Dynamic Programming. Example - multi-stage graph. sink. source. Data Structures &Algorithms II

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1 Dynamc Programmng Example - mult-stage graph 1 source snk Data Structures &Algorthms II

2 A labeled, drected graph Vertces can be parttoned nto k dsjont sets u, v E, u V, v V, 1 k k 1 V (source) V (snk) 1 1 Fnd the mn cost path from source to snk Data Structures &Algorthms II

3 Q: Wll dvde-and-conquer fnd the mnmum cost path? A: Probably not best left-half path best overall path best rght-half path Data Structures &Algorthms II

4 Best paths found ndependently may not form a path Best overall paths may be suboptmal at dfferent subproblem stages Dvde-and-Conquer requres subproblems to be ndependent! Data Structures &Algorthms II

5 Q: Wll the greedy method fnds the mnmum cost path? A: May not (f you are not Djkstra) Choose the shortest lnk frst Solve the problem stage-by-stage Cost may be very low mnmum cost at stage 1 Cost may be very hgh Data Structures &Algorthms II

6 A low cost edge may be followed by paths of a very hgh cost A hgh cost edge may be followed by paths of a very low cost Based on local nformaton (one stage at a tme) t mght not be possble to lookahead Pckng the remanng lowest cost edge may not generate a path Data Structures &Algorthms II

7 Q: Is there an applcaton? A: Yes, e.g., resource allocaton n unts of resources to be allocated to r projects N(,j) proft earned f j unts of resources are allocated to project goal s to maxmze the proft earned Data Structures &Algorthms II

8 V(project beng consdered, resources commtted) V(2,0) N(2,0) V(3,0) 3 projects 3 PCs N(1,0) V(2,1) N(2,1) V(3,1) V(1,0) N(1,1) N(1,2) V(2,2) N(2,2) V(3,2) V(4,3) N(1,3) V(2,3) N(2,3) V(3,3) Data Structures &Algorthms II

9 Surface Generaton n Tomography

10 p1 p2 m by n lattce Vertcal edge: an uprght trangle Horzontal edge: an nverted trangle Closed surfaces correspond to paths of length m+n Best path (surface) has the lowest cost

11 Q: What should we do? A: Enumerate all possbltes Q: How much s the cost of enumeraton? A: Hgh, for complete connecton between two adjacent stages n stages m vertces per stages O( m n ) Data Structures &Algorthms II

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13 S D S D S S S S S S S D D D D D D D A lot of repettons: buld tables to remember partal solutons (reuse) S S S S S S S S D D D D D D D D A lot of alternatves: buld tables to remember optmal partal solutons (prncple of optmalty)

14 Q: Is there a more effcent method of enumeraton? A: Yes, dynamc programmng Underlyng prncples: Prncple of optmalty Early elmnaton of suboptmal subsolutons Recurson and reuse Construct solutons by reusng optmal subsolutons Early termnaton By feasblty or optmalty Data Structures &Algorthms II

15 Prncple of Optmalty Left half Rght half source snk (source to ) ( to snk) Data Structures &Algorthms II

16 The optmal soluton (f t go through node ) must contans the best left path from source to and best rght path from to snk Any other left paths from source to and any other rght paths from to snk need not be extended any further Data Structures &Algorthms II

17 Recurson and Reuse Identfy subproblems Record the optmal solutons of subproblems Buld larger and larger solutons source e 1 e 2 snk Any path that ncludes a porton of (source to ), the cost of that partcular porton s known

18 Feasblty Is a partal soluton stll feasble? Based on the current path alone Optmalty Is a partal soluton gong to be optmal? Based on comparson of the current path wth others

19 ( 1 6) Backward approach 9(mn( 4 9, 2 7)) ( 1 9) ( 1 2) ( 1 3) ( 1 4) ( 1 5) 9 ( 1 7) 7 11(mn( 2 9, 7 7, 11 2)) ( 110) ( 1 8) 15(mn( 9 6, 4 11)) 14(mn( 5 9, 3 11, 5 10)) ( 112) BCost(, j): 16(mn( 4 15, 2 14, 5 16)) (mn( 1 9, 11 3, 8 2)) ( 111) 16( 6 10) the cost of the optmal path from the source to vertex j at stage BCost(, j) mn { BCost(, l) c( j, l)} lv j, le source snk Data Structures &Algorthms II

