Greedy Algorithms. Kye Halsted. Edited by Chuck Cusack. These notes are based on chapter 17 of [1] and lectures from CSCE423/823, Spring 2001.

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1 #! Greedy Algrithms Kye Halsted Edited by Chuk Cusak These ntes are based n hapter 17 f [1] and letures frm CCE423/823, pring Greedy algrithms slve prblems by making the hie that seems best at the partiular mment. any ptimizatin prblems an be slved using a greedy algrithm. me prblems have n effiient slutin, but a greedy algrithm may prvide a slutin that is lse t ptimal. We will begin ur explratin with a simple example. 1 Greedy Algrithm Definitin A greedy algrithm wrks if a prblem exhibits the fllwing tw prperties: 1. Greedy hie prperty: A glbally ptimal slutin an be arrived at by making a lally ptimal slutin. In ther wrds, an ptimal slutin an be btained by making greedy hies. 2. Optimal substruture: Optimal slutins ntain ptimal subslutins. In ther wrds, slutins t subprblems f an ptimal slutin are ptimal. 2 Ativity eletin Prblem Prblem 1 The ativity seletin prblem invlves a set f ativities, eah with a start time and finish time, with. We an view eah ativity as an interval "!. Tw ativities and are mpatible if the intervals d nt verlap. That is, # r. We wish t selet the largest set f mpatible ativities. In what rder shuld we selet the ativities? me pssibilities are: 1. Pik shrtest interval first. 2. rt by start time, and pik arding t earliest start time. 3. rt by finish time, and pik arding t earliest finish time. Example: Let $% & ')(*,+-.')/, with intervals,+, (*)., +*0, 21 )34, 0' 654, and 3* 7 &. We will try several methds t slve the prblem. 1

2 1. hrtest interval: Pik, in rder, 21 34, (*., and 3' Earliest start time: Pik, in rder,,+, +-0, and Earliest finish time: Pik, in rder,,+, +-0&, 81 )3, and 3' 7. tie in the last ase, we get the mst ativities. In fat, this is an ptimal slutin. We will prve that seleting the task with the earliest finish time first will yield an ptimal slutin. First, we will assume that the ativities are srted in rder f inreasing finish times. That is, If they are nt srted, they an be srted in?a :9 =- <; CBED&FGH <> time. The algrithm is as fllws: GreedyAtivity(,F,n) A=1} // Add the task with the earliest finish time j=1 // j hlds the value f the last task added fr(i=2, i<=n, i++) // Try eah remaining task if ([i] >= F[j]) // If task i is mpatible A=A+i // Add i t the set A j=i // et j t the element just added return A // Return the maximal set f mpatible tasks During eah iteratin, the algrithm will hse the ativity with the earliest finish time that an be sheduled withut nflit with the ativities already sheduled. We will nw shw that the GreedyAtivity algrithm mputes an ptimal slutin. Therem 1 The GreedyAtivity algrithm gives an ptimal slutin t the ativity seletin prblem. Prf: We need t demnstrate the greedy hie prperty and ptimal substruture. Greedy Chie Prperty: ine ativity 1 has the earliest finish time, it is the greedy hie. Thus, we must shw that there exists an ptimal slutin ntaining 1. Let I be an ptimal slutin and assume ativity 1 is nt in I. Let be the first ativity in I. Then K9 L, s OQPR & I is a valid set f ativities. ine IT, is als ptimal. ine ntains ativity 1, we have shwn the greedy hie prperty. Optimal ubstruture: Let be the first element in an ptimal slutin I. We need t shw that X Z IV IW is an ptimal slutin t the subprblem invlving the set f tasks V! V <L ( is the set f tasks mpatible with task ). Assume I[ is nt ptimal. Then there exists a slutin fr, suh that \] I. The first ativity in has t be mpatible with whih means P^ P^ b is a valid slutin fr. Hwever, \` I a IT, whih ntradits that I is ptimal fr. Therefre, I[ is ptimal. 2

