Programmazione dinamica: Selezione di intervalli pesati. 4 novembre 2014

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1 Programmazione dinamica: Selezione di intervalli pesati 4 novembre 2014

2 6.1 Weighted Interval Scheduling

3 Interval scheduling problem. Interval Scheduling Job j starts at s j, finishes at f j. Two jobs compatible if they don't overlap. Goal: find biggest subset of mutually compatible jobs. a b c d e f g h Time

4 Weighted Interval Scheduling Weighted interval scheduling problem. Job j starts at s j, finishes at f j, and has weight or value v j. Two jobs compatible if they don't overlap. Goal: find maximum weight subset of mutually compatible jobs. weight = 9 a weight = 10 b weight = 7 c weight = 2 weight = 8 d e weight = 12 f 4 weight = 2 weight = g h Time

5 Weighted Interval Scheduling Weighted interval scheduling problem. Job j starts at s j, finishes at f j, and has weight or value v j. Two jobs compatible if they don't overlap. Goal: find maximum weight subset of mutually compatible jobs. Copyright 2005 Pearson Addison-Wesley. All rights reserved. 5 Insiemi compatibili: {1,3} di peso 2 {2} di peso 3 è la soluzione ottimale

6 Unweighted Interval Scheduling Review Note: Greedy algorithm works if all weights are 1. Consider jobs in ascending order of finish time. Add job to subset if it is compatible with previously chosen jobs. Observation. Greedy algorithm can fail spectacularly if arbitrary weights are allowed. weight = 999 weight = 1 6 a b Time

7 Esercizio Immaginate un problema reale (o concreto) che si possa formalizzare in un problema computazionale di Interval Scheduling pesato.

8 Weighted Interval Scheduling Notation. Label jobs by finishing time: f 1 f 2... f n. Def. p(j) = largest index i < j such that job i is compatible with j. Ex: (independently from weights) p(8) = 5, p(7) = 3, p(2) = Time

9 Soluzione = {5, 3, 1} Valore = = 8 Copyright 2005 Pearson Addison-Wesley. All rights reserved. 6-9

10 Dynamic Programming: Binary Choice Notation. OPT(j) = value of optimal solution to the problem consisting of job requests 1, 2,..., j. Case 1: OPT selects job j. can't use incompatible jobs { p(j) + 1, p(j) + 2,..., j - 1 } must include optimal solution to problem consisting of remaining compatible jobs 1, 2,..., p(j) Case 2: OPT does not select job j. optimal substructure must include optimal solution to problem consisting of remaining compatible jobs 1, 2,..., j-1 OPT ( j) 0 max v j OPT ( p( j)), OPT ( j 1) if j 0 otherwise 10

11 Weighted Interval Scheduling: Brute Force Brute force recursive algorithm. Input: n, s 1,,s n, f 1,,f n, v 1,,v n Sort jobs by finish times so that f 1 f 2... f n. Compute p(1), p(2),, p(n) Compute-Opt(j) { if (j = 0) return 0 else return max(v j + Compute-Opt(p(j)), Compute-Opt(j-1)) } 11

12 OPT ( j) 0 max v j OPT ( p( j)), OPT ( j 1) if j 0 otherwise Copyright 2005 Pearson Addison-Wesley. All rights reserved. 6-12

13 Weighted Interval Scheduling: Brute Force Observation. Recursive algorithm fails spectacularly because of redundant sub-problems exponential algorithms. Ex. Number of recursive calls for family of "layered" instances grows like Fibonacci sequence p(1) = 0, p(j) = j-2 T(n)= O(2 n ): too much!

14 Weighted Interval Scheduling: Memoization Memoization. Store results of each sub-problem in a cache; lookup as needed. Input: n, s 1,,s n, f 1,,f n, v 1,,v n Sort jobs by finish times so that f 1 f 2... f n. Compute p(1), p(2),, p(n) for j = 1 to n M[j] = empty M[0] = 0 M-Compute-Opt(j) { if (M[j] is empty) M[j] = max(w max(v j + M-Compute-Opt(p(j)), M-Compute-Opt(j-1)) return M[j] } 14

15 Weighted Interval Scheduling: Running Time Claim. Memoized version of algorithm takes O(n log n) time. Sort by finish time: O(n log n). Computing p( ) : O(n) after sorting by start time (exercise). M-Compute-Opt(j): each invocation takes O(1) time and either (i) returns an existing value M[j] (ii) fills in one new entry M[j] and makes two recursive calls Progress measure = # nonempty entries of M[]. initially = 0, throughout n. (ii) increases by 1 at most 2n recursive calls. Overall running time of M-Compute-Opt(n) is O(n). Remark. O(n) if jobs are pre-sorted by start and finish times. 15