20 ( 2 12) Forward approach 1 ( 6 12) ( 9 12) ( 10 12) ( 1112) 2 ( 7 12) ( 8 12) 7(mn( 6 4, 5 2)) 5(mn( 4 4, 3 2)) 7(mn( 5 2, 6 5)) ( 3 12) ( 4 12) ( 5 12) 7(mn( 4 7, 2 51, 7)) 9(mn( 2 7, 7 5)) ( 112) 4 Cost(, j): 18( 11 7) 16((mn( 9 7, 7 9, 3 18, 2 15)) 5 the cost of the optmal path from vertex j at stage to snk 15(mn( 11 5, 8 7)) Cost(, j) mn { c( j, l) Cost(, l)} lv j, le source Data Structures &Algorthms II snk

21 Intuton on DP Dynamc programmng sometmes can be confusng because t s bascally recurson but s slghtly more than recurson A general soluton pattern s dentfy stages and all possble alternatves n a stage recurson to generate all possble solutons need to combne and elmnate partal solutons usng prncple of optmalty

22 For mult-stage graph steps are defned by stages straght recurson generates brushy tree O(node^stage)

23 The mportant thng s to trm the tree by combnng and coalescng nodes by prncple of optmalty

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25 Fancy Recursve Equatons Cost (node 1 at level 1) Mn (c(1,2)+cost(node 2 at level2), c(1,3)+cost(node 3 at level2), c(1,4)+cost(node 4 at level2),) Fancy equaton descrbes the recurson DP says that all such Cost functons should be reused!!

26 Remember the costs here Don t compute agan and agan

27 Important Characterstcs of Dynamc Programmng A sequence of decsons to be made Decsons are nter-dependent (Dvde-and- Conquer not applcable) Local nformaton not suffcent (Greedy not work) Data Structures &Algorthms II

28 Important Characterstcs of Dynamc Programmng (cont.) It examnes all solutons n an effcent manner It nvolves buldng solutons recursvely Prncple of optmalty s used to elmnate suboptmal solutons Table of some sort are usually used to store optmal partal solutons Some reuse of optmal partal solutons Mathematcally as recurson Data Structures &Algorthms II

29 Tme Complexty of DP DP == buldng tables of partal solutons 1. How many tables? 2. How many entres per table? 3. How much effort to fll n entres? 1*2*3 gves the complexty Data Structures &Algorthms II

30 Tme complexty of mult-stage graph One table s bult There are V entres n the table The cost of generatng an entry s proportonal to the ncdent edges O( V + E ): a sgnfcant savng over exponent runtme Data Structures &Algorthms II

31 0/1-Knapsack Input: a set of n objects a knapsack of capacty M Output: fll the knapsack (no partal ncluson) to maxmze the total proft earned Data Structures &Algorthms II

32 Greedy method can fal P ( 9, 7, 7), W ( 6, 5, 5), M 10 Greedy Optmal P 1 P 2 P 3 Dvde-and-conquer may not apply a a a a a 1 2 n n n subproblem (X?) subproblem (M-X) Data Structures &Algorthms II

33 How to buld solutons recursvely? One object at a tme How does prncple of optmalty apply? x x x x 1 2 n ( 1, xx x):( xx x) must be optmal for M W ( 0, y y y):( y y y) must be optmal for M 1 How to dentfy sub-optmal solutons? f two solutons: (1,,0,x,,x), and (0,,1,x,, x) are such that one acheves better proft wth less weght, then the other cannot be optmal Data Structures &Algorthms II

34 How to buld table? Proft Weght,,,, P=(1,2,5), W=(2,3,4),M=6 Boundng possble! (0,0) (Proft earned, Weght used) (1,2) (0,0) (3,5) (2,3) (1,2) (0,0) (8,9) (7,7) (6,6) (5,4) (3,5) (2,3) (1,2) (0,0) Data Structures &Algorthms II

