CSE 417, Winter Greedy Algorithms

Transcription

1 CSE 417, Witer 2012 Greedy Algorithms Be Birbaum Widad Machmouchi Slides adapted from Larry Ruzzo, Steve Taimoto, ad Kevi Waye 1

2 Chapter 4 Greedy Algorithms Slides by Kevi Waye. Copyright 2005 Pearso-Addiso Wesley. All rights reserved. 2

3 4.1 Iterval Schedulig

4 Iterval Schedulig Iterval schedulig. Job j starts at s j ad fiishes at f j. Two jobs compatible if they do't overlap. Goal: fid maximum subset of mutually compatible jobs. a b c d e f g h Time 4

5 Iterval Schedulig: Greedy Algorithms Greedy template. Cosider jobs i some atural order. Take each job provided it's compatible with the oes already take. [Earliest start time] Cosider jobs i ascedig order of s j. [Earliest fiish time] Cosider jobs i ascedig order of f j. [Shortest iterval] Cosider jobs i ascedig order of f j - s j. [Fewest coflicts] For each job j, cout the umber of coflictig jobs c j. Schedule i ascedig order of c j. 5

6 Iterval Schedulig: Greedy Algorithms Greedy template. Cosider jobs i some atural order. Take each job provided it's compatible with the oes already take. couterexample for earliest start time couterexample for shortest iterval couterexample for fewest coflicts 6

7 Iterval Schedulig: Greedy Algorithm Greedy algorithm. Cosider jobs i icreasig order of fiish time. Take each job provided it's compatible with the oes already take. Sort jobs by fiish times so that f 1 f 2... f. set of jobs selected A φ for j = 1 to { if (job j compatible with A) A A {j} } retur A Implemetatio. O( log ). Remember job j* that was added last to A. Job j is compatible with A if s j f j*. 7

8 Iterval Schedulig: Aalysis Theorem. Greedy algorithm is optimal. Pf. (by cotradictio) Assume greedy is ot optimal, ad let's see what happes. Let i 1, i 2,... i k deote set of jobs selected by greedy. Let j 1, j 2,... j m deote set of jobs i the optimal solutio with i 1 = j 1, i 2 = j 2,..., i r = j r for the largest possible value of r. job i r+1 fiishes before j r+1 Greedy: i 1 i 2 i r i r+1 OPT: j 1 j 2 j r j r+1... why ot replace job j r+1 with job i r+1? 8

9 Iterval Schedulig: Aalysis Theorem. Greedy algorithm is optimal. Pf. (by cotradictio) Assume greedy is ot optimal, ad let's see what happes. Let i 1, i 2,... i k deote set of jobs selected by greedy. Let j 1, j 2,... j m deote set of jobs i the optimal solutio with i 1 = j 1, i 2 = j 2,..., i r = j r for the largest possible value of r. job i r+1 fiishes before j r+1 Greedy: i 1 i 2 i r i r+1 OPT: j 1 j 2 j r i r+1... solutio still feasible ad optimal, but cotradicts maximality of r. 9

10 4.1 Iterval Partitioig

11 Iterval Partitioig Iterval partitioig. Lecture j starts at s j ad fiishes at f j. Goal: fid miimum umber of classrooms to schedule all lectures so that o two occur at the same time i the same room. Ex: This schedule uses 4 classrooms to schedule 10 lectures. 4 e j 3 c d g 2 b h 1 a f i 9 9: : : :30 1 1:30 2 2:30 3 3:30 4 4:30 Time 11

12 Iterval Partitioig Iterval partitioig. Lecture j starts at s j ad fiishes at f j. Goal: fid miimum umber of classrooms to schedule all lectures so that o two occur at the same time i the same room. Ex: This schedule uses oly 3. 3 c d f j 2 b g i 1 a e h 9 9: : : :30 1 1:30 2 2:30 3 3:30 4 4:30 Time 12

13 Iterval Partitioig: Lower Boud o Optimal Solutio Def. The depth of a set of ope itervals is the maximum umber that cotai ay give time. Key observatio. Number of classrooms eeded depth. Ex: Depth of schedule below = 3 schedule below is optimal. a, b, c all cotai 9:30 Q. Does there always exist a schedule equal to depth of itervals? 3 c d f j 2 b g i 1 a e h 9 9: : : :30 1 1:30 2 2:30 3 3:30 4 4:30 Time 13

14 Iterval Partitioig: Greedy Algorithm Greedy algorithm. Cosider lectures i icreasig order of start time: assig lecture to ay compatible classroom. Sort itervals by startig time so that s 1 s 2... s. d 0 umber of allocated classrooms for j = 1 to { if (lecture j is compatible with some classroom k) schedule lecture j i classroom k else allocate a ew classroom d + 1 schedule lecture j i classroom d + 1 d d + 1 } Implemetatio. O( log ). For each classroom k, maitai the fiish time of the last job added. Keep the classrooms i a priority queue. 14

