Data structures and algorithms

Nadar saraswathi college of arts & science,theni.
Department of cs & it
data structures and algorithms
PRESENTED BY
G.KAVIYA
M.SC(IT)
TOPIC:TRAVELLING SALESMAN
PROBLEM.

Travelling Salesman Problem using Branch and
Bound
Travelling salesman problem:
 A set of some cities.
 Distance between every pair of cities.
Travelling salesman problem states-
 A salesman has to visit every city exactly once.
 He has to come back to the city from where he starts his journey.
 What is the shortest possible route that the salesman must
follow to complete his tour?

EXAMPLE:
The following graph shows a set of cities and distances between every pair of
cities-
If salesman starting city is A, then a TSP tour in the given is-
A  B  D  C  A
Cost of the tour
= 10 + 25 + 30 + 15
=80 units.

PRACTICE PROBLEM BASED ON TRAVELLING
SALESMAN PROBLEM USING BRANCH AND BOUND
APPROACH-
Problem-
Solve Travelling Salesman Problem using Branch and Bound Algorithm in the
following graph-

Solution-
Step-01:
Write the initial cost matrix and reduce it-
Row Reduction-
Consider the rows of above matrix one by one.
If the row already contains an entry ‘0’, then-
 There is no need to reduce that row.

If the row does not contains an entry ‘0’, then-
 Reduce that particular row.
 Select the least value element from that row.
 Subtract that element from each element of that row.
 This will create an entry ‘0’ in the row, thus reducing that row.
Following this, we have-
• Reduce the elements of row-1 by 4.
Cont;

Cont;
Performing this, we obtain the following row-reduced matrix-
Column Reduction-
Consider the columns of above row-reduced matrix one by one.
If the contains already contains an entry ‘0’, then-
 There is no need to reduce that column.

Cont;
If the column does not contains an entry ’0’, then-
 Reduce that particular column.
 Select the least value element from that column.
 Subtract that element from each element of that column.
 This will create an entry ‘0’ in the row, thus reducing that column.
Following this, we have-
• There is no need to reduce column-1.
• Reduce the elements of column-3 by -1.

Cont;
Performing this, we obtain the following column-reduced matrix-
Finally, the initial distance matrix is completely reduced.
Now, we calculate the cost of node-1 by adding all the reduction elements.
Cost(1)
=sum of all reduction elements
=4 + 5 + 6 + 2 + 1
=18

Cont;
Step-02:
o We consider all other vertices one by one.
o We select the best vertex where we can land upon to minimize the tour cost.
Choosing To Go To Vertex-B: Node-2(Path A  B)
• From the reduced matrix of step-1, M[A,B] = 0
• Set row-A and column-B to ∞
• Set M[B,A] = ∞

Cont;
Now, resulting cost matrix is-
Now,
• We reduce this matrix.
• Then, we find out the cost of node-02.

Cont;
Row Reduction-
 We can not reduce row-1 as all its elements are ∞.
 Reduce all the elements of row-2 by 13.
 There is no need to reduce row-3.

Cont;
Column Reduction-
 Reduction the elements of column-1 by -5.
 We can not reduce column-2 as all its elements are ∞.
 There is no need to reduce column-3.
Performing this, we obtain the following column-reduced matrix-

Cont;
Finally, the matrix is completely reduced.
Now, we calculate the cost of node-2.
Cost(2)
=Cost(1) + Sum of reduction elements + M[A,B]
=18 + (13 + 5) +0
=36
Choosing To Go To Vertex-C: Node-3 (Path A  C)
 From the reduced matrix of step-01,M[A,C]=7
 Set row-A and column-C to ∞.
 Set M[C,A] = ∞.

Cont;
Now,
• We reduce this matrix.
• Then, we find out the cost of node-03.

Cont;
Row Reduction-
 We can not reduce row-1 as all its elements are ∞.
Thus, the matrix is already row-reduced.
Column Reduction-
 We can not reduce column-3 as all its elements are ∞.
Thus, the matrix is already column-reduced.

Cont;
Cost(3)
=Cost(1) + Sum of reduction elements + M[A,C]
=18 + 0 + 7
=25
Choosing To Go To Vertex-D: Node-4 (Path A  D)
• From the reduced matrix of step-01,M[A,D] = 3
• Set row-A and column-D to ∞.
• Set M[D,A] = ∞.