35 How to wrte the recursve equaton? Knap(,X): current proft wth objects to n left to be processed wth a remanng capacty of X Intally, we have Knap(1,M) Knap( 1, M ) max{ Knap( 2, M ), Knap( 2, M W ) P} 1 1 Knap(, X ) max{ Knap( 1, X ), Knap( 1, X W ) P} Tme complexty An brushy tree for the table Constant tme to generate entres O( 2 n ) DP can help, but the complexty wll depend on actual problem nstance Data Structures &Algorthms II

36 Three Useful Trcks Feasblty If a branch s over capacty, don t expand t anymore Optmalty If a branch s worse than another branch (more capacty used wth smaller proft), don t expand t anymore Feasblty and optmalty are problem nstance specfc, cannot guarantee worst runtme n general Reuse Remember optmal partal solutons, don t regenerate over and over agan

37 Reuse Examples Weght = (3, 2, 3, 1, 4, 5), knapsack capacty = 10 Proft = (2, 3, 4, 1, 5, 1) Used capacty Max(2,4) Max(6,7) The complexty s O(n) wth reuse!

38 Relable Desgn Input: A system composed of several devces n seral Each devce (D) has a fxed relablty ratng (r) Multple copes of the same devce can be used n parallel to ncrease relablty Output: A system wth hghest relablty ratng subject to a cost constrant Data Structures &Algorthms II

39 D 1 D 1 D 1 D 2 D 2 D D D D n D D m max copes wth a 1n subject to of ( m 1n devces c m D relablty ratng ) C and at stage of m, 11, r 1 (1 : say,90% r ) n m Connected n parallel At least one should work Connected n seres All of them have to work Data Structures &Algorthms II

40 Greedy method may not be applcable Strategy to maxmze relablty: Buy more less relable unts (Costs may be hgh) Strategy to mnmze cost: Buy more less expensve unts (Relablty may not mprove sgnfcantly) Dvde-and-Conquer may fal D D D D D 1 2 n n n subproblem (X?) subproblem (C-X) Data Structures &Algorthms II

41 Comparson A knapsack of capacty C Objects of sze c and proft p Fll up the knapsack wth 0 or 1 copy of Maxmze proft Total expendture of C Stages of cost c and relablty r Construct a system wth 1 or more copes of Maxmze relablty u 1 C c n j1 c j Data Structures &Algorthms II

42 How to buld solutons recursvely? One stage and one devce at a tme How does prncple of optmalty apply? m m m m 1 2 n ( 1, xx x):( xx x) must be optmal for C c ( 2, y y y):( y y y) must be optmal for C 2 c ( u, y y y):( y y y) must be optmal for C u c How to dentfy sub-optmal solutons? f two solutons: (m1,,m,x,,x), and (n1,,n,x,, x) are such that one acheves hgher relablty wth a smaller cost, then the other cannot be optmal Data Structures &Algorthms II

43 How to buld table? r=(0.9,0.8,0.5), c=(30,15,20),c=105 (1,65) (relablty, cost) 1 ( ) ( ) ( ) 0. 8 (0.9,65) (0.99,95) 1 ( ) ( ) (0.72,65) (0.864,80) (0.893,95) (0.792,95) (0.95,110)(0.98,125) 1 ( ) 0. 5 (0.36,65) (0.63,105) (0.54,85) 1 ( ) ( 1 05) (0.432,80) (0.648,95) (0.756,110) (0.446,95) (0.781,135) (0.70,115)

44 How to wrte the recursve equaton? f ( X ): max ( m ) 1 j j subject to c m X, 1 m, 1 j 1 j j j f ( C) max { ( m ) f ( C c m )} n 1 m f ( X ) max{ ( m ) f ( X c m )} n 1 m n j j jn n n n1 n n 1 Tme complexty An brushy tree for the table Constant tme to generate entres O( 2 n ) Data Structures &Algorthms II

45 Chan Matrx Multplcaton Input: A sequence of n matrces Output Ther products

46 Even though not so obvous, greedy algorthms (e.g., keep ndvdual multplcatons small) do not always produce optmal solutons Furthermore, the costs can vary qute a bt dependng on the orderng A 135 B 589 (( AB) C) D ( AB)( CD) ( A( BC) D A(( BC) D) A( B( CD)) C D 334

47 Can dynamc programmng be used? Does the prncple of optmalty apply? Are there small problems? Can the subsolutons be reused and how?

