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CSC401 – Analysis of Algorithms Chapter 5 -- 3 Dynamic Programming

CSC401 – Analysis of Algorithms Chapter 5 -- 3 Dynamic Programming. Objectives: Present the Dynamic Programming paradigm Review the Matrix Chain-Product problem Discuss the General Technique Solve the 0-1 Knapsack Problem Introduce pseudo-polynomial-time algorithms. Dynamic Programming.

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CSC401 – Analysis of Algorithms Chapter 5 -- 3 Dynamic Programming

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  1. CSC401 – Analysis of Algorithms Chapter 5 -- 3Dynamic Programming Objectives: Present the Dynamic Programming paradigm Review the Matrix Chain-Product problem Discuss the General Technique Solve the 0-1 Knapsack Problem Introduce pseudo-polynomial-time algorithms CSC401: Analysis of Algorithms

  2. Dynamic Programming • Dynamic Programmingis a general algorithm design technique • Invented by American mathematician Richard Bellman in the 1950s to solve optimization problems • “Programming” here means “planning” • Main idea: • solve several smaller (overlapping) subproblems • record solutions in a table so that each subproblem is only solved once • final state of the table will be (or contain) solution CSC401: Analysis of Algorithms<footer>

  3. Example: Fibonacci numbers • Recall definition of Fibonacci numbers: • f(0) = 0 • f(1) = 1 • f(n) = f(n-1) + f(n-2) • Computing the nth Fibonacci number recursively (top-down): • f(n) • f(n-1) + f(n-2) • f(n-2) + f(n-3) f(n-3) + f(n-4) • ...... • Time efficiency: T(n) = T(n-1)+T(n-2)+1 O(2n) CSC401: Analysis of Algorithms<footer>

  4. Example: Fibonacci numbers • Computing the nth fibonacci number using bottom-up iteration: • f(0) = 0 • f(1) = 1 • f(2) = 0+1 = 1 • f(3) = 1+1 = 2 • f(4) = 1+2 = 3 • f(5) = 2+3 = 5 • f(n-2) = • f(n-1) = • f(n) = f(n-1) + f(n-2) • Time efficiency: O(n) CSC401: Analysis of Algorithms<footer>

  5. Examples of Dynamic Programming Algorithms • Computing binomial coefficients • Optimal chain matrix multiplication • Constructing an optimal binary search tree • Warshall’s algorithm for transitive closure • Floyd’s algorithms for all-pairs shortest paths • Some instances of difficult discrete optimization problems: • travelling salesman • knapsack CSC401: Analysis of Algorithms<footer>

  6. f B j e e A C d i i,j d f Optimal chain matrix multiplication • Review: Matrix Multiplication. • C = A×B • Aisd × eandBise × f • O(def)time CSC401: Analysis of Algorithms<footer>

  7. Matrix Chain-Product: Compute A=A0*A1*…*An-1 Ai is di × di+1 Problem: How to parenthesize? Example B is 3 × 100 C is 100 × 5 D is 5 × 5 (B*C)*D takes 1500 + 75 = 1575 operations B*(C*D) takes 1500 + 2500 = 4000 operations Matrix Chain-Products CSC401: Analysis of Algorithms<footer>

  8. Matrix Chain-Product Algorithm: Try all possible ways to parenthesize A=A0*A1*…*An-1 Calculate number of operations for each one Pick the best one Running time: The number of paranethesizations is equal to the number of binary trees with n nodes This is exponential! It is called the Catalan number, and it is almost 4n. This is a terrible algorithm! A Brute-force Approach CSC401: Analysis of Algorithms<footer>

  9. Idea #1: repeatedly select the product that uses (up) the most operations. Counter-example: A is 10 × 5 B is 5 × 10 C is 10 × 5 D is 5 × 10 Greedy idea #1 gives (A*B)*(C*D), which takes 500+1000+500 = 2000 operations A*((B*C)*D) takes 500+250+250 = 1000 operations Greedy Approaches • Idea #2: repeatedly select the product that uses the fewest operations. • Counter-example: • A is 101 × 11 • B is 11 × 9 • C is 9 × 100 • D is 100 × 99 • Greedy idea #2 gives A*((B*C)*D)), which takes 109989+9900+108900=228789 operations • (A*B)*(C*D) takes 9999+89991+89100=189090 operations • The greedy approach is not giving us the optimal value. CSC401: Analysis of Algorithms<footer>

  10. A “Recursive” Approach • Define subproblems: • Find the best parenthesization of Ai*Ai+1*…*Aj. • Let Ni,j denote the number of operations done by this subproblem. • The optimal solution for the whole problem is N0,n-1. • Subproblem optimality: The optimal solution can be defined in terms of optimal subproblems • There has to be a final multiplication (root of the expression tree) for the optimal solution. • Say, the final multiply is at index i: (A0*…*Ai)*(Ai+1*…*An-1). • Then the optimal solution N0,n-1 is the sum of two optimal subproblems, N0,i and Ni+1,n-1 plus the time for the last multiply. • If the global optimum did not have these optimal subproblems, we could define an even better “optimal” solution. CSC401: Analysis of Algorithms<footer>

