Outline DP paradigm Discrete optimisation Viterbi algorithm DP: 0 1 Knapsack. Dynamic Programming. Georgy Gimel farb

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1 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Dynamic Programming Georgy Gimel farb (with basic contributions by Michael J. Dinneen) COMPSCI 69 Computational Science /

2 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Dynamic Programming (DP) Paradigm Discrete Optimisation with DP Viterbi algorithm Knapsack Problem: Dynamic programming solution Learning outcomes: Understand DP and problems it can solve Be familiar with the edit-distance problem and its DP solution Be familiar with the Viterbi algorithm Additional sources: programming problem Dynamic programming solution /

3 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Main Algorithmic Paradigms Greedy: Building up a solution incrementally, by optimising at each step some local criterion Divide-and-conquer: Breaking up a problem into separate subproblems, solving each subproblem independently, and combining solution to subproblems to form solution to original problem Dynamic programming (DP): Breaking up a problem into a series of overlapping subproblems, and building up solutions to larger and larger subproblems Unlike the divide-and-conquer paradigm, DP typically involves solving all possible subproblems rather than a small portion DP tends to solve each sub-problem only once, store the results, and use these later again, thus reducing dramatically the amount of computation when the number of repeating subproblems is exponentially large /

4 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack History of Dynamic Programming Etymology: Richard E. Bellman [ ]: Famous applied mathematician (USA) who pioneered the systematic study of dynamic programming in the 95s when he was working at RAND Corporation Dynamic programming = planning over time Secretary of Defense was hostile to mathematical research Bellman sought an impressive name to avoid confrontation It s impossible to use dynamic in a pejorative sense Something not even a Congressman could object to Reference: Bellman, R. E.: Eye of the Hurricane, An Autobiography. /

5 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Bellman s Principle of Optimality R. E. Bellman: Dynamic Programming. Princeton Univ. Press, 957, Ch.III. An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision state s Optimal policy s t w.r.t. s s i s Can t be an optimal policy w.r.t. s i time t i n The optimal policy w.r.t. s after any its state s i cannot differ from the optimal policy w.r.t. the state s i! See equation 5 /

6 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state t 6 /

7 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state 7 Greedy algorithm: the total cost c = 7 t 6 /

8 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state DP: Step t 6 /

9 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state DP: Step t 6 /

10 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state DP: Step t 6 /

11 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state 6 DP: Step 6 t 6 /

12 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Simple Example: Find the Cheapest Route state 6 DP: Backtracking the cheapest route (c = ) 6 t 6 /

13 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Discrete Optimisation with DP Problem: (s,..., s n ) = arg min F (s,..., s n ) (s i S i : i=,...,n ) where an objective function F (s,..., s n ) depends on states s i ; i =,..., n, having each a finite set S i of values Frequently, an objective function to apply DP is additive: n F (s, s,..., s n ) = ψ o (s ) + ϕ i (s i, s i ) Generally, each state s i takes only a subset S i (s i+ ) S i of values, which depends on the state s i+ S i+ Overlapping subproblems are solved for all the states s i S i at each step i sequentially for i =,..., n i= 7 /

14 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Computing DP Solution to Problem on Slide 7 Bellman Equation: For i =,..., n and each s i S i, Φ i (s i ) = min i (s i ) + ϕ i (s i, s i )} s i S i (s i ) B i (s i ) = arg min i (s i ) + ϕ i (s i, s i )} s i S i (s i ) Φ i (s i ) is a candidate decision for state s i at step i and B i (s i ) is a backward pointer for reconstructing a candidate sequence of states s,..., s i, s i, producing Φ i (s i ) Backtracking to reconstruct the solution: min Φ n (s n ) s n S n min F (s,..., s n ) s,...,s n s n = arg min Φ n (s n ) s n S n s i = B i (s i ) for i = n,..., /

15 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Computing DP Solution to Example on Slide 6 Step i Set of states S i {} {,, } {,, } {,, } State constraints S i (s i ) {} {} {} {, } {} {, } {, } {} {, } Cost functions: ψ () = s >< ϕ (, s ) = >: >< ϕ (s, s ) = >: >< ϕ (s, s ) = >: s \s = s \s = Step i = : Φ (s = ) = ψ () = Step i = : Φ (s ) = s = Φ () + ϕ (, ) {z } {z } >< s = Φ () + ϕ (, ) {z } {z } s = Φ () + ϕ (, ) >: {z } {z } B (s ) = for s =,, s i = s i = s i = 9 = ; = = = 9 /

16 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Computing DP Solution to Example on Slide 6 (continued) Step i = : >< Φ (s ) = >: s = min{φ () + ϕ (, ), Φ () + ϕ (, ) } = ; {z } {z } B () = + + s = Φ () + ϕ (, ) = ; {z } B () = + s = min{φ () + ϕ (, ), Φ () + ϕ (, ) } = ; {z } {z } B () = + + state i /