48 Yes! There are many possble ways to apply DP, as long as do thngs n stages merge and reuse nodes based on prncple of optmalty We wll show some examples below

49 Based on the number of multplcatons performed (AB) (BC) (CD) (( AB ) C) ( AB )( CD) ( A ( BC)) (( BC ) D) ( B ( CD)) ( AB)( CD) ((( AB ) C) D) (( AB )( CD)) (( A ( BC)) D) ( A (( BC) D)) ( A ( B( CD))) (( AB)( CD))

50 An obvous DP algorthm need to multply all the matrces ndvdual steps wll be multplyng two adjacent matrces and reduce the number of matrces by one at each step, choose any two adjacent matrces to multply n n-1 steps, we wll be done reuse: (ABC) = ((AB) C), reuse results of (AB) prncple of optmalty: ((AB) C) and (A (BC)) produce the same results, keep one A perfectly legal DP algorthm!

51 The problem s that t s not a good DP algorthm wth n matrces frst multplcaton: n-1 possbltes second multplcaton:??? thrd multplcaton:??? even wth reuse and prncple of optmalty the numbers of ntermedate stages are large many multplcatons are repeated many tmes AB ((CD)(EF)) and ((AB)C)D(EF), (EF) s done more than once

52 Based on the number of matrces multpled together (the range of ndces) (A) (B) (C) (D) (AB) (BC) (CD) (ABC) (BCD) (ABCD)

53 Does the prncple of optmalty apply? Yes, whatever the last step n the chan multplcaton, the steps leadng to those two matrces must be optmal Are there small problems? Yes, multplcatons of two adjacent matrces Can the subsolutons be reused and how? Yes, 1 1,2,..., 1 1 1, n d d d m s n d d d m m m s k s k k s k s,..., 1,2 ) ( mn 1 1,,,

54 s s s s j D C B A ) , , 5 13 mn( ) , 89 5 mn( ) , 5 13 mn( m m m m m m m m m m m m m m m m m

55 Longest Increasng Subsequence Gven an array of n numbers [0..n-1], fnd a subset of numbers that are ncreasng [ ] [0 8 15] [ ] [2 3 7 ] [ ] <- longest one [ ] [ ]

56 Brute Force Method Every number can be ether n or not n n LIS Wth n numbers, there are 2 n subsequences Generate all, dscard those that are not ncreasng subsequences Complexty O(2 n )

57 DP Key Idea An partal LIS soln (head) must end at some ndex The same tal porton can be added to all these solns Only best soln s kept tal

58 0: not n the IS 1: n the IS

59 Include 1 Include 2 Include 3 Include 4 LIS k = best 0<=<k (LIS +1)

60 0, 8, 4, 1, 2, 2, 10, 6, 14, 1, 9, 5, 13 LIS k = best 0<=<k (LIS +1) 0 1 ( 0 ) 10 4 ( 0, 1, 2, 10 ) (0, 1, 2, 10) 8 2 ( 0, 8 ) 4 2 ( 0, 4 ) 1 2 ( 0, 1 ) 2 3 ( 0, 1, 2 ) 2 3 ( 0, 1, 2 ) 6 4 ( 0, 1, 2, 6 ) (0, 1, 2, 6) 14 5 ( 0, 1, 2, 10, 14 ) (0, 1, 2, 10, 14 (0,1,2,6, 14) (0, 1,2,6,14) 1 2 ( 0, 1 ) 9 5 ( 0, 1, 2, 6, 9 ) (0, 1, 2, 6, 9) 5 4 ( 0, 1, 2, 5) (0, 1, 2, 5) 13 6 ( 0, 1, 2, 6, 9, 13 ) (0, 1, 2, 6, 9