16 Weighted Interval Scheduling: Bottom-Up Bottom-up dynamic programming. Unwind recursion. Input: n, s 1,,s n, f 1,,f n, v 1,,v n Sort jobs by finish times so that f 1 f 2... f n. Compute p(1), p(2),, p(n) Iterative-Compute-Opt { M[0] = 0 for j = 1 to n M[j] = max(v j + M[p(j)], M[j-1]) } 16

17 Iterative-Compute-Opt { M[0] = 0 for j = 1 to n M[j] = max(w j + M[p(j)], M[j-1]) } M[1] = max ( 2+M[0], M[0]) = max (2+0, 0) = 2 M[2] = max ( 4+M[0], M[1]) = max (4+0, 2) = 4 M[3] = max ( 4+M[1], M[2]) = max (4+2, 4) = 6 M[4] = max ( 7+M[0], M[3]) = max (7+0, 6) = 7 M[5] = max ( 2+M[3], M[4]) = max (2+6, 7) = 8 M[6] = max ( 1+M[3], M[5]) = max (1+6, 8) = Copyright 2005 Pearson Addison-Wesley. All rights reserved.

18 Weighted Interval Scheduling: Finding a Solution Q. Dynamic programming algorithms computes optimal value. What if we want the solution itself (the set of intervals)? A. Do some post-processing. 18 # of recursive calls n O(n). Run M-Compute-Opt(n) Run Find-Solution(n) Find-Solution(j) { if (j = 0) output nothing else if (v j + M[p(j)] > M[j-1]) print j Find-Solution(p(j)) else Find-Solution(j-1) }

19 Esempio del calcolo di una soluzione OPT ( j) 0 max v j OPT ( p( j)), OPT ( j 1) if j 0 otherwise M[1] = max ( 2+M[0], M[0]) = max (2+0, 0) = 2 M[2] = max ( 4+M[0], M[1]) = max (4+0, 2) = 4 M[3] = max ( 4+M[1], M[2]) = max (4+2, 4) = 6 M[4] = max ( 7+M[0], M[3]) = max (7+0, 6) = 7 M[5] = max ( 2+M[3], M[4]) = max (2+6, 7) = 8 M[6] = max ( 1+M[3], M[5]) = max (1+6, 8) = 8 M[6]=M[5]: 6 non appartiene a OPT M[5]=v 5 +M[3]: OPT contiene 5 e una soluzione ottimale al problema per {1,2,3} M[3]=v 3 +M[1]: OPT contiene 5, 3 e una soluzione ottimale al problema per {1} M[1]=v 1 +M[0]: OPT contiene 5, 3 e 1 (e una soluzione ottimale al problema vuoto) Soluzione = {5, 3, 1} Valore = = 8

20 Dynamic Programming Summary Recipe. Characterize structure of problem. Recursively define value of optimal solution. Compute value of optimal solution. Construct optimal solution from computed information. Dynamic programming techniques. Binary choice: weighted interval scheduling. Multi-way choice: segmented least squares. Adding a new variable: knapsack. Dynamic programming over intervals: RNA secondary structure. Top-down vs. bottom-up: different people have different intuitions. 20

21 Tale proprietà è detta: Proprietà della Sottostruttura Ottima.

22 Proprietà della sottostruttura ottima: Scheduling di attività Il punto cruciale qui è che O {n} : è una soluzione, cioè è un insieme di attività compatibili di {1,, p(n)} (perché le attività di O sono compatibili) è ottima, cioè ha cardinalità massima fra le soluzioni al sottoproblema relativo alle attività {1,, p(n)} (infatti, se così non fosse, esisterebbe una soluzione O per {1,, p(n)} di cardinalità strettamente superiore a quella di O {n} e allora O {n} sarebbe una soluzione strettamente migliore di O.

23 Riepilogo e riferimenti Primi esempi della programmazione dinamica: calcolo dei numeri di Fibonacci (la volta scorsa), selezione di intervalli pesati (oggi: [KT] par. 6.1) Domani mercoledì 5: ancora programmazione dinamica e breve test di valutazione su algoritmi ricorsivi e analisi, relazioni di ricorrenza, tecnica divide et impera e in particolare MergeSort, Partition, QuickSort.