15 Iterval Partitioig: Greedy Aalysis Observatio. Greedy algorithm ever schedules two icompatible lectures i the same classroom. Theorem. Greedy algorithm is optimal. Pf. Let d = umber of classrooms that the greedy algorithm allocates. Classroom d is opeed because we eeded to schedule a job, say j, that is icompatible with all d-1 other classrooms. These d jobs each ed after s j. Sice we sorted by start time, all these icompatibilities are caused by lectures that start o later tha s j. Thus, we have d lectures overlappig at time s j + ε. Key observatio all schedules use d classrooms. 15

16 4.2 Schedulig to Miimize Lateess

17 Schedulig to Miimizig Lateess Miimizig lateess problem. Sigle resource processes oe job at a time. Job j requires t j uits of processig time ad is due at time d j. If j starts at time s j, it fiishes at time f j = s j + t j. Lateess: l j = max { 0, f j - d j }. Goal: schedule all jobs to miimize maximum lateess L = max l j. Ex: t j d j lateess = 2 lateess = 0 max lateess = 6 d 3 = 9 d 2 = 8 d 6 = 15 d 1 = 6 d 5 = 14 d 4 =

18 Miimizig Lateess: Greedy Algorithms Greedy template. Cosider jobs i some order. [Shortest processig time first] Cosider jobs i ascedig order of processig time t j. [Earliest deadlie first] Cosider jobs i ascedig order of deadlie d j. [Smallest slack] Cosider jobs i ascedig order of slack d j - t j. 18

19 Miimizig Lateess: Greedy Algorithms Greedy template. Cosider jobs i some order. [Shortest processig time first] Cosider jobs i ascedig order of processig time t j. d j 1 t j couterexample [Smallest slack] Cosider jobs i ascedig order of slack d j - t j. 1 t j 1 2 d j couterexample 19

20 Miimizig Lateess: Greedy Algorithm Greedy algorithm. Earliest deadlie first. Sort jobs by deadlie so that d 1 d 2 d t 0 for j = 1 to Assig job j to iterval [t, t + t j ] s j t, f j t + t j t t + t j output itervals [s j, f j ] max lateess = 1 d 1 = 6 d 2 = 8 d 3 = 9 d 4 = 9 d 5 = 14 d 6 =

21 Miimizig Lateess: No Idle Time Observatio. There exists a optimal schedule with o idle time. d = 4 d = d = d = 4 d = 6 d = Observatio. The greedy schedule has o idle time. 21

22 Miimizig Lateess: Iversios Def. Give a schedule S, a iversio is a pair of jobs i ad j such that: i < j but j scheduled before i. iversio f i before swap j i [ as before, we assume jobs are umbered so that d 1 d 2 d ] Observatio. Greedy schedule has o iversios. Observatio. If a schedule (with o idle time) has a iversio, it has oe with a pair of iverted jobs scheduled cosecutively. 22

23 Miimizig Lateess: Iversios Def. Give a schedule S, a iversio is a pair of jobs i ad j such that: i < j but j scheduled before i. iversio f i before swap j i after swap i j f' j Claim. Swappig two cosecutive, iverted jobs reduces the umber of iversios by oe ad does ot icrease the max lateess. Pf. Let l be the lateess before the swap, ad let l ' be it afterwards. l ' k = l k for all k i, j l ' i l i If job j is late: l j = f j d j (defiitio) = f i d j ( j fiishes at time f i ) f i d i (i < j) l i (defiitio) 23

24 Miimizig Lateess: Aalysis of Greedy Algorithm Theorem. Greedy schedule S is optimal. Pf. Defie S* to be a optimal schedule that has the fewest umber of iversios, ad let's see what happes. Ca assume S* has o idle time. If S* has o iversios, the S = S*. If S* has a iversio, let i-j be a adjacet iversio. swappig i ad j does ot icrease the maximum lateess ad strictly decreases the umber of iversios this cotradicts defiitio of S* 24

25 Greedy Aalysis Strategies Greedy algorithm stays ahead. Show that after each step of the greedy algorithm, its solutio is at least as good as ay other algorithm's. Structural. Discover a simple "structural" boud assertig that every possible solutio must have a certai value. The show that your algorithm always achieves this boud. Exchage argumet. Gradually trasform ay solutio to the oe foud by the greedy algorithm without hurtig its quality. Other greedy algorithms. Kruskal, Prim, Dijkstra, Huffma, 25

26 4.5 Miimum Spaig Tree

27 Miimum Spaig Tree Miimum spaig tree. Give a coected graph G = (V, E) with realvalued edge weights c e, a MST is a subset of the edges T E such that T is a spaig tree whose sum of edge weights is miimized G = (V, E) T, Σ e T c e = 50 Cayley's Theorem. There are -2 spaig trees of K. ca't solve by brute force 27

