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  • 1. Dynamic Programming Lecture 16
  • 2. Dynamic Programming Dynamic programming is typically applied to optimization problems. In such problems there can be many possible solutions. Each solution has a value, and we wish to find a solution with the optimal (minimum or maximum) value.
  • 3. Dynamic Programming Like divide and conquer, Dynamic Programming solves problems by combining solutions to sub problems. Unlike divide and conquer, sub problems are not independent.  Subproblems may share subsubproblems,  However, solution to one subproblem may not affect the solutions to other subproblems of the same problem. (More on this later.) Dynamic Programming reduces computation by  Solving subproblems in a bottom-up fashion.  Storing solution to a subproblem the first time it is solved.  Looking up the solution when subproblem is encountered again
  • 4. Dynamic ProgrammingThe development of a dynamic-programming algorithm can bebroken into a sequence of four steps.1.Characterize the structure of an optimal solution.2.Recursively define the value of an optimal solution.3.Compute the value of an optimal solution in a bottom-upfashion.4.Construct an optimal solution from computed information.
  • 5. Matrix-chain Multiplication Suppose we have a sequence or chain A1, A2, …, An of n matrices to be multiplied  That is, we want to compute the product A1A2…An There are many possible ways (parenthesizations) to compute the product
  • 6. Matrix-chain Multiplication Example: consider the chain A1, A2, A3, A4 of 4 matrices  Let us compute the product A1A2A3A4 There are 5 possible ways: 1. (A1(A2(A3A4))) 2. (A1((A2A3)A4)) 3. ((A1A2)(A3A4)) 4. ((A1(A2A3))A4) 5. (((A1A2)A3)A4)
  • 7. Algorithm to Multiply 2 MatricesInput: Matrices Ap×q and Bq×r (with dimensions p×q and q×r)Result: Matrix Cp×r resulting from the product A·BMATRIX-MULTIPLY(Ap×q , Bq×r)1. for i ← 1 to p2. for j ← 1 to r3. C[i, j] ← 04. for k ← 1 to q5. C[i, j] ← C[i, j] + A[i, k] · B[k, j]6. return C
  • 8. Matrix-chain Multiplication Example: Consider three matrices A10×100, B100×5, and C5×50 There are 2 ways to parenthesize  ((AB)C) = D10×5 · C5×50  AB ⇒ 10·100·5=5,000 scalar multiplications  DC ⇒ 10·5·50 =2,500 scalar multiplications  (A(BC)) = A10×100 · E100×50  BC ⇒ 100·5·50=25,000 scalar multiplications  AE ⇒ 10·100·50 =50,000 scalar multiplications
  • 9. Matrix-chain Multiplication Problem Matrix-chain multiplication problem  Given a chain A1, A2, …, An of n matrices, where for i=1, 2, …, n, matrix Ai has dimension pi-1×pi  Parenthesize the product A1A2…An such that the total number of scalar multiplications is minimized Brute force method of exhaustive search takes time exponential in n
  • 10. Step 1: The structure of an optimalparenthesization The optimal substructure of this problem is as follows. Suppose that the optimal parenthesization of AiAi+1…Aj splits the product between Ak and Ak+1. Then the parenthesization of the subchain AiAi+1…Ak within this optimal parenthesization of AiAi+1…Aj must be an optimal parenthesization of AiAi+1…Ak. A similar observation holds for the parenthesization of the subchain Ak+1Ak+2…Aj in the optimal parenthesization of AiAi+1…Aj.
  • 11. Step 2: A recursive solution Let m[i, j] be the minimum number of scalar multiplications needed to compute the matrix AiAi+1…Aj; the cost of a cheapest way to compute A1A2…An would thus be m[1, n]. 0  i= j m[i, j ] =  min{m[i, k ] + m[k + 1, j ] + pi −1 pk p j } i < j  i ≤k < j 
  • 12. Step 3: Computing the optimal costsMATRIX-CHAIN-ORDER(p)1 n←length[p] - 12 for i←1 to n3 do m[i, i]←04 for l←2 to n5 do for i←1 to n - l + 16 do j ←i + l-17 m[i, j]←∞8 for k←i to j - 19 do q←m[i, k] + m[k + 1, j] +pi-1pkpj10 if q < m[i, j]11 then m[i, j]←q12 s[i, j]←k13 return m and s It’s running time is O(n3).
