Dynamic Programming 1

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Item Weight Value1 6 $302 3 $143 4 $164 2 $9W = 10! Will greedy always work?! Exponential number of item combinations– 2 n for Knapsack without repetition – why?– Many more for Knapsack with repetition! How about DP?– Always ask what are the subproblemsCS 312 – Dynamic Programming 40
! Two types of subproblems possible– consider knapsacks with less capacity– consider fewer items! Define K(w) = maximum value achievable with a knapsackof capacity w– Final answer is K(W)! Subproblem relation – if K(w) includes item i, thenremoving i leaves optimal solution K(w-w i )– Can only contain i if w i ≤ w! Thus K(w) = max i:wi ≤w [K(w – w i ) + v i ]! Note that it is not dependent on a n-1 type recurrence likeedit distance)CS 312 – Dynamic Programming 41
Page 4 and 5: A!B! C!B!E: solution EF: solution F
Page 6 and 7: Fn⎧⎧F if 1n−1 + Fn−2 n >⎪
Page 8: ! 5 2 8 6 3 6 9 7- 2 3 6 7! Conside
Page 11 and 12: ! Why not use divide and conquer st
Page 13 and 14: ! DP is optimal when the optimality
Page 16 and 17: ! How many ways to choose k items f
Page 18 and 19: C(5,3)"C(4,2)"C(4,3)"C(3,1)" C(3,2)
Page 20 and 21: n k: 0 1 2 30 1 0 0 01 1 1 0 02 1 1
Page 22 and 23: n k: 0 1 2 30 1 0 0 01 1 1 0 02 1 2
Page 24 and 25: n k: 0 1 2 30 1 0 0 01 1 1 0 02 1 2
Page 26 and 27: ! A natural measure of similarity b
Page 28 and 29: ! Assume two strings x and y of len
Page 30 and 31: ! Start building a table- What are
Page 32 and 33: ! E(i,j) = min[1 + E(i-1,j), 1 + E(
Page 34 and 35: ! This is a weighted DAG with weigh
Page 36 and 37: X=ACGCTC !!Y=ACTTG!CS 312 - Dynamic
Page 38 and 39: ! You will implement two versions (
Page 42 and 43: K(0) = 0for w = 1 to WK(w) = max i:
Page 44 and 45: function K(w)if w = 0 return(0)K(w)
Page 46 and 47: ! Our relation now has to track wha
Page 48 and 49: Item Weight Value1 6 $302 3 $143 4
Page 50 and 51: w=0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0
Page 52 and 53: w=0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0
Page 54 and 55: ! Want to multiply A 1 × A 2 × ·
Page 56 and 57: ! Each subtree breaks the problem i
Page 58 and 59: ! We used BFS, Dijkstra's and Bellm
Page 60 and 61: ! Arbitrarily number the nodes from
Page 62 and 63: ! Assume n cities (nodes) and an in
Page 64 and 65: ! Table is n × 2 n! Algorithm has
Page 66 and 67: ! sdfsdfS1 2 3 41,1 01,2 ∞ 31,3
Page 68 and 69: ! sdfsdfS1 2 3 41,1 01,2 ∞ 31,3
Page 70: ! Example Applications- Fibonacci-

Dynamic Programming 1

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