Dynamic Programming - Part II

Dynamic Programming - Part II

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  Design and Analysis of Algorithms DYNAMICPROGRAMMING PART II Maximum Value Contiguous Subarray Maximum Increasing Subsequence Coin Change Algorithms Dynamic Programming - Part II 1
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   Instructor Prof. AmrinderArora amrinder@gwu.edu Please copy TA on emails Please feel free to call as well   Available for study sessions Science and Engineering Hall GWU Algorithms Dynamic Programming - Part II 2 LOGISTICS
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  Algorithms Analysis Asymptotic NP- Completeness Design D&C Greedy DP Graph B&B Applications Algorithms Dynamic Programming- Part II 3 WHERE WE ARE  Done  Done  Finishing today..  Done
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   http://video.google.com/videoplay?docid=16073867 32227802988#  http://people.csail.mit.edu/bdean/6.046/dp/ AlgorithmsDynamic Programming - Part II 4 DP (CONT.)
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   Given anArray A(1..n)  To find A(i..j) such that the sum of the terms in the subarray is maximized.  Insights  If no negative terms in the array, then of course, we just select the entire array.  So, this problem is meaningful only when we have negative numbers  We can find sum of all contiguous subarrays in O(n3) time.  Can we do better? Algorithms Dynamic Programming - Part II 5 MAXIMUM VALUE CONTIGUOUS SUBSEQUENCE
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  MaxValue = -infinity for i = 1 to n { for j = i to n { currSum = findSubArraySum(i,j) if CurrSum > MaxValue, then MaxValue = CurrSum } } Procedure FindSubArraySum(i,j) Double sum = 0 For int k = i to j { sum = sum + A[k] } return sum } Algorithms Dynamic Programming - Part II 6 ALGORITHM MVCS1 Brute Force Search! O(n3) time
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  Procedure InitSubArraySums() { forint i = 1 to n { double sum = 0 for int k = i to n { sum = sum + A(k) SubArraySum[i][k] = sum } } } InitSubArraySums(); MaxValue = - infinity for i = 1 to n { for j = i to n { currSum = SubArraySum[i][j] If currSum > MaxValue, then MaxValue = currSum } } Algorithms Dynamic Programming - Part II 7 ALGORITHM MVCS2 Still a full search, but don’t recompute sums O(n2) time
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   What canbe a greedy variation of MVCS algorithm?  What can be a D&C version of MVCS algorithm?  How about Dynamic Programming? Algorithms Dynamic Programming - Part II 8 USING OTHER TECHNIQUES FOR MVCS
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  Using Dynamic ProgrammingTemplate 1. N: MVCS(i): value of MVCS ending at position i. 2. O: Prove Optimal Substructure Holds (What is our claim?) 3. R: MVCS(i) = max {MVCS(i-1) + A[i],A[i]} 4. A: Start from left, and keep track of maximum MVCS(i) encountered. 1. For I = 1 to n Algorithms Dynamic Programming - Part II 9 MVCS3 O(n) time
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   Using theArray used in class  A[]: 22, -17, -71, 12, -22, 81, -10, 63  MVCS[]: 22, 5, -66, 12, -10, 81, 71, 134 Algorithms Dynamic Programming - Part II 10 SAMPLE RUN OF MVCS3
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   Given arrayA(1..n)  Find a subsequence (not necessarily contiguous) that is strictly increasing.  For example  {1, 7, 2, 8, 4, 1, 6, 11, 3, 15, 5, 12, 14} Algorithms Dynamic Programming - Part II 11 LONGEST INCREASING SUBSEQUENCE Longest subsequence is of length 7: {1, .., 2, .., 4, .., 6, 11, .., .., .., 12, 14}
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   Given arrayA(1..n)  LIS(i) = longest strictly increasing subsequence that ends at i.  LIS(i) = max {LIS(j)} + 1, such that j < i, and A[j] < A[i] [Or A[j] ≤ A[i] if we are looking for that condition] Algorithms Dynamic Programming - Part II 12 LONGEST INCREASING SUBSEQUENCE
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   What isthe time complexity of this algorithm? Algorithms Dynamic Programming - Part II 13 LIS
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   Using theArray used in Class  {1, 7, 2, 8, 4, 1, 6, 11, 3, 15, 5, 12, 14}  {1, 2, 2, 3, 3, 1, 4, 5, 3, 6, 4, 6, 7}  What are the LIS values? Algorithms Dynamic Programming - Part II 14 SAMPLE RUN OF LIS
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   Given aset of coins with values v1, v2, … vn such that v1 = 1, v1 < v2 < v3 < v4 < … < vn  We want to make change for a given value S using as few coins as possible Algorithms Dynamic Programming - Part II 15 COIN CHANGE
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   http://people.csail.mit.edu/bdean/6.046/dp/ Algorithms DynamicProgramming - Part II 16 AN INTERESTING SET OF PROBLEMS
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  Algorithms Dynamic Programming- Part II 17 80% OF SUCCESS IS SHOWING UP.
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  Algorithms Dynamic Programming- Part II 18 REST 80% OF SUCCESS IS BEING PRESENT!
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  CS 6212 Analysis Asymptotic NP- Completeness Design D&C Greedy DP Graph B&B Applications Algorithms DynamicProgramming - Part II 19 WHERE WE ARE  Done  Done  Done  Done