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Session 15

 Dynamic Programming:Multistage Graph Problem

Course : T0034 !lgorithm Design " !nalysis

#ear : $013

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MULTISTAGE GRAPH

% Multistage Graph is a graph &ith special characteristics:1' Directe( Graph

$' )ach e(ge has &eight

3' *as only 1 source +calle( as s, an( 1 sin- +calle( as t,

4' Path .rom source to sin- consists some stages /1 to /-

5' !ll stages connect no(e in /i to no(e /i01 &here 1 i -

2' There are - stages &here - $

' )ach path .rom s to t is conse6uence o. choice -7$'

% Multistage Graph is a mo(eling that can be use( to sol8esome real problems'  )9ample: choosing proect to get ma9imum pro.it; inclu(ing

selecting steps to per.orm each tas-'

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MULTISTAGE GRAPH PROBLEM

% Multistage Graph Problem :

  Sortest path .in(ing problem .rom source to sin- in Multistage Graph'

  The problem is one o. goo( implementation o. Dynamic Programming'

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DP IN MULTISTAGE GRAPH PROBLEM

% Sol8ing o. Multistage Graph problem usingDynamic Programming in shortest path .rom ano(e to another is a shortest path o. pre8ious

stage a((e( by (istance o. one o. an e(geconnects to a stage'

% <or&ar( Metho(  Calculate (istance to .ront +to sin-,

% =ac-&ar( Metho(

  Calculate (istance to bac- +.rom source,

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FORWARD METHOD

%  !nalysis by calculating path .rom a node to sink 

% <ormula: cost(i,j) = min{c(j,k) + cost(i+1,k)}

% Calculation starts .rom no(es in stage -7$% cost+i, is (istance o. path .rom node  in stage i

to sin-+t,

% c+l, is (istance o. path .rom node  to node l

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FORWARD METHOD

cost(4,I) = c(I,L) = 7

cost(4,J) = c(J,L) = 8

cost(4,K) = c(K,L) = 11

cost(3,F) = min { c(F,I) + cost(4,I) | c(F,J) + cost(4,J) }

cost(3,F) = min { 12 + 7 | 9 + 8 } = 17

cost(3,G) = min { c(G,I) + cost(4,I) | c(G,J) + cost(4,J) }

cost(3,G) = min { 5 + 7  | 7 + 8 } = 12

cost(3,H) = min { c(H,J) + cost(4,J) | c(H,K) + cost(4,K) }

cost(3,H) = min { 1 + 8 | 8 + 11 } = 18

cost(2,!) = min { c(!,F) + cost(3,F) | c(!,G) + cost(3,G) | c(!,H) + cost(3,H) }

cost(2,!) = min { 4 + 17 | 8 + 12 | 11 + 18 } = 2cost(2,") = min { c(",F) + cost(3,F) | c(",G) + cost(3,G) }

cost(2,") = min { 1 + 17 | 3 + 12 } = 15

cost(2,#) = min { c(#,H) + cost(3,H) }

cost(2,#) = min { 9 + 18 } = 27

cost(2,$) = min { c($,G) + cost(3,G) | c($,H) + cost(3,H) }

cost(2,$) = min { % + 12 | 12 + 18 } = 18

cost(1,&) = min { c(&,!) + cost(2,!) | c(&,") + cost(2,") | c(&,#) + cost(2,#) | c(&,$) + cost(2,$) }

cost(1,&) = min { 7 + 2 | % + 15  | 5 + 27 | 9 + 18 } = 21

'ot*st t is &-"-G-I-L .it /istnc* 21

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BACKWARD METHOD

%  !nalysis by calculating path .rom source to a

node

% <ormula: bcost(i,j) = min{bcost(i–1,l) + c(l,j)}

% Calculation starts .rom no(es in stage 3

% bcost+i, is (istance o. path backward  .rom source 

+s, to node  in stage i

% c+l, is (istance o. path .rom node  to node l

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METODE BACKWARD

 0cost(2,!) = c(&,!) = 7

 0cost(2,") = c(&,") = %

 0cost(2,#) = c(&,#) = 5

 0cost(2,$) = c(&,$) = 9

 0cost(3,F) = min { c(!,F) + 0cost(2,!) | c(",F) + 0cost(2,") }

 0cost(3,F) = min { 4 + 7 | 1 + % } = 11

 0cost(3,G) = min { c(!,G) + 0cost(2,!) | c(",G) + 0cost(2,") | c($,G) + 0cost(2,$) }

 0cost(3,G) = min { 8 + 7 | 3 + % | % + 9 } = 9

 0cost(3,H) = min { c(!,H) + 0cost(2,!) | c(#,H) + 0cost(2,#) | c($,H) + 0cost(2,$) }

 0cost(3,H) = min { 11 + 7 | 9 + 5 | 12 + 9 } = 14

 0cost(4,I) = min { c(F,I) + 0cost(3,F) | c(G,I) + 0cost(3,G) }

 0cost(4,I) = min { 12 + 11 | 5 + 9 } = 14

 0cost(4,J) = min { c(F,J) + 0cost(3,F) | c(G,J) + 0cost(3,G) | c(H,J) + 0cost(3,H) }

 0cost(4,J) = min { 9 + 11 | 7 + 9 | 1 + 14 } = 1%

 0cost(4,K) = min { c(H,K) + cost(3,H) }

 0cost(4,K) = min { 8 + 14 } = 22

 0cost(5,L) = min { c(I,L) + 0cost(4,I) | c(J,L) + 0cost(4,J) | c(K,L) + 0cost(4,K) }

 0cost(5,L) = min { 7 + 14 | 8 + 1% | 11 + 22 } = 21

'ot*st t is &-"-G-I-L .it /istnc* 21

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SHORTEST PATH IN MULTISTAGE GRAPH

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EXERCISE

% <in( shortest path .rom no(e ! to no(e > usingDynamic Programming +.or&ar( metho( an(bac-&ar( metho(, ?

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REVIEW

% M@>TAST!G) GB!P*

% M@>TAST!G) GB!P* PB=>)M

% DA!MAC PBGB!MMAG A M@>TAST!G)GB!P*

% <BE!BD M)T*D !D =!CFE!BD M)T*D

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Books References

% Be.erences:

  Computer !lgorithms C

% )llis *oro&itH Sarta Sahni Sanguthe8ar Baase-aran'

% Computer Science Press' +1IIJ,

  Antro(uction to !lgorithms

% Thomas * Cormen Charles ) >eiserson Bonal( >'

% 3n( )(ition' The MAT Press' e& #or-' +$00I,

  !lgoritma Atu Mu(ah% Bobert Setia(i'

% PT Prima An.osarana Me(ia Felompo- Grame(ia'

% Ka-arta' +$00J,

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END