4.2 Scheduling to Minimize Maximum Lateness

Transcription

1 4. Schedulng to Mnmze Maxmum Lateness

2 Schedulng to Mnmzng Maxmum Lateness Mnmzng lateness problem. Sngle resource processes one ob at a tme. Job requres t unts of processng tme and s due at tme d. If starts at tme s, t fnshes at tme f = s + t. Lateness: = max { 0, f - d }. Goal: schedule all obs to mnmze maxmum lateness L = max. 1 ob number Ex: t 3 tme requred d 1 3 deadlne d = 3 d 1 = 1 lateness = 4 Q. Whch schedule s better? d 1 = 1 d = 3 lateness =

3 Schedulng to Mnmzng Maxmum Lateness Mnmzng lateness problem. Sngle resource processes one ob at a tme. Job requres t unts of processng tme and s due at tme d. If starts at tme s, t fnshes at tme f = s + t. Lateness: = max { 0, f - d }. Goal: schedule all obs to mnmze maxmum lateness L = max. 1 ob number Ex: t 3 tme requred d 1 3 deadlne d = 3 d 1 = lateness = 4 Q. Whch schedule s better? A. The lower; lateness. d 1 = 1 d = 3 lateness =

4 Schedulng to Mnmzng Maxmum Lateness Mnmzng lateness problem. Sngle resource processes one ob at a tme. Job requres t unts of processng tme and s due at tme d. If starts at tme s, t fnshes at tme f = s + t. Lateness: = max { 0, f - d }. Goal: schedule all obs to mnmze maxmum lateness L = max ob number Ex: t tme requred d deadlne lateness = lateness = 0 max lateness = 6 d 3 = 9 d = 8 d 6 = 15 d 1 = 6 d 5 = 14 d 4 =

5 Mnmzng Maxmum Lateness: Greedy Algorthms Greedy template. Consder obs n some order. [Shortest processng tme frst] Consder obs n ascendng order of processng tme t (least work frst). [Earlest deadlne frst] Consder obs n ascendng order of deadlne d (nearest deadlne). [Smallest slack] Consder obs n ascendng order of slack d t (least tme to start to make deadlne). Q. Whch one do you thnk may work? (1 mn) 5

6 Mnmzng Maxmum Lateness: Greedy Algorthms Greedy template. Consder obs n some order. [Shortest processng tme frst] Consder obs n ascendng order of processng tme t (least work frst). d 1 t counterexample [Smallest slack] Consder obs n ascendng order of slack d - t (least tme to start to make deadlne). 1 t 1 d counterexample 6

7 Mnmzng Maxmum Lateness: Greedy Algorthm Greedy algorthm. Earlest deadlne frst. Sort n obs by deadlne so that d 1 d d n t 0 for = 1 to n Assgn ob to nterval [t, t + t ]: s t f t + t t t + t output ntervals [s, f ] 7

8 Mnmzng Maxmum Lateness: Greedy Algorthm Greedy algorthm. Earlest deadlne frst ob number t tme requred d deadlne max lateness = 1 d 1 = 6 d = 8 d 3 = 9 d 4 = 9 d 5 = 14 d 6 = Observaton. The greedy schedule has no dle tme. 8

9 Mnmzng Maxmum Lateness: No Idle Tme Observaton. There exsts an optmal schedule wth no dle tme. d = 4 d = d = d = 4 d = 6 d = Q*. How to prove that earlest-deadlne-frst greedy algorthm s optmal? 9

10 Mnmzng Maxmum Lateness: No Idle Tme Observaton. There exsts an optmal schedule wth no dle tme. d = 4 d = d = d = 4 d = 6 d = Q*. How to prove that earlest-deadlne-frst greedy algorthm s optmal? A. Idea of proof: exchange argument: Take an optmal schedule. Change nto greedy schedule wthout losng optmalty. but how? 10

11 Towards provng greedy s optmal Idea. Change optmal schedule to greedy wthout losng optmalty. Greedy Optmal Problems to solve frst: What do we know about the greedy schedule? How can we change the optmal to be more lke that wthout losng optmalty? 11

12 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap We ll now study a number of propertes of such an nverson. Q. How many nversons can a schedule from our Greedy algorthm have? (0, 1, or more than 1) 1

13 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Q. What s the dfference n maxmum lateness between two schedules wthout nversons and wthout dle tme? (1 mn) 13

14 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Q. What s the dfference n maxmum lateness between two schedules wthout nversons and wthout dle tme? Pf. Only dfference s an nverson of and wth equal deadlne (d =d ). Maxmum lateness of and s only nfluenced by last ob (f - d ). Maxmum lateness of and s the same f and are swapped. S S 14

15 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Observaton. All schedules wthout nversons have same lateness. Q. If a schedule (wth no dle tme) has an nverson, how can we fnd t? 15

16 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Observaton. All schedules wthout nversons have same lateness. Observaton. If a schedule (wth no dle tme) has an nverson, then t has one wth a par of nverted obs scheduled consecutvely. Pf. Q. How do we proof If, then? 16

17 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Observaton. All schedules wthout nversons have same lateness. Observaton. If a schedule (wth no dle tme) has an nverson, then t has one wth a par of nverted obs scheduled consecutvely. Pf. Suppose there s an nverson. gong through schedule, at some pont deadlne decreases. 17

18 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson before swap Observaton. Greedy schedule has no nversons. Observaton. All schedules wthout nversons have same lateness. Observaton. If a schedule (wth no dle tme) has an nverson, then t has one wth a par of nverted obs scheduled consecutvely. Pf. Suppose there s an nverson. There s a par of obs and such that: d < d but scheduled before. Walk through the schedule from to. Increasng deadlnes (= no nversons), at some pont deadlne decreases. 18