61 DP Key Idea (Reuse) How to buld table? LIS k = best 0<=<k (LIS +1) Fnal soluton? LIS = best 0<=k<n (LIS k ) Complexty One table n entres LIS k Most expensve entry O(n) O(n 2 ) More effcent (O(nlogn)) exsts

62 Sequence Algnment Gven two sequences a, b of length m, n Algn them to match Use n DNA matchng: a: AGCTTCGA b: GATCGA Deleton (nserton): 1 st A, 4 th or5 th T Change: 3 rd C->A AGCTTCGA GAT CGA AGCTTCGA GA TCGA

63 Constrants Lnear orderng If a matches wth b j, a k, k< must match only wth b l, l<j a k, k> must match only wth b l, l>j Deleton, nserton, and change all have assocated costs (doman dependent) Also called the longest common subsequence (LCS) problem (GTCGA) AGCTTCGA GA TCGA

64 Brute Force Method Agan, thnk about tree At each tree node, lookng at some a and some b j (ntally, a o, b o ) Match a and b j No change necessary Change a <-> b j Skp (delete) a, but keep b j Skp (delete) b j, but keep a Skp (delete) both a and b j Max fan out s 4, Max tree depth s m+n, bad

65 Prncple of Optmalty Smlar to LIS Head: some partal results (match, delete, nsert, change, etc.) up to a and b j Tal: (match, delete, nsert, change) results for a +1 and b j a tal b j

66 Reuse Buld a table of sze m by n to store the partal results (,j)th entry s results up to a b j How to fll the table? Fll them n dagonally C(,j) (W) = mn among R: C(-1,j-1) + (match, change, skp) a and b j G: C(,j-1) + skp b j j B: C(-1,j) + skp a Complexty: O(n 2 )

67 a a: AGCTTCGA b: GATCGA Match: 0 Delete (nsert): 1 Change: 1 b

68 a a: AGCTTCGA b: GATCGA Match: 0 Delete (nsert): 1 Change: 1? b

69 Polygon Trangulaton Gven: A convex polygon wth n sdes, a trangulaton s a set of chords of the polygon that dvde the polygon nto dsjont trangles There are n-2 trangles wth n-3 chords

70 Not all trangulatons are equally good Need a cost functon to evaluate the cost of a trangulaton The cost of a trangulaton s the total costs of ts component trangles The cost of a partcular trangle s the sum of some dstance measure (e.g., Eucldean) of all ts sdes,,, ),, ( k k j j k j v v v v v v v v v

71 Agan, dvde-and-conquer mght not work say, chose a chord to dvde the polygon nto two parts and perform trangulaton for both parts ndependently the sad chord wll be n the fnal trangulaton however, the optmal trangulaton may not nclude that partcular chord Greedy? the polygon of the smallest cost may not be n the fnal trangulaton Need to look at all possble combnatons

72 Intutvely, when we consder the frst step n trangulaton, say, usng v(0) and v(n-1) as base, the vertex can be v(1), v(2),, v(n-2) we do not know whch one s the best, should consder all possbltes Furthermore, f we pck, say v(j), then trangulaton of v(0) to v(j) and v(j) to v(n-1) must be optmal w.r.t each sub problems (prncple of optmalty) v v j v 0 v n1

73 Lke Matrx Multplcaton Multple two matrces Create one trangle Multply n-1 tmes Create n-2 trangles Randomly pck two matrces to multply Randomly pck a trangle to add Better to do t by gradually enlarge the chan Better to do t by gradually enlarge the trangle area Adjacency Adjacency

74 Prncple of Optmalty v v n1 v 0 v v1 2 v v n 2 v1 v 1 v 1 v n 2

75 Reuse f we need to trangulate an area nsde the orgnal polygon spanned by, say k, vertces f we already know the best way to trangulate an area spanned by k-1 vertces or less then we can take advantage of that!

