Dynamic Programming 1

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! Anytime we have a collection of subproblems such that:! There is an ordering on the subproblems, and a relation thatshows how to solve a subproblem given the answers to"smaller" subproblems, that is, subproblems that appear earlierin the ordering! Problem becomes an implicit DAG with each subproblemrepresented by a node, with edges giving dependencies– Just one order to solve it? - Any linearization! Does largest increasing subsequence algorithm fit this?– Ordering is in the for loop – an appropriate linearization, finish L(1)before starting L(2), etc.– Relation is L(j) = 1 + max{L(i) : (i,j) ∈ E}CS 312 – Dynamic Programming 12
! DP is optimal when the optimality property is met– First make sure solution is correct! The optimality property: An optimal solution to a problemis built from optimal solutions to sub-problems! Question to consider: Can we divide the problem into subproblemssuch that the optimal solutions to each of thesub-problems combine into an optimal solution for theentire problem?CS 312 – Dynamic Programming 13
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: ! Why not use divide and conquer st
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 40 and 41: Item Weight Value1 6 $302 3 $143 4
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-
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Dynamic Programming 1

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