19 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson f before swap after swap Q. What happens to the number of nversons when we swap two adacent, nverted obs? (reduces, stays equal, ncreases) Q. Can we swap two adacent, nverted obs wthout ncreasng the maxmum lateness? f' 19

20 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson f before swap after swap Clam. Swappng two adacent, nverted obs reduces the number of nversons by one and does not ncrease the maxmum lateness. f' Pf. 0

21 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson f before swap after swap Clam. Swappng two adacent, nverted obs reduces the number of nversons by one and does not ncrease the maxmum lateness. f' Pf. Let be the max lateness before the swap, and let ' be t afterwards. Q. What happens wth the lateness of other obs? Q. What happens wth the lateness of? Q. What happens wth the lateness of? 1

22 Mnmzng Maxmum Lateness: Inversons Def. An nverson n schedule S s a par of obs and such that: d < d but scheduled before. nverson f before swap after swap Clam. Swappng two adacent, nverted obs reduces the number of nversons by one and does not ncrease the maxmum lateness. f' Pf. Let be the max lateness before the swap, and let ' be t afterwards. ' k = k for all k, (lateness other obs the same) ' (new lateness for smaller) If ob s late: l f d (defnton) f d ( fnshes at tme f ) f d (d d ) l (defnton)

23 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Idea of proof: Suppose S s not optmal. Take a specfc optmal schedule S*. Change to look lke greedy schedule (less nversons) wthout losng optmalty. 3

24 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Suppose S s not optmal. Defne S* to be an optmal schedule that has the fewest number of nversons (of all optmal schedules) and has no dle tme. Clearly S S*. 4

25 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Suppose S s not optmal. Defne S* to be an optmal schedule that has the fewest number of nversons (of all optmal schedules) and has no dle tme. Clearly S S*. Case analyss: If S* has no nversons If S* has an nverson 5

26 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Suppose S s not optmal. Defne S* to be an optmal schedule that has the fewest number of nversons (of all optmal schedules) and has no dle tme. Clearly S S*. Case analyss: If S* has no nversons. Q. How can we derve a contradcton? If S* has an nverson 6

27 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Suppose S s not optmal. Defne S* to be an optmal schedule that has the fewest number of nversons (of all optmal schedules) and has no dle tme. Clearly S S*. If S* has no nversons, then L S = L S*. Contradcton. If S* has an nverson, Q. How can we derve a contradcton? Greedy has no nversons. All schedules wthout nversons have same lateness. 7

28 Mnmzng Lateness: Analyss of Greedy Algorthm Theorem. Greedy schedule S s optmal. Pf. (by contradcton) Suppose S s not optmal. Defne S* to be an optmal schedule that has the fewest number of nversons (of all optmal schedules) and has no dle tme. Clearly S S*. If S* has no nversons, then maxl(s) = maxl(s*). Contradcton. If S* has an nverson, let - be an adacent nverson. swappng and does not ncrease the maxmum lateness and strctly decreases the number of nversons ths contradcts defnton of S* So S s an optmal schedule. Greedy has no nversons. All schedules wthout nversons have same lateness. Ths proof can be found on pages

29 Greedy Analyss Strateges Greedy algorthm stays ahead. Show that after each step of the greedy algorthm, ts soluton s at least as good as any other algorthm's. Exchange argument. Gradually transform an optmal soluton to the one found by the greedy algorthm wthout hurtng ts qualty. Structural. Dscover a smple "structural" bound assertng that every possble soluton must have a certan value. Then show that your algorthm always acheves ths bound. Q. Whch strategy dd we use for the problems n ths chapter (nterval schedulng, nterval parttonng, mnmzng lateness)? 9

30 Greedy Analyss Strateges Greedy algorthm stays ahead. Show that after each step of the greedy algorthm, ts soluton s at least as good as any other algorthm's. [Interval schedulng] Exchange argument. Gradually transform an optmal soluton to the one found by the greedy algorthm wthout hurtng ts qualty. [Mnmzng lateness, Interval schedulng] Structural. Dscover a smple "structural" bound assertng that every possble soluton must have a certan value. Then show that your algorthm always acheves ths bound. [Interval parttonng] 30

31 Example exam exercse Plannng a mn-trathlon: 1. swm 0 laps (one at a tme). bke 10 km (can be done smultaneously) 3. run 3 km (can be done smultaneously) expected tmes are gven for each contestant Def. The completon tme s the earlest tme all contestants are fnshed. Q. In what order should they start to mnmze the completon tme? Q. Proof that ths order s optmal (mnmal). Ex. 1 3 contestant s 3 4 tme requred for swmmng b tme requred for bkng r tme requred for runnng Come to the nstructon (Frday 13:45) f you do not know how to answer ths. 31

32 Varant: Schedulng to Mnmzng Total Lateness Mnmzng total lateness problem. Sngle resource processes one ob at a tme. Job requres t unts of processng tme and s due at tme d. If starts at tme s, t fnshes at tme f = s + t. Lateness: = max { 0, f - d }. Goal: schedule all obs to mnmze total lateness L = Σ. 1 ob number Ex: t 3 tme requred d 1 3 deadlne d = 3 d 1 = total lateness = 0+4 No polynomal algorthm can compute optmal schedule (unless P=NP) (see Master s course Advanced Algorthms) d 1 = 1 d = 3 total lateness =