76 That allows us to wrte the followng recurrence relaton let c(,j) be the cost of an optmal trangulaton of polygon <v(), v(+1), v(j)>, then j v v v j k c k c j c j or j j c j k j k 1 )),, ( ), ( ), ( mn( ), ( 1 0 ), ( k j

77 j j-=0 j-=1 C(,)=0 C(,+1)=0 C(,+2) C(k,j) C(,j) C(,k) = mn{ + + } j v v v j k c k c j c j or j j c j k j k 1 ),, ( ), ( ), ( mn( ), ( 1 0 ), ( startng endng

78 Optmal Bnary Search Tree Input: A set of n dentfers Output: An optmal bnary search tree that mnmzes the average search effort Data Structures &Algorthms II

79 else do n dentfers { a a,..., a } P( ): probablty that Q( ): probablty that search s for a symbol n E that a E a n P( ) Q( ) end f successful faled swtch 1 2 a 1 whle n s searched n successful: P( ) level( a ) n 1 faled: Q( ) ( level( E ) 1) 0

80 Convnce yourself that dvde-and-conquer and greedy methods are not sutable Dynamc Programmng Identfy small problems Progressvely buld larger problems Reuse optmal sub-solutons (table buldng) Data Structures &Algorthms II

81 How to buld soluton recursvely (reuse)? p( 1) Q( 0) Q( 1) p( 2) Q( 1) Q( 2) p( 3) Q( 2) Q( 3) a 1 a 2 a 3 p(2) Q(2) p(1) Q(0) p(3) Q(3) p(2) Q(1) ( p(1) Q(0) Q(1)) 2 ( p(2) Q(1) Q(2)) 2 ( p(2) Q(1) Q(2)) 2 ( p(3) Q(2) Q(3)) 2 a 2 a 1 a 3 a 2 a 1 a 2 a 2 a 3 Data Structures &Algorthms II

82 p(2) Q(2) ( p(1) Q(0) Q(1)) 2 a 2 a 1 p(2) ( p(1) Q(0) Q(1)) 2 ( p(3) Q(2) Q(3)) 2 p(1) Q(0) (( p(2) Q(1)) 2 ( p(3) Q(2) Q(3)) 3) a 1 p(3) Q(3) (( p(2) Q(2)) 2 ( p(1) Q(0) Q(1)) 3) a 3 a 2 a 2 a 1 a 3 a 2 a 1 a 3 Data Structures &Algorthms II

83 How does prncple of optmalty apply? p(2) Q(2) p(1) Q(0) p(3) Q(3) p(2) Q(1) ( p(1) Q(0) Q(1)) 2 ( p(2) Q(1) Q(2)) 2 ( p(2) Q(1) Q(2)) 2 ( p(3) Q(2) Q(3)) 3 a 2 a 1 a 3 a 2 a 1 a 2 a 2 a 3 Only one of the two confguratons should be kept! Data Structures &Algorthms II

84 Only one of the fve confguratons should be kept! a 1 a 3 a 2 a 2 a1 a 3 a 2 a 3 a 1 a 1 a 3 a3 a 1 a 2 a 2 Data Structures &Algorthms II

85 Lke Matrx Multplcaton Multple two matrces Pull one node up to root Multply n-1 tmes Create n-2 subtrees Randomly pck two matrces to multply Randomly pck a node up to be root Better to do t by gradually enlarge the chan Better to do t by gradually enlarge the subtree sze Adjacency Adjacency

86 a k R L E, a E a..., a E 0 1, 1, 2, k1, k1 L s optmal w.r.t. all bnary search trees wth the above elements C( L) p( ) level( a ) 1 k Q( ) ( level( E ) 1) 0k E, a E a..., a E k k1, k1, k2, n, n R s optmal w.r.t. all bnary search trees wth the above elements C( R) p( ) level( a ) kn Q( ) ( level( E ) 1) kn

