Lecture16 - Dynamic Programming CSE 421 Algorithms Richard...

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1 CSE 421 Algorithms Richard Anderson Lecture 16 Dynamic Programming Dynamic Programming Weighted Interval Scheduling Given a collection of intervals I 1 ,…,I n with weights w 1 ,…,w n , choose a maximum weight set of non-overlapping intervals 4 6 3 5 7 6 Optimality Condition Opt[ j ] is the maximum weight independent set of intervals I 1 , I 2 , . . ., I j Opt[ j ] = max( Opt[ j – 1], w j + Opt[ p[ j ] ]) – Where p[ j ] is the index of the last interval which finishes before I j starts Algorithm MaxValue(j) = if j = 0 return 0 else return max( MaxValue(j-1), w j + MaxValue(p[ j ])) Worst case run time: 2 n A better algorithm M[ j ] initialized to -1 before the first recursive call for all j MaxValue(j) = if j = 0 return 0; else if M[ j ] != -1 return M[ j ]; else M[ j ] = max(MaxValue(j-1), w j + MaxValue(p[ j ])); return M[ j ]; Iterative Algorithm Express the MaxValue algorithm as an iterative algorithm MaxValue { }
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2 Fill in the array with the Opt values Opt[ j ] = max (Opt[ j – 1], w j + Opt[ p[ j ] ]) 4 7 4 6 7 6 2 Computing the solution Opt[ j ] = max (Opt[ j – 1], w j
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