weighted digraph */

T = {1}; /* initialize T to source vertex */ for i = 2 to N do DISTANCE [i] = COST [1,i];

/* initialize DISTANCE vector to the cost of the edges connecting vertex I with the source vertex 1. If there is no edge then COST[1,i] = ∞ */

for i = 1 to N-1 do Choose a vertex u in V – T such that DISTANCE [u] is a

minimum; Add u to T; for each vertex w in V – T do DISTANCE [w] = minimum ( DISTANCE [w] , DISTANCE [u] + COST [u ,w]); end end end DIJKSTRA_SSSP

EXAMPLE• Consider the weighted graph of cities and its cost matrix

given below.

3

4

5

2

1

60 2030

20 40

70

1 2 3 4 5

1 0 20 ∞ 40 110

2 ∞ 0 60 ∞ ∞

3 ∞ ∞ 0 ∞ 20

4 ∞ ∞ 30 0 70

5 ∞ ∞ ∞ ∞ 0

WEIGHTED GRAPH COST MATRIX C5X5

SOURCE

Trace of The Table following that shows the trace of the Dijkstra’s algorithm

• The DISTANCE vector in the last iteration records the shortest distance of the vertices {2,3,4,5} from the source vertex 1.

• To reconstruct the shortest path from the source vertex to all other vertices, a vector PREDECESSOR [1:N] where PREDECESSOR [v] records the predecessor of vertex v in the shortest path, is maintained.

• PREDECESSOR [1:N] is initialized to source for all v != source

• PREDECESSOR [1:N] is updated by

if (( DISTANCE [u] + COST [u, w]) < DISTANCE [w]) then PREDECESSOR [w] = u

soon after DISTANCE [w] = minimum ( DISTANCE [w] , DISTANCE [u] + COST [u ,w]) is computed in procedure DIJKSTRA_SSS

• To trace the shortest path we move backwards from the destination vertex, hopping on the predecessors recorded by the PREDECESSOR vector until the source vertex is reached

EXAMPLE• To trace the paths of the vertices from vertex 1 using

Dijkstra’s algorithm, inclusion of the statement updating PREDECESSOR vector results in

• To trace the shortest path from source 1 to vertex 5, we move in the reverse direction from vertex 5 hopping on the predecessors until the source vertex is reached. The shortest path is given by

PREDECESSOR(5)=3 PREDECESSOR(3)=4 PREDECESSOR(4)=1

Vertex 5 Vertex 3 Vertex 4 Source Vertex 1 Thus the shortest path between vertex 1 and vertex 5 is 1-4-3-5 and the distance is given by DISTANCE [5] is 90

FINGERPRINT RECOGNITION USING GRAPH REPRESENTATION

The three characteristics of FINGER PRINTS are:

1. There are no similar fingerprints in the world.2. Fingerprints are unchangeable.3. Fingerprints are one of the unique features for

identification systems.

FINGERPRINT TYPES

The lines that flow in various patterns across fingerprints are called Ridges and the space between ridges are Valleys.

Cont..

Types of patterns in Fingerprint.

1 and 2 are terminations. 3 is bifurcation.

MINUTE, CORE AND DELTA

Minute – The places at which the ridges intersects or ends.

Core – The places where the ridges form a half circle.Delta – The places where the ridges form a triangle.

DIFFERENT CLASSIFICATION

OLD METHOD

NEW METHOD

PROCESS FLOW

FINGERPRINT CAPTURE DEVICES

SEGMENTATION OF DIRECTIONAL IMAGE

SEGMENTATION

Graph can be shown by G index including four parameters of G= (V, E, μ,υ).

whereV is number of nodes.E is number of edges.μ is weight of nodes.υ is weight of edges.

CONSTRUCTION OF RELATED WEIGHT GRAPH

Some information can be used in constructing the graph related to a finger print like:

• Centre of gravity of regions.• The direction related to the elements of the various

regions.• The area of all the regions.• The distance between centres of gravity.• The perimeter of regions.

Wn = Area (Ri) where i = 1,2,3,……,n. Wn is the weight of nodes .

Ri is the specified region in block directional image.

We = (Adj − p) × (Node − d) × (Diff −v) where Adj-p is the boundary of two adjacent regions linking with an edge. Node-d is the distance difference between nodes that links by an

edge Diff-v is the phase difference or direction difference between two

regions of block directional image.

WEIGHTAGE TO NODES AND EDGES

The nodes are placed in the centre of gravity of each region. The nodes size vary according to the weight. The edges are shown by the lines and the thickness is proportional to the weight .

We combine the properties of the Graph and model of the Super Graph to form,

• A node for region with similar directions.• Its co-ordinates is the centre of gravity related to those

regions of the graph.

CONSTRUCTING SUPER GRAPH

Wsn = nΣi=1 Area (Ri)Where Area (Ri) is the area of all regions with similar directions. Ri includes regions with similar directions.

Wse = dis(sn) + Σ Adj-p(Ri,Rj)Where Wse is the weight of edges in super graph. dis(sn) is the distance between nodes of a super graph.

Adj-p is the sum of the adjacent perimeter between two regions.

WEIGHTAGE TO sNODES AND sEDGES

The obtained block directional image have four directions, so we have four nodes.

All the nodes are connected with the other three edges.

If the number of nodes are high, its takes much time to find a match.

SUPER GRAPHSUPER GRAPH

• Fingerprints are classified according to their structure.• A sample from each structure is taken for comparison.

Cost Function = (Σi (Wi.node – *Wi.node )) * (Σj (Wj.edge - *Wj.edge))

whereWi.node and Wj.edge are the node weight and edge weight ofthe Super graph.*Wi.node and *Wj.edge are the node weight and edge weight of model super graph.

RETRIEVING THE FINGERPRINTS

• Different classifications give different cost value.Among that the lowest cost value function is taken and the comparison of the fingerprint proceeds in that class.

• By this method high accuracy is achieved in short comparison time.

• Previously FBI used 3 major classifications for matching the fingerprints and they had many sub-classifications.

• But now they use around 10 major classifications and many sub-classifications which gives them a fast result.

SOME MORE INTERESTING APPLICATIONS…..

EPIDEMOLOGY• Networks model used to represent the spread of infectious diseases and design prevention and response strategies.

• Vertices represent individuals, and edges their possible contacts. It is useful to calculate how a particular individual is connected to others.

• Knowing the shortest path lengths to other individuals can be a relevant indicator of the potential of a particular individual to infect others.

SOCIAL NETWORKS

NETWORKS (ROADS, FLIGHTS, COMMUNICATIONS)

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JFK

HJK

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HNL

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