  11. A Characterizing Equation • The global optimal has to be defined in terms of optimal subproblems, depending on where the final multiply is at. • Let us consider all possible places for that final multiply: • Recall that Ai is a di × di+1 dimensional matrix. • So, a characterizing equation for Ni,j is the following: • Note that subproblems are not independent--the subproblems overlap. CSC401: Analysis of Algorithms<footer>

  12. A Dynamic Programming Algorithm • Since subproblems overlap, we don’t use recursion. • Instead, we construct optimal subproblems “bottom-up.” • Ni,i’s are easy, so start with them • Then do length 2,3,… subproblems, and so on. • Running time: O(n3) AlgorithmmatrixChain(S): Input:sequence S of n matrices to be multiplied Output:number of operations in an optimal paranethization of S fori 1 to n-1 do Ni,i0 forb 1 to n-1 do fori 0 to n-b-1 do ji+b Ni,j+infinity fork i to j-1 do Ni,jmin{Ni,j , Ni,k +Nk+1,j +di dk+1 dj+1} CSC401: Analysis of Algorithms<footer>

  13. answer N n-1 … 0 1 2 j 0 1 … i n-1 A Dynamic Programming Algorithm • The bottom-up construction fills in the N array by diagonals • Ni,j gets values from pervious entries in i-th row and j-th column • Filling in each entry in the N table takes O(n) time. • Total run time: O(n3) • Getting actual parenthesization can be done by remembering “k” for each N entry CSC401: Analysis of Algorithms<footer>

  14. The General Dynamic Programming Technique • Applies to a problem that at first seems to require a lot of time (possibly exponential), provided we have: • Simple subproblems: the subproblems can be defined in terms of a few variables, such as j, k, l, m, and so on. • Subproblem optimality: the global optimum value can be defined in terms of optimal subproblems • Subproblem overlap: the subproblems are not independent, but instead they overlap (hence, should be constructed bottom-up). CSC401: Analysis of Algorithms<footer>

  15. The 0/1 Knapsack Problem • Given: A set S of n items, with each item i having • bi - a positive benefit • wi - a positive weight • Goal: Choose items with maximum total benefit but with weight at most W. • If we are not allowed to take fractional amounts, then this is the 0/1 knapsack problem. • In this case, we let Tdenote the set of items we take • Objective: maximize • Constraint: CSC401: Analysis of Algorithms<footer>

  16. Example • Given: A set S of n items, with each item i having • bi - a positive benefit • wi - a positive weight • Goal: Choose items with maximum total benefit but with weight at most W. “knapsack” • Solution: • 5 (2 in) • 3 (2 in) • 1 (4 in) Items: 1 2 3 4 5 Weight: 4 in 2 in 2 in 6 in 2 in 9 in Benefit: $20 $3 $6 $25 $80 CSC401: Analysis of Algorithms<footer>

  17. A 0/1 Knapsack Algorithm, First Attempt • Sk: Set of items numbered 1 to k. • Define B[k] = best selection from Sk. • Problem: does not have subproblem optimality: • Consider S={(3,2),(5,4),(8,5),(4,3),10,9)} weight-benefit pairs Best for S4: Best for S5: CSC401: Analysis of Algorithms<footer>

  18. A 0/1 Knapsack Algorithm, Second Attempt • Sk: Set of items numbered 1 to k. • Define B[k,w] = best selection from Sk with weight exactly equal to w • Good news: this does have subproblem optimality: • I.e., best subset of Sk with weight exactly w is either the best subset of Sk-1 w/ weight w or the best subset of Sk-1 w/ weight w-wk plus item k. CSC401: Analysis of Algorithms<footer>

  19. The 0/1 Knapsack Algorithm • Recall definition of B[k,w]: • Since B[k,w] is defined in terms of B[k-1,*], we can reuse the same array • Running time: O(nW). • Not a polynomial-time algorithm if W is large • This is a pseudo-polynomial time algorithm Algorithm01Knapsack(S,W): Input:set S of items w/ benefit biand weight wi; max. weight W Output:benefit of best subset with weight at most W forw 0to W do B[w]0 fork 1 to n do forwWdownto wkdo ifB[w-wk]+bk > B[w]then B[w]B[w-wk]+bk CSC401: Analysis of Algorithms<footer>

  20. Pseudo-Polynomial Time Algorithms • An algorithm is a pseudo-polynomial time algorithm, if the running time of the algorithm depends on the magnitude of a number given in the input, not its encoding size • Example: The 0/1 Knapsack algorithm depends on W • Not a true polynomial-time algorithm • But usually it’s better than the brute-force algorithm CSC401: Analysis of Algorithms<footer>

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