3 P e n \ 9 9 P 3 inimum panning Trees Given a weighted graph d, we wish t nstrut a minimum spanning tree (T) 1 f d. Kruskal s algrithm is a greedy algrithm algrithm t find an T f a graph d. Briefly it wrks as fllws: Kruskal(G) T=} rt Edges in asending rder f weight Fr eah edge e (in srted rder) If T+e des nt ntain a yle T=T+e Return T imply, we nsider eah edge in rder f weight, adding it t the tree as lng as it des nt reate a yle. We will prve: d e Therem 2 Let be a graph. The the set f edges returned by Kruskal s algrithm is a T. Prf: Greedy-hie prperty: We need t shw that the minimum weight edge f d is ntained in sme T f d. Let e be a T f d, and assume sme minimum weight edge, f 7g hikj Pm le. We will find an T that des have this edge. tie that e f must ntain a yle. ine f is a minimum weight edge, every ther edge f the yle is at least the same weight as f. Als, if we remve any edge frm that yle, the resulting graph will be a tree. PR Let n be ne suh edge, and let e f. Then e is a spanning tree with weight qp sr st qp, s e is a T. ine f qe, we are dne. Optimal substruture: Let e be a T and f g h' be a minimum weight edge. Cnsider e e$ f. There are tw ases t nsider: Case 1: Either g r h has degree 1 in e (withut lss f generality, assume it is g ). Then e is a 7gH 7gH spanning tree fr dm (dm is the graph d less the pint g and all the edges nneted t it). Case 2: Remving f frm e partitins e int tw trees, eu9 and e;. Let dv9 be the graph indued by the verties in e9 and dw; the graph indued by the verties in eh;. Then e<9 is a spanning tree fr dx9 and e; is a spanning tree fr dy;. In either ase the trees e, e<9, and e; are T fr their respetive subgraphs. Fr instane, if e 9 is nt an T fr dv9, then there is an T ez 9, with Pw ae<9 aez. In this ase, ez e; f is a spanning tree f d e 9 e; Ps f O: with weight ae 9 ntraditing that e we have shwn ptimal substruture. e; f OK e 9 e; f OG is an T fr d. Thus, e9 is a T fr dv9 (and similar fr e and e; ). Thus, 1 ee hapter 24 f [1] fr a mplete verage f minimum spanning trees. e 3

4 I 4 Greedy Algrithms: Thery Definitin 1 A matrid is an rdered pair } ~Oui suh that 1. is a finite nn-empty set. 2. is a family f subsets f all subsets I. if Iƒ 3. Fr I Example: Let } Š = ayli subgraphs f G Vague prf that } is a matrid., alled independent subsets suh that if I, then there exists sme q )Šx fr a graph d ˆ 1. is learly finite and nn-empty if d is. suh that I be defined as P, then. fr 2. If I is ayli, then I is learly ayli t. 3. Let I with Iƒ. Clearly, I and are frests and has less trees than A. 7g hi Therefre, there is an edge suh that g and h are in different trees f I, s P$7g hi (beause this edge nnets tw different trees in I, it annt make a yle). We are still lking t renile this with inimum panning Trees. Definitin 2 Fr a matrid } u' Definitin 3 I is nt ntained in any larger subsets f., an element j ŒI, is an extensin f I~ is alled maximal if I has n extensins. Put anther way, I is maximal if I if I PV. Therem 3 All maximal independent subsets f a matrid have the same size. Prf: Let I P sme ŽI suh that I whih ntradits I all maximal independent subsets have the same size. Definitin 4 A matrid }. By prperty 3, there exists, and be maximal independent subsets with Iƒ being maximal. Thus, IT u' with = eah element a psitive weight a. Definitin 5 A subset I~ is ptimal if is weighted if there is a weight funtin I. Given a subset I is maximal. that assiates, define I 4

