Input: n jobs (associated start time s j, finish time f j, and value v j ) for j = 1 to n M[j] = empty M[0] = 0. M-Compute-Opt(n)

Transcription

1 Objec&ves Dnamic Programming Ø Wrapping up: weighted interval schedule Ø Ø Subset Sums Summar: Proper&es of Problems for DP Polnomial number of subproblems Solu&on to original problem can be easil computed from solu&ons to subproblems Natural ordering of subproblems, eas to compute recurrence ar 25, 216 SI211 - Sprenkle 1 ar 25, 216 SI211 - Sprenkle 2 Review: Weighted Interval Scheduling Job j starts at s j, finishes at f j, and has weight or value v j Two jobs are compa&ble if the don't overlap Goal: find maimum weight subset of mutuall compa&ble jobs ar 25, 216 SI211 - Sprenkle 3 Weighted Interval Scheduling: emoiza&on nalsis Input: n jobs (associated start time s j, finish time f j, and value v j ) Sort jobs b finish times so that f 1 f 2... f n O(n log n) ompute p(1), p(2),, p(n) O(n log n) [j] = empt [] = -ompute-opt(j): if [j] is empt: [j] = ma(v j + -ompute-opt(p(j)), -ompute-opt(j-1)) return [j] -ompute-opt(n) ar 25, 216 SI211 - Sprenkle 4 Weighted Interval Scheduling: Finding a Solu&on Dnamic programming algorithms compute op#mal value What if we want the solu#on itself (not simpl the value)? Do some post-processing -ompute-opt(n) Find-Solution(n) Runtime? def Find-Solution(j): if j = : output nothing elif v j + [p(j)] > [j-1]: print j Find-Solution(p(j)) else: ar 25, 216 Find-Solution(j-1) SI211 - Sprenkle 5 Turning it round We solved the Fibonacci problem as both recursive/memoized and an itera+ve algorithm an we write this algorithm as an iterative solution? Input: n jobs (associated start time s j, finish time f j, and value v j ) Sort jobs b finish times so that f 1 f 2... f n ompute p(1), p(2),, p(n) [j] = empt [] = -ompute-opt(j): if [j] is empt: [j] = ma(v j + -ompute-opt(p(j)), -ompute-opt(j-1)) return [j] -ompute-opt(n) ar 25, 216 SI211 - Sprenkle 6 1

2 Towards Itera&ve Solu&on p(j) D F G H Itera&ve Solu&on uild up solu&on from subproblems instead of breaking down Input: n, s 1,,s n, f 1,,f n, v 1,,v n Sort jobs b finish times so that f 1 f 2... f n. ompute p(1), p(2),, p(n) [] = [j] = ma(v j + [p(j)], [j-1]) Runtime? Tpicall, we ll take itera&ve approach ar 25, 216 SI211 - Sprenkle 7 ar 25, 216 SI211 - Sprenkle 8 ample: Itera&vel p(j) ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) D F G H ar 25, 216 SI211 - Sprenkle 9 D F G H ar 25, 216 SI211 - Sprenkle 1 ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) D F G H 1 ar 25, 216 SI211 - Sprenkle 11 D F G H 1 ar 25, 216 SI211 - Sprenkle 12 2

3 ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) D F G H 1 2 ar 25, 216 SI211 - Sprenkle 13 D F G H ar 25, 216 SI211 - Sprenkle 14 ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) D F G H ar 25, 216 SI211 - Sprenkle 15 D F G H ar 25, 216 SI211 - Sprenkle nd so on. 16 ample: Itera&vel [j] = ma(v j + [p(j)], [j-1]) Find the solution? D F G H ar 25, 216 SI211 - Sprenkle 17 Review: Weighted Interval Scheduling 1. Determine op&mal substructure of problem Ø Define the recurrence rela&on 2. Define algorithm to find the value of op&mal solu&on 3. Op&onall, change algorithm to an itera#ve rather than recursive solu&on 4. Define algorithm to find op#mal solu#on 5. nalze running &me of algorithms ap to weighted-interval scheduling ar 25, 216 SI211 - Sprenkle 18 3

4 SGNTD LST SQURS Founda&onal problem in sta&s&c and numerical analsis Given n points in the plane: ( 1, 1 ), ( 2, 2 ),..., ( n, n ) Find a line = a + b that minimizes the sum of the squared error Ø line of best fit Sum of squared error n SS = ( i a i b) 2 i=1 ar 25, 216 SI211 - Sprenkle 19 ar 25, 216 SI211 - Sprenkle 2 Founda&onal problem in sta&s&c and numerical analsis Given n points in the plane: ( 1, 1 ), ( 2, 2 ),..., ( n, n ) Find a line = a + b that minimizes the sum of the squared error Ø line of best fit What happens to the error if we tr to fit one line to these points? Sum of squared error n SS = ( i a i b) 2 i=1 losed form solu&on. alculus min error is achieved when a = n i i i ( i i ) ( i i ) i, b = n 2 i ( i ) 2 i i i a i i ar 25, 216 SI211 - Sprenkle 21 n What paiern does it seem like these points have? ar 25, 216 SI211 - Sprenkle 22 What happens to the error if we tr to fit one line to these points? Ø Large error Points lie roughl on a sequence of line segments Given n points in the plane ( 1, 1 ), ( 2, 2 ),..., ( n, n ) with 1 < 2 <... < n, find a sequence of line segments that minimizes f() If I want the best fit, how man lines should I use? Paiern: ore like 3 lines ar 25, 216 SI211 - Sprenkle 23 ar 25, 216 SI211 - Sprenkle 24 4