Cont;
Now,
 We reduce this matrix.
 Then, we find out the cost of node-04.

Cont;
Row Reduction-
• We can not reduce row-1 as all its elements are ∞.
• There is no need to reduce row-2.
• Reduce all the elements of row-3 by 5.

Cont;
Column Reduction-
• We can not reduce column-4 as all its elements are ∞.
Thus, the matrix is already column-reduced.
Cost(4)
=Cost(1) + Sum of reduction elements + M[A,D]
=18 + 5 + 3
=26

Cont;
Thus, we have-
 Cost (2) = 36 (for path A  B)
 Cost (3) = 25 (for path A  C)
 Cost (4) = 26 (for path A  D)
We choose the node with the lowest cost.
Since cost for node-3 is lowest, so we prefer to visit node-3.
Thus, we choose node-3 i.e, path A  C

Cont;
Step-03:
We explore the vertices B and D from node-3.
We now start from the cost matrix at node-3 which is-
Cost(3) = 25

Cont;
Choosing To Go To Vertex-B: Node-5 (Path A  C  B)
• From the reduce matrix of step-02,M[C,B] = ∞.
• Set row-C and column-B to ∞.
• Set M[B,A] = ∞.

Cont;
Now,
Row Reduction:

Cont;
Column Reduction:

Cont;
Thus, the matrix is already column reduced.
Cost(5)
=cost(3) + Sum of reduction elements + M[C,B]
=25 + (13 + 8) + ∞
= ∞.

Cont;
Choosing To Go To Vertex-D: Node-6 (Path A  C  D)
• From the reduced matrix of step-02,M[C,D] = ∞.
• Set row-C and column-D to ∞.
• Set M[D,A] = ∞.

Cont;
Now,
Row Reduction:
Thus, the matrix is already row reduced.

Cont;
Column Reduction:
Thus, the matrix is already column reduced.

Cont;
Cos(6)
=cost(3) + Sum of reduction elements + M[C,D]
=25 + 0 + 0
=25
Thus, we have-
 Cost(5) = ∞ (for path A  C  B)
 Cost(6) = 25 (for path A  C  D)
We choose the node with the lowest cost.
Since cost for node-6 is lowest, so we prefer to visit node-6.
Thus, we choose node-6 i.e, path C  D

Cont;
Step-04:
We explore vertex B from node-6.
We start will the cost matrix at node-6 which is-
Cost(6) = 25

Cont;
Choosing To Go To Vertex-B: Node-7 (Path A  C  D  B)
• From the reduced matrix of step-03, M[D,B] = 0.
• Set row-D and column-B to ∞.
• Set M[B,A] = ∞.

Cont;
Now,
 We reduce the matrix.
 Then, we find out the cost of node-7.
Row Reduction:
• We can not reduce row reduce row-1 as all its elements are ∞.

Cont;
Column Reduction:
Thus, the matrix is already column reduce.
All the entries have become ∞.

Cont;
Now, we calculate the cost node-7
Cost(7)
=cost(6) + Sum of reduction elements + M[D,B]
=25 + 0 + 0
=25
Thus,
 Optimal path is: A C  D  B  A
 Cost of Optimal path is = 25 units.

EFFICIENCY CONSIDERATION:
 When doing branch and bound, efficiency considerations usually
demand that a priority queue be used to hold the nodes that have
been generated but not yet processed.
 The heap is often an ideal structure for implementing a large priority
queue.
 Branch and bound algorithms may be less than straightforward to
program. Also, it may not be tractable to calculate precise estimates of
the big -0 or big -0 of a branch and bound algorithm.
 Nevertheless, branch and bound techniques are well worth
considering and using in many applications, because they have the
potential to greatly reduce the amount of work involved in computing.

 A good bounding function is one that can be calculated quickly and
which has a high probability of diverting the search toward the
better solutions.
 It is something of an “art” to design a good bounding function.
Sometimes it requires considerable expertise. Consider the problem
of evaluating the worth of a board position in the game of chess, or of
evaluating the standing of a nation in the world’s economy.

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