5 I Ñ \ \ I Fat: An ptimal subset must be maximal (have n extensins), sine extending it wuld inrease its weight. Example: T turn the T prblem int a weighted matrid, we define the weight funtin " s a f š * -œ a \ž f š - *œ a fr all. We nw present a greedy algrithm t find an ptimal subset f a weighted matrid. Greedyatrid(,I,W) A=} // tart with the empty set rt nn-inreasing by W Fr eah x in in srted rder if A+x in I // If A unin x is in I A=A+x // Add x t A Return A // A is ptimal Assuming determining whether r nt I CBED FGŸ? ž?a H G nu CBED&FG?A PR H nu Ÿ takes time nq, then the mplexity is. We will nw shw that the algrithm is rret. Lemma 4 (atrids exhibit the greedy hie prperty) u' Let } be a matrid with a weight funtin, and assume is srted, nn-inreasing arding t. Let Œ be the first (thus largest) element suh that, if ne exists. Then is ntained in sme ptimal slutin. Prf: If there is n suh, there is n ptimal slutin (and there is nthing t prve). Let be an ptimal slutin, and assume W. tie that if ž `, and has maximum weight. Let I. While P ' Iƒ and add ~ P t I. Eventually, we will have I G P OQ ši a then, sine, find ` Ñ I suh that š! fr sme A I } I I Thus, is an ptimal slutin ntaining. Lemma 5 Let u' be a matrid. If j, then P j fr any subset. P sine y P Prf: By prperty 2, if I, then q, with. Lemma 6 (atrids exhibit ptimal substruture) Let be the first element hsen by Greedyatrid u' l zp fr a weighted matrid }. Let I Ib is ptimal fr }. Then Ib is ptimal fr }m IV, where ~ žz* ª and ~ OZ Prf: Let I«be ptimal with I. Let I I }. Then there exists a slutin fr } suh that slutin fr } with P O: I Ps i and assume I is nt ptimal fr I. Thus, P is a ntraditing that I is ptimal. Thus I is ptimal. G ši 5

6 Z Therem 7 (The Greedyatrid algrithm is rret) If } is a weighted matrid with weight funtin, then Greedyatrid returns an ptimal subset. Prf: By Lemma 5, if we skip ver any element as a first hie beause it is nt in, then it is nt in any ptimal slutin, s it is n lss. One we pik the first element, Lemma 4 ensures there exists an ptimal slutin ntaining, s is safe t add. Arding t Lemma 6, we need t find an ptimal slutin t }. ine iff P we an view the algrithm as ating n }. Thus the algrithm will ntinue by finding an ptimal slutin t the subprblem. Therefre it finds an ptimal slutin t the prblem. 5 Task sheduling Prblem 2 In the task sheduling prblem we have a set f tasks, suh that: 1. Eah task takes unit time. 2. Eah has a deadline 3. Eah has a penalty by whih it shuld be dne. if it is late. We wish t find a shedule f all tasks in time units s that the verall penalty is minimum. We an easily see a few fats abut this prblem: Fat 1: We an list the tasks s that all the early tasks are sheduled befre any f the late tasks are. A shedule in this frm is alled early first. Fat 2: We an list the tasks in annial frm. That is, early first suh that the early tasks are in rder f deadline. Why? tie that if is sheduled befre but \$ 7#, swithing and makes earlier (s it is still n time) and puts where was, and sine has a later deadline, is still n time. Fat 3: inimizing the penalty f the late tasks is equivalent t maximizing the penalties f the n-time tasks. u' Therem 8 Let } where = the set f tasks, = subsets f suh that there exists an n-time shedule (Thse subsets that an be sheduled n time.) The } is a matrid. Prf: We need t shw the 3 matrid prperties hld. u' 1. is learly nn-empty and finite. 2. If I~, every subset f I~ 3. If I and IT have the latest deadline is lear. then there exists = k. Let ~ ŒIR³! suh that I and P. Let ±I 6

7 ! Z ~ j j A and tie that if, then sine therwise it wuld have been hsen instead f. Pµ Thus, sine this is a set f tasks % P whih all must y be mpleted by time, there an be at mst f them). tie that I ši and P± bp. ine IT, P y Ps Pq ši O P I ~ O P I Ps ¹j s I Assume I. Then there exists tasks in I with deadline at mst Ps. Thus, I, ntraditing that I. Therefre I Figure 1 gives a graphial representatin f the prf f Therem 8. evº'»7 I ( +. / ½ ¾= À Á 65 ½ ¾= À & 7 6( + 7. ½ ¾= À Â Figure 1: A graphial example f the prf f Therem 8. The tasks are srted by inreasing deadline. tie that is the largest task in tasks t its left have deadline. that is nt in I, and sine it has deadline, thse Referenes [1] Crmen, Leisersn, and Rivest, Intrdutin t Algrithms, Graw Hill,