87 C( combned ) p( k) C( L) C( R) p( ) Q( ) 1 k 0k p( ) Q( ) kn kn C( L) C( R) W( combned ) W( combned ) p( ) Q( ) p( k) p( ) Q( ) p( ) Q( ) 1 n 0n Recurrence relaton 1 k 0k kn kn C( 0, n) mn{ C( 0, k 1) C( k, n) W( 0, n)} 0kn C(, j) mn{ C(, k 1) C( k, j) W(, j)} k j Red: left got one deeper Blue: rght got one deeper Green: root Data Structures &Algorthms II

88 C(, j) mn{ C(, k 1) C( k, j) W(, j)} k j j C(,j) C(+1,j) C(+2,j) C(,j-1) C(j-1,j) j-=1 j-=0 C(,+2) C(,+1) C(,) = mn{ + + } Data Structures &Algorthms II

89 Table buldng j j-=n j-=2 j-=1 j-=0 order of computaton C(, ) 0 W(, ) P( k) Q( k) Q( ) 1 k k Data Structures &Algorthms II

90 How to compute W(,j)? Why not recursvely? E, a, E, a, E, a,..., a, E, a, E j1 j1 j j W(, j 1) W(, j) W(, j) p( j) Q( j) W(, j 1) Data Structures &Algorthms II

91 n 4 ( a, a, a, a ) ( do, f, read, whle) ( p, p, p, p ) ( 3, 311,, ) ( Q, Q, Q, Q, Q ) ( 2, 3111,,, ) j j-= j-= W(, j) p( j) Q( j) W(, j 1) =p(1)+Q(1)

92 C(0,4) mn{ C(0, k 1) C( k,4) W (0,4)} 0k4 mn{ C(0,0) C(1,4),(0 19) C(0,1) C(2,4),(8 8) C(0,2) C(3,4),(19 3) C(0,3) C(4,4),(25 0)} W (0,4),(16) C(, j) mn{ C(, k 1) C( k, j) W(, j)} k j j j-= j-= C( 1, 2) mn{ C( 1, k 1) C( k, 2) W( 1, 2)} 1k 2 C( 11, ) C( 2, 2) W( 1, 2) C( 0, 1) mn{ C( 0, k 1) C( k, 1) W( 0, 1)} 0k1 C( 0, 0) C( 11, ) W( 0, 1) Data Structures &Algorthms II

93 Tme Complexty - C(,j) j-=m, there are n-m+1 of them Each one takes mnmum of m quanttes n j ( n m 1) m O( n ) 0mn 3 C(0,n) C(n-1,n) C(n,n) j-=1 j-=0... C(2,3) C(.,.) 2 C(1,2) C(2,2) 1 C(0,1) C(1,1) 0 C(0,0) n Data Structures &Algorthms II

94 DP-based Graph Algorthms Graph = (vertces, edges) Edges Buld a long path wth many edges wth short paths wth fewer edges Stop when the path s longer than mn(e,n) Vertces Buld a subgraph wth wth many vertces wth smaller subgraphs wth fewer vertces Stop when all vertces are consdered Data Structures &Algorthms II

95 All Pars of Shortest Paths Input: a labeled graph G=(V,E) Output: the shortest path from very vertex to very other vertces Data Structures &Algorthms II

96 Soluton 1: Iteraton on the number of edges a drect path (length=1) paths of length=2 paths of length=mn(e,n) Soluton 2: Iteraton on the number of vertces a drect path (no ntervenng vertces) paths wth one ntervenng vertex paths wth (n-2) ntervenng vertces Data Structures &Algorthms II

97 Both solutons have ths recurrence relaton: C t (, j) mn{ C t1 (, j),mn{ C k, j 1 (, k) C t1 ( k, j)}} Gong through no other vertex V V 3 2 V 2 A nf 0 mn A Gong through one vertex A (0) (5) nf+6(3) (4) 11+nf (0) 2+nf 3+4 nf+0(7) 0+nf (6) (2) nf+2(0) Data Structures &Algorthms II

98 Tme Complexty How many tables? O(mn(e,n)) 2 How many entres per table? O( n ) How much effort to generate each entry? O(n) 3 4 O( n O(mn( e, n)) O( n ) OK solutons, but not great Try Floyd algorthm teratng on vertex s cardnal number Data Structures &Algorthms II

