Greedy Algorithms. Interval Scheduling. Greedy Algorithm. Optimality. Greedy Algorithm (cntd) Greed is good. Greed is right. Greed works.

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1 Algorithm Grdy Algorithm 5- Grdy Algorithm Grd i good. Grd i right. Grd work. Wall Strt Data Structur and Algorithm Andri Bulatov Algorithm Grdy Algorithm 5- Algorithm Grdy Algorithm 5- Intrval Schduling Conidr th following problm (Intrval Schduling) Thr i a group of propod talk to b givn. W want to chdul a many talk a poibl in th main lctur room. Lt t, t, K, t m b th talk, talk t j bgin at tim b j and nd at tim j. (No two lctur can procd at th am tim, but a lctur can bgin at th am tim anothr on nd.) W aum that K. t t t m 9:00 0:00 :00 :00 Grdy Algorithm Grdy algorithm: At vry tp choo a talk with th arlit nding tim among all tho talk that bgin aftr all talk alrady chduld nd. t t t 9:00 0:00 :00 :00 Algorithm Grdy Algorithm 5-5 Algorithm Grdy Algorithm 5-6 Grdy Algorithm (cntd) Input: St R of propod talk Output: St A of talk chduld in th main lctur hall t A:= whil R choo a talk i R that ha th mallt finihing tim t A:=A {i} dlt all talk from R that ar not compatibl with i rturn A Th grdy algorithm i optimal in th n that it alway chdul th mot talk poibl in th main lctur hall. Optimality By induction on n w prov that if th grdy algorithm chdul n talk, thn it i not poibl to chdul mor than n talk. Bai tp. Suppo that th grdy algorithm ha chduld only on talk, t. Thi man that vry othr talk tart bfor, and nd aftr. Hnc, at tim ach of th rmaining talk nd to u th lctur hall. No two talk can b chduld bcau of that. Inductiv tp. Suppo that if th grdy algorithm chdul k talk, it i not poibl to chdul mor than k talk. W prov that if th algorithm chdul k + talk thn thi i th optimal numbr.

2 Algorithm Grdy Algorithm 5-7 Algorithm Grdy Algorithm 5-8 Optimality (cntd) Suppo that th algorithm ha lctd k + talk. Firt, w how that thr i an optimal chduling that contain t Indd, if w hav a chdul that bgin with th talk t i, i >, thn thi firt talk can b rplacd with t. To thi, not that, inc i, all talk chduld aftr t till can b chduld. t t Optimality (cntd) Onc w includd t, chduling th talk o that a many a poibl talk ar chduld i rducd to chduling a many talk a poibl that bgin at or aftr tim. Th grdy algorithm alway chdul t, and thn chdul k talk chooing thm from tho that tart at or aftr. By th induction hypothi, it i not poibl to chdul mor than k uch talk. Thrfor, th optimal numbr of talk i k +. t 9:00 0:00 :00 :00 Algorithm Grdy Algorithm 5-9 Algorithm Grdy Algorithm 5-0 Shortt Path Suppo that vry arc of a digraph G ha lngth (or cot, or wight, or ) ln() Thn w can naturally dfin th lngth of a dirctd path in G, and th ditanc btwn any two nod Th -t-shortt Path Problm Intanc: Digraph G with lngth of arc, and nod,t Objctiv: Find a hortt path btwn and t Singl Sourc Shortt Path Th Singl Sourc Shortt Path Problm Intanc: Digraph G with lngth of arc, and nod Objctiv: Find hortt path from to all nod of G Grdy algorithm: Attmpt to build an optimal olution by mall tp, optimizing locally, on ach tp Algorithm Grdy Algorithm 5- Algorithm Grdy Algorithm 5- Dijktra Algorithm Exampl Input: digraph G with lngth ln, and nod Output: ditanc d( from to vry nod u lt S b th t of xplord nod for ach v S lt d(v) b th ditanc from to v a b t S:={} and d():=0 pick a nod v not from S uch that th valu d' ( v) : = min = ( u, i minimal c g t S:=S {v}, and d(v):=d (v)

3 Algorithm Grdy Algorithm 5- Algorithm Grdy Algorithm 5- Qution What if G i not connctd? thr ar vrtic unrachabl from? How can w find hortt path from to nod of G? Dijktra Algorithm Input: digraph G with lngth ln, nod Output: ditanc d( from to vry nod u and prdcor P( in th hortt path t S:={}, d():=0, and P():=null pick a nod v not from S uch that th valu d' ( v) : = min = ( u, i minimal t S:=S {v} and d(v):=d (v) t P(v):= u (providing th minimum) Algorithm Grdy Algorithm 5-5 Algