tsp presentation

8/13/2019 TSP Presentation

1/15

Data Structure and Algorithms

Presentation on

Travelling Salesman Problem Using Dynamic

Programing

Submitted To: Submitted By:

Ms. Smriti Sehgal Nawneet a!" Amit #umar Atul

Tyagi


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$ntroduction:

Definition: Given a set of cities and the distance

between each possible pair, the Travelling

Salesman Problem is to find the best possibleway of visiting all the cities exactly once and

returning to the starting point

In terms of graph theory, it means finding

the shortestHamiltonian cycleof a graph G.


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A Brie% &istory: The problem was first defined in the 1800s by

the Irish mathematician W.R. Hamilton and theBritish mathematicianThomas Kirkman.

The name Travelling Salesman Problem wasintroduced by mericanHassler Whiteney.

!esearchers from !"# $orporation create a

method to optimally sol%e T&' for () cities. In 1)*+, arp pro%es that T&' is "' complete.


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A''lications: Transportation

&chool Bus !outes Tra%eling &alesman

Technology -icrochip -anufacturing "etwor /ngineering

Biological Genome &euencing

T&' appears as a subproblem in many areas,such as DNA sequencing.


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Finding a Solution:

There are a few general methods for finding solutionsfor the TSP:

Brute Force Dynamic Programming Approximation


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ecurrence elation:

TSP (A") * The Minimum cost re+uired to go %rom ,erte- A to all

other emaining ,ertices is e-actly once and

coming bac to Source (S).

/ (A"S) $% *

TSP (A") *

/ (A"#) 0 TSP (#"12#3)

min

%or all # 4

5et Sbe the Source.

Dynamic Programming


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TSP Example: Let we have with us a Adjency Matrix. 1 2 !

1 " 1" 1# 2"

2 # " $ 1"

% 1 " 12

! & & $ "

Let 1 'e the S(urce). 1" 2#

*+1,2- TSP +2, /,!0-

# 1# 2#

TSP +1,/2,,!0- min *+1,- TSP +, /2,!0-

2" 2

*+1,!- TSP +!, /2,0-


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$ 2"

2# *+2,- TSP+, /!0-

TSP+2, /,!0- min

*+2,!- TSP+!, /0-

1" 1#

12 &

TSP/,!0 *+,!- *+!,1- 2"

$ %

TSP/!,0 *+!,- *+,1- 1#


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1 1&

2# *+,2- TSP+2, /!0-

TSP+, /2,!0- min

*+,!- TSP+!, /20-

12 1

1" &

TSP/2,!0 *+2,!- *+!,1- 1&

& #

TSP/!,20 *+!,2- *+2,1- 1


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& 1# 2 *+!,2- TSP+2, /0-

TSP+!, /2,0- min

*+!,- TSP+, /20-

$ 1&

$ %

TSP/2,0 *+2,- *+,1- 1#

1 # TSP/!,20 *+!,2- *+2,1- 1&


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S(, the Minimum ei3ht c(mes t( 'e #.

And the Path 4(ll(wed is).

1" 1" $ %

1 2 ! 1

S(, the t(tal ei3ht 1" 1" $ % #


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Time /om'le-city

1

+ 2 (

2 ( + ( 2 +

( 2 ( + + 2

!ecursi%e Tree ha%e 3n le%els4 where e%ery nodeha%e ma5imum 3n children4. &o, total number of

nodes are 67nn .


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S'ace /om'le-city

&pace $omple5city 9 n : nn

;here, n 9 Table.nn9 #ifferent function calls.

&o, &pace $omple5city 9 67nn.


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Summary

The TSP is a gra'h theory minimi6ation'roblem.

The com'le-ity o% the 'roblem (NP /om'lete) maes

it di%%icult to solve o'timally.

Solving TSP o'timally sheds light on other 'roblems

(P*NP).

7'timal solution carries a 'ri6e o% 89 Million

(Millennium Pri6e).


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Thank you.

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