99 Travelng Salesperson Input: a drected labeled graph G=(V,E) Output: a tour of the mnmum cost a tour vsts all vertces a tour vst any vertex exactly once Data Structures &Algorthms II

100 Mult-stage graph Travelng salesman source snk source snk dffcult dffcult source snk source snk Cost(, j) mn { c( j, l) Cost(, l)} lv j, le canddate: n-1 vertces canddate: n-2 vertces 1 g(, S) mn{ c(, j) g( j, S { j})} 1 js Data Structures &Algorthms II

101 Dfference For travellng salesman source = snk Every vertex can possbly be at every stage O((n-1)!) complexty n-1 vertces!! source snk

102 Does the prncple of optmalty apply? Small problems wth reuse? source Only one need be kept! source

103 g(1, V {1}) mn{ c(1, k) 2kn g( k, V {1, k})} g(, S) mn{ c(, j) g( j, S { j})} js g(,s): the length of the shortest path startng at vertex, gong through all vertces n S, then back to the source Data Structures &Algorthms II

104 Data Structures &Algorthms II /5 9/13 12/9 20/8 10/8 15/ ) 25,20 25,15 mn(10 (1,{2,3,4}) 23 18) 15,9 mn(8 (4,{2,3}) 25 13) 18,12 mn(13 (3,{2,4}) 25 15) 20,10 mn(9 (2,{3,4}) 6) 15(9 (4,{3}) 5) 13(8 (4,{2}) 8) 20(12 (3,{4}) 5) 18(13 (3,{2}) 8) 18(10 (2,{4}) 6) 15(9 (2,{3}) 8 ) (4, 6 ) (3, 5 ) (2, g g g g g g g g g g g g g

105 Tme Complexty : there are n-1 vertces to vst at each level S: there are n 2 2 n ( n 1) O( n 2 ) k n2 k0 n 2 k choces C 12 C 13 g(1,v-{1}) C 1n g(2,v-{1,2}) g(3,v-{1,3}) g(,s) g(n,v-{1,n}) C 23 C 32 C 34 C 3n g(2,v-{1,2,3}) g(4,v-{1,3,4}) g(n,v-{1,3,n})

106 World Seres Odds DP may be used to solve problems where prncple of optmalty s not applcable Input: two teams A and B play a maxmum of 7 games whchever team frst wns 4 wns the seres Output: P(,j): condtonal probablty(a wns the seres A needs more games and B need j more games) Data Structures &Algorthms II

107 Even though prncple of optmalty does not apply here, but the problem does possess recursve nature solutons can be constructed by reuse P(, j) 1 0, j 0 0 0, j 0 P(, j) ( p( 1, j) p(, j 1)) / 2, 0, j 0 P(,j): condtonal probablty(a wns the seres A needs more games and B need j more games) Data Structures &Algorthms II

108 j P(, j) ( p( 1, j) p(, j 1)) / 2, 0, j /1613/16 21/32 1/2 j /8 3/4 11/16 1/2 11/32 1/2 5/16 3/16 (-1,j) (,j) /2 1/4 1/8 1/ (,j-1) Data Structures &Algorthms II

109 Brute force method A wns p(4,3) p(4,4) B wns A wns p(3,4) B wns A wns B wns +j=8 +j=7 p(3,3) p(4,2) p(2,4) p(3,3) +j=6 A wns B wns p(2,3) p(3,2) p(3,2) p(4,1) p(1,4) p(2,3) p(2,3) p(3,2) n O( 2 ) for brute force vs. O( n ) 2 for DP Data Structures &Algorthms II

110 Lessons Learned Basc prncples (Mult-stage graphs, 0/1-knapsack, Relable desgn) Brute force Reuse, Feasblty, Optmalty Table buldng (recurson) Beng Smart (Matrx multplcaton, polygon trangulaton) There are dfferent tables and dfferent recursons Beng flexble (World seres odds) Reuse regardless of optmalty constrant (more later) Nothng really matter much (Travelng Salesperson) There are hard problems n the unverse