  • 13. matrix dimension---------------------- A1 30 X 35 A2 35 X 15 A3 15 X 5 A4 5 X 10 A5 10 X 20 A6 20 X 25
  • 14. Step 4: Constructing an optimal solutionPRINT-OPTIMAL-PARENS(s, i, ,j)1 if j =i2 then print “A”,i3 else print “(”4 PRINT-OPTIMAL-PARENS(s, i, s[i, j])5 PRINT-OPTIMAL-PARENS(s, s[i, j] + 1, j)6 print “)” In the above example, the call PRINT-OPTIMAL- PARENS(s, 1, 6) computes the matrix-chain product according to the parenthesization ((A1(A2A3))((A4A5)A6)) .
  • 15. Longest common subsequence Formally, given a sequence X ={x1, x2, . . . , xm}, another sequence Z ={z1, z2, . . . , zk} is a subsequence of X if there exists a strictly increasing sequence i1, i2, . . . , ik of indices of X such that for all j = 1, 2, . . . , k, we have xij = zj. For example, Z ={B, C, D, B} is a subsequence of X ={A, B, C, B, D, A, B } with corresponding index sequence {2, 3, 5, 7}.
  • 16.  Given two sequences X and Y, we say that a sequence Z is a common subsequence of X and Y if Z is a subsequence of both X and Y. In the longest-common-subsequence problem, we are given two sequences X = {x1, x2, . . . , xm} and Y ={y1, y2, . . . , yn} and wish to find a maximum-length common subsequence of X and Y.
  • 17. Step 1: Characterizing a longest commonsubsequence Let X ={x1, x2, . . . , xm} and Y ={y1, y2, . . . , yn} be sequences, and let Z ={z1, z2, . . . , zk} be any LCS of X and Y. 1. If xm = yn, then zk = xm = yn and Zk-l is an LCS of Xm-l and Yn-l. 2. If xm≠yn, then zk≠xm implies that Z is an LCS of Xm-1 and Y. 3. If xm≠yn, then zk≠yn implies that Z is an LCS of X and Yn-l.
  • 18. Step 2: A recursive solution to subproblems Let us define c[i, j] to be the length of an LCS of the sequences Xi and Yj. The optimal substructure of the LCS problem gives the recursive formula 0 i = 0orj = 0  c[i, j ] = c[i − 1, j − 1] + 1 i, j > 0andxi = y j max(c[i, j − 1], c[i − 1, j ]) i, j > 0andx ≠ y  i j
  • 19. Step 3: Computing the length of an LCSLCS-LENGTH(X,Y)1 m←length[X]2 n←length[Y]3 for i←1 to m4 do c[i,0]←05 for j←0 to n6 do c[0, j]←07 for i←1 to m8 do for j←1 to n9 do if xi = yj10 then c[i, j]←c[i - 1, j -1] + 111 b[i, j] ←”↖”12 else if c[i - 1, j]←c[i, j - 1]13 then c[i, j]←c[i - 1, j]14 b[i, j] ←”↑”15 else c[i, j]←c[i, j -1]16 b[i, j] ←”←”17 return c and b
  • 20. The running time of the procedure isO(mn).x=ABCBDAB and y=BDCABA,
  • 21. Step 4: Constructing an LCSPRINT-LCS(b,X,i,j)1 if i = 0 or j = 02 then return3 if b[i, j] = “↖"4 then PRINT-LCS(b,X,i - 1, j - 1)5 print xi6 elseif b[i,j] = “↑"7 then PRINT-LCS(b,X,i - 1,j)8 else PRINT-LCS(b,X,i, j - 1)
  • 22. Elements of dynamic programming For dynamic programming, Optimization problem must have following to be applicable optimal substructure overlapping subproblems Memoization.
  • 23. Optimal substructure The first step in solving an optimization problem by dynamic programming is to characterize the structure of an optimal solution. We say that a problem exhibits optimal substructure if an optimal solution to the problem contains within it optimal solutions to subproblems. Whenever a problem exhibits optimal substructure, it is a good clue that dynamic programming might apply.
  • 24. Overlapping subproblems When a recursive algorithm revisits the same problem over and over again, we say that the optimization problem has overlapping subproblems
  • 25. Memoization A memoized recursive algorithm maintains an entry in a table for the solution to each subproblem. Each table entry initially contains a special value to indicate that the entry has yet to be filled in. When the subproblem is first encountered during the execution of the recursive algorithm, its solution is computed and then stored in the table. Each subsequent time that the subproblem is encountered, the value stored in the table is simply looked up and returned.