5 Points lie roughl on a sequence of line segments Given n points in the plane ( 1, 1 ), ( 2, 2 ),..., ( n, n ) with 1 < 2 <... < n, find a sequence of line segments that minimizes f() What's a reasonable choice for f() to balance accurac and parsimon? goodness of fit number of lines Points lie roughl on a sequence of several line segments. Given n points in the plane ( 1, 1 ), ( 2, 2 ),..., ( n, n ) with 1 < 2 <... < n, find a sequence of line segments that minimizes: Ø : sum of the sums of the squared errors in each segment Ø L: the number of lines Tradeoff func+on: + c L, for some constant c >. How should we define an optimal solution? ar 25, 216 SI211 - Sprenkle 25 ar 25, 216 SI211 - Sprenkle 26 What made it seem like the points were in 3 lines? What happened? What made it seem like the points were in 3 lines? What happened? ar 25, 216 SI211 - Sprenkle 27 rror increased Looking for change in linear approima&on Ø Where to par&&on points into line segments ar 25, 216 SI211 - Sprenkle 28 Recall: Proper&es of Problems for DP Polnomial number of subproblems Solu&on to original problem can be easil computed from solu&ons to subproblems Natural ordering of subproblems, eas to compute recurrence We need to: Figure out how to break the problem into subproblems Figure out how to compute solution from subproblems Define the recurrence relation between the problems Toward a Solu&on onsider just the first or last point What do we know about those points? their segments? cost of a segment? ar 25, 216 SI211 - Sprenkle 29 ar 25, 216 SI211 - Sprenkle 3 5

6 Toward a Solu&on p n can onl belong to one segment Ø Segment: p i,, p n Ø ost: c (cost for segment) + error of segment What is the remaining problem? Toward a Solu&on p n can onl belong to one segment Ø Segment: p i,, p n Ø ost: c (cost for segment) + error of segment What is the remaining problem? Ø Solve for p 1,, p i-1 Net: Formulate as a recurrence ar 25, 216 SI211 - Sprenkle 31 ar 25, 216 SI211 - Sprenkle 32 Dnamic Programming: ul&wa hoice Nota&on. Ø OPT(j) = minimum cost for points p 1, p i+1,, p j. Ø e(i, j) = minimum sum of squares for points p i, p i+1,, p j. How do we compute OPT(j)? Ø Last problem: binar decision (include job or not) Ø This &me: mul+wa decision Which op&on do we choose? ar 25, 216 SI211 - Sprenkle 33 Dnamic Programming: ul&wa hoice Nota&on. Ø OPT(j) = minimum cost for points p 1, p i+1,, p j. Ø e(i, j) = minimum sum of squares for points p i, p i+1,, p j. To compute OPT(j): Ø Last segment contains points p i, p i+1,, p j for some i Ø ost = e(i, j) + c + OPT(i-1). $ & if j = OPT( j) = % min { e(i, j) + c + OPT(i 1) } otherwise '& 1 i j ar 25, 216 SI211 - Sprenkle 34 : lgorithm INPUT: n, p 1,,p N, c Segmented-Least-Squares() [] = e[][] = # needed? for i = 1 to j e[i][j] = least square error for the segment p i,, p j [j] = min 1 i j (e[i][j] + c + [i-1]) return [n] osts? ar 25, 216 SI211 - Sprenkle 35 : lgorithm nalsis INPUT: n, p 1,,p N, c can be improved to O(n 2 ) b Segmented-Least-Squares() pre-computing various statistics [] = e[][] = for i = 1 to j O(n 3 ) e[i][j] = least square error for the segment p i,, p j O(n 2 ) [j] = min 1 i j (e[i][j] + c + [i-1]) return [n] How do we find the solution? can be improved to O(n 2 ) b pre-computing various statistics oileneck: compu&ng e(i, j) for O(n 2 ) pairs, per pair using previous formula ar 25, 216 SI211 - Sprenkle 36 6

7 Post-Processing: Finding the Solu&on FindSegments(j): if j = : output nothing else: Find an i that minimizes e i,j + c + [i-1] Output the segment {p i,, p j } FindSegments(i-1) ost? O(n 2 ) Dnamic Programming Process Determine op&mal substructure of problem Ø Define the recurrence rela&on Define algorithm to find the value of op&mal solu&on Op&onall, change algorithm to an itera#ve rather than recursive solu&on Define algorithm to find op#mal solu#on nalze running &me of algorithms ar 25, 216 SI211 - Sprenkle 37 ar 25, 216 SI211 - Sprenkle 38 Looking head Wiki onda Ø Sec&ons Problem Set 8 due Frida ar 25, 216 SI211 - Sprenkle 39 7