28 Applicatios MST is fudametal problem with diverse applicatios. Network desig. telephoe, electrical, hydraulic, TV cable, computer, road Approximatio algorithms for NP-hard problems. travelig salesperso problem, Steier tree Idirect applicatios. max bottleeck paths LDPC codes for error correctio image registratio with Reyi etropy learig saliet features for real-time face verificatio reducig data storage i sequecig amio acids i a protei model locality of particle iteractios i turbulet fluid flows autocofig protocol for Etheret bridgig to avoid cycles i a etwork 28

29 Greedy Algorithms Kruskal's algorithm. Start with T = φ. Cosider edges i ascedig order of cost. Isert edge e i T uless doig so would create a cycle. Prim's algorithm. Start with some root ode s ad greedily grow a tree T from s outward. At each step, add the cheapest edge e to T that has exactly oe edpoit i T. Uses the same approach as Dijkistra s algorithm that you ve see before. Remark. All these algorithms produce a MST. 29

30 Greedy Algorithms Simplifyig assumptio. All edge costs c e are distict. Cut property. Let S be ay subset of odes, ad let e be the mi cost edge with exactly oe edpoit i S. The the MST cotais e. Cycle property. Let C be ay cycle, ad let f be the max cost edge belogig to C. The the MST does ot cotai f. f C S e e is i the MST f is ot i the MST 30

31 Cycles ad Cuts Cycle. Set of edges the form a-b, b-c, c-d,, y-z, z-a Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, Cutset. A cut is a subset of odes S. The correspodig cutset D is the subset of edges with exactly oe edpoit i S Cut S = { 4, 5, 8 } Cutset D = 5-6, 5-7, 3-4, 3-5,

32 Cycle-Cut Itersectio Claim. A cycle ad a cutset itersect i a eve umber of edges Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, 6-1 Cutset D = 3-4, 3-5, 5-6, 5-7, 7-8 Itersectio = 3-4, Pf. (by picture) C S V - S 32

33 Greedy Algorithms Simplifyig assumptio. All edge costs c e are distict. Cut property. Let S be ay subset of odes, ad let e be the mi cost edge with exactly oe edpoit i S. The the MST T* cotais e. Pf. (exchage argumet) Suppose e does ot belog to T*, ad let's see what happes. Addig e to T* creates a cycle C i T*. Edge e is both i the cycle C ad i the cutset D correspodig to S there exists aother edge, say f, that is i both C ad D. T' = T* { e } - { f } is also a spaig tree. Sice c e < c f, cost(t') < cost(t*). This is a cotradictio. S f e T* 33

34 Greedy Algorithms Simplifyig assumptio. All edge costs c e are distict. Cycle property. Let C be ay cycle i G, ad let f be the max cost edge belogig to C. The the MST T* does ot cotai f. Pf. (exchage argumet) Suppose f belogs to T*, ad let's see what happes. Deletig f from T* creates a cut S i T*. Edge f is both i the cycle C ad i the cutset D correspodig to S there exists aother edge, say e, that is i both C ad D. T' = T* { e } - { f } is also a spaig tree. Sice c e < c f, cost(t') < cost(t*). This is a cotradictio. S f e T* 34

35 Kruskal's Algorithm: Proof of Correctess Kruskal's algorithm. [Kruskal, 1956] Cosider edges i ascedig order of weight. Case 1: If addig e to T creates a cycle, discard e accordig to cycle property. Case 2: Otherwise, isert e = (u, v) ito T accordig to cut property where S = set of odes i u's coected compoet. v e S e u Case 1 Case 2 35

36 Lexicographic Tiebreakig To remove the assumptio that all edge costs are distict: perturb all edge costs by tiy amouts to break ay ties. Impact. Kruskal ad Prim oly iteract with costs via pairwise comparisos. If perturbatios are sufficietly small, MST with perturbed costs is MST with origial costs. e.g., if all edge costs are itegers, perturbig cost of edge e i by i / 2 Implemetatio. Ca hadle arbitrarily small perturbatios implicitly by breakig ties lexicographically, accordig to idex. Ruig Time: O(m log ) 36

37 MST Algorithms: Theory Determiistic compariso based algorithms. O(m log ) [Jarík, Prim, Dijkstra, Kruskal, Boruvka] O(m log log ). [Cherito-Tarja 1976, Yao 1975] O(m β(m, )). [Fredma-Tarja 1987] O(m log β(m, )). [Gabow-Galil-Specer-Tarja 1986] O(m α (m, )). [Chazelle 2000] Holy grail. O(m). Notable. O(m) radomized. [Karger-Klei-Tarja 1995] O(m) verificatio. [Dixo-Rauch-Tarja 1992] Euclidea. 2-d: O( log ). compute MST of edges i Delauay k-d: O(k 2 ). dese Prim 37