17 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Computing DP Solution to Example on Slide 6 (continued) Step i = : >< Φ (s ) = >: s = min{φ () + ϕ (, ), Φ () + ϕ (, ) } = 6; {z } {z } B () = + + s = Φ () + ϕ (, ) = 6; {z } B () = + s = min{φ () + ϕ (, ), Φ () + ϕ (, ) } = ; {z } {z } B () = + + state 6 6 i /

18 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Computing DP Solution to Example on Slide 6 (continued) Solution: Optimal solution: min F (s,..., s ) min Φ (s ) = min{6, 6, } = s,...,s s S Ending optimal state: s = min Φ (s ) = s S Backtracking preceding states for the optimal solution: state s = B () = s = B () = s = B () = 6 6 i /

19 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack DP: Applications and Algorithms Areas: Control theory Signal processing Information theory Operations research Bioinformatics Computer science: theory, AI, graphics, image analysis, systems,... Algorithms Viterbi: error correction coding, hidden Markov models Unix diff: comparing two files Smith-Waterman: gene sequence alignment Bellman-Ford: shortest path routing in networks Cocke-Kasami-Younger: parsing context free grammars /

20 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: Minimum Cost of Editing Applications: Unix diff, speech recognition, computational biology Different penalties for insertion deletion, >, and for mismatch between two characters x and y: α xy ; α xx = c l a i m l i m e α ll = α ii = α mm = cost(claim lime) = c l a i m l i m e α cl α ii = α mm = cost(claim lime) = α cl + distance /

21 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Two Strings C and C : Levenshtein, or Edit Distance Minimum number D(C, C ) of edit operations to transform C into C : insertion (weight ), deletion (), or character substitution ( if the same character, otherwise α.. > ); e.g. = α.. = delete c i substitute c j for c i insert c j D(claim, lime) = ) claim laim ) laim lim ) lim lime c l a i m l i m e c l a i m l i m e distance 5 /

22 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation Strings: x x [m] = x x... x m and y y [n] = y y... y n Substrings: x [i] = x... x i ; i m, and y j] = y... y j ; j n j x i Distance d(i, j) = D(x [i], y [j] ) i Recurrent computation: if i = ; j = i d(i, ) + if i > ; j = j d(, j ) + if i = ; j > d(i, j) = d(i, j) +, d(i, j ) +, min d(i, j ) + α otherwise xiy j }{{} ; α xx= j y j i 6 /

23 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation e α ce α le α ae α ie α me m α cm α lm α am α im i α ci α li α ai α mi l α cl α al α il α ml c l a i m 7 /

24 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

25 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

26 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

27 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

28 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

29 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

30 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

31 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

32 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

33 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

34 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

35 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

36 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

37 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

38 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

39 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

40 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

41 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

42 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

43 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

44 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

45 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

46 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

47 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

48 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

49 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

50 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

51 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

52 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

53 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

54 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Levenshtein, or Edit Distance: DP Computation c l a i m l i m e 5 /

55 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Sequence Alignment Given two strings x = x x... x m and y = y y... y n, find their alignment of minimum cost Alignment M: a set of ordered pairs (x i y j, such that each item occurs in at most one pair and there are no crossings Pairs x i y j and x i y j cross if i < i, but j > j cost(m) = (x i,y y) M α xiy j } {{ } mismatch + i:x i unmatched + j:y j unmatched } {{ } gaps Example : x = claim; y = lime M = {x y, x y, x 5 y }; cost = Example : x = ctaccg; y = tacatg M = {x y, x y, x y, x 5 y, x 5 y, x 6 y 6 }; cost = α CA + x x x x x 5 c l a i m x x x x x 5 C T A C C G l i m e T A C A T G y y y y y y y y y 5 y 6 9 /

56 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Sequence Alignment: Algorithm Sequence-Alignment ( x x... x m, y y... y n,, α) { for i = to m D[i, ] = i for j = to n D[, j] = j for i = to m for j = to n D[i, j] = min( α[x i, y j ] + D[i, j ], + D[i, j], + D[i, j ] ) } Time and space complexity Θ(mn) English words: m, n Computational biology: m = n, ( billion operations is fine, but GB array?) /

57 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Viterbi Algorithm: Probabilistic Model DP search for the most likely sequence of unobserved (hidden) states from a sequence of observations (signals) Each signal depends on exactly one corresponding hidden state The hidden states are produced by a first-order Markov model: Set of the hidden states S = {s,..., s n} Transitional probabilities P s(s i s j) : i, j {,..., n} Given the states, the signals are statistically independent Set of the signals V = {v,..., v m} Observational probabilities P o(v j s i) : v j V; s i S v [] v [] v [] v [] Log-likelihood of a sequence of states s = s [] s []... s [K], given a sequence of s [] s [] s [] s [] signals v = v [] v []... v [K] : L(s v) log Pr(s v) Pr(s v) Pr(s, v) = Pr s (s) Pr o (v s) algorithm /

58 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Maximum (Log-)Likelihood s = arg max s S K L(s v) s = s [] s []... s [K] a hidden (unobserved) Markov chain of states at steps k =,..., K with joint probability Pr s (s) = π ( ) K s [] k= P ( ) s s[k] s [k ] π (s) prior probability of state s S at step k = P s (s s ) probability of transition from state s to the next one, s v = v [] v []... v [K] an observed sequence of conditionally independent signals with probability Pr(v s) = K k= P ( ) o v[k] s [k] P o (v s) probability of observing v V in state s S at step k K ( ( ) ( )) s = arg max ψk s[k] + ϕ s[k] s [k ] s S K k= { ( log π (s) + log Po v[k] s ) k = ; s S ψ k (s) = ( log P o v[k] s ) k > ; s S { k = ; s S ϕ (s s ) = log P s (s s ) k > ; s S /

59 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Probabilistic State Transitions and Signals for States Example for S = {a, b, c}, V = {A, B, C} P s (c c) P s (c a) P s (a c) c Ps(c b) Ps(b c) Probabilistic signal generator at each P step k; P o(v s) = for all s S v V s a P s (b a) b P s (a b) P s (a a) P s (b b) Non-deterministic (probabilistic) finite automaton (NFA) P for state transitions at each step k; P s(s s ) = for all s S s S P o (A s) P o (C s) P o (B s) A B C /

60 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Graphical Model for S = {a, b, c} and Given Signals v v [] = A v [] = A v [] = B v [] = A c ψ (c) ϕ(c c) ψ (c) ψ (c) ψ (c) ϕ(a c) ϕ(b c) b ψ (b) ϕ(c b) ϕ(b b) ϕ(b c) ψ (b) ψ (b) ψ (b) a ϕ(c a) ϕ(b a) ψ (a) ϕ(a a) ψ (a) ψ (a) ψ (a) k = k = k = k = /

61 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Maximum (Log-)Likelihood via Dynamic Programming Viterbi DP algorithm: Initialisation: k = ; Φ (s [] ) = ψ (s [] ) for all s [] S Forward pass for k =,..., K and all s [k] S: Φ k ( s[k] ) { ( )} = ψ k (s [k] ) + max ϕ(s[k] s [k ] ) + Φ k s[k ] s [k ] S ( ) { ( )} B k s[k] = arg max ϕ(s[k] s [k ] ) + Φ k s[k ] s [k ] S k = K: the maximum log-likelihood state s [K] = arg max Φ K(s [K] ) s [k] S Backward pass for k = K,..., : s [k] = B k+ ( ) s [k+] 5 /

62 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b 6 6 a k= v [] =A k= v [] =A k= v [] =B k= v [] =A k=5 v [5] =B 6 /

63 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b 6 6 a k= v [] =A k= v [] =A Step k = : Initialisation k= v [] =B k= v [] =A k=5 v [5] =B 6 /

64 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b 6 6 a k= v [] =A Step k = k= v [] =A k= v [] =B k= v [] =A k=5 v [5] =B 6 /

65 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b a k= v [] =A Step k = k= v [] =A k= v [] =B k= v [] =A k=5 v [5] =B 6 /

66 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b a k= v [] =A Step k = k= v [] =A k= v [] =B k= v [] =A k=5 v [5] =B 6 /

67 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b a k= v [] =A Step k = 5 k= v [] =A k= v [] =B k= v [] =A k=5 v [5] =B 6 /

68 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Example: S = {a, b}; V = {A, B}; v = AABAB ϕ(s s ) = ψ k (s) = { s = s s s ; for s, s S { 6 v s {A a, B b, C c} ; for s S otherwise b a k= v [] =A k= v [] =A Backtracking: s = aaaaa k= v [] =B k= v [] =A k=5 v [5] =B 6 /

69 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem Maximise n x i v i subject to n x i s i S; x i {, }; s i, v i, S > i= i= DP solution of pseudo-polynomial time complexity, O(nS) No contradiction to the NP-completeness of the problem: S is not polynomial in the length n of the problem s input The length of S is proportional to the number of bits, i.e. log S Space complexity is O(nS) (or O(S) if rewriting from µ(s) to µ() for each i) µ(i, s) the maximum value that can be obtained by placing up to i items to the knapsack of size less than or equal to s DP solution uses a table µ(i, s) or µ(s) to store previous computations 7 /

70 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Recursive definition of µ(i, s): µ(, s) = µ(i, ) = ; { µ(i, s) if si > s µ(i, s) = max {µ(i, s), µ(i, s s i ) + v i } if s i s i µ(i, ŝ) i s =... ŝ s i ŝ s = S µ(i, ŝ s i ) µ(i, ŝ) /

71 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

72 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

73 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

74 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

75 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

76 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = i = i = 9 i = i = i = s = /

77 Outline DP paradigm Discrete optimisation Viterbi algorithm DP: Knapsack Knapsack Problem: DP Solution Example: n=5; S = ; i 5 v i 5 s i 6 7 i = x 5 = i = x = i = 9 x = i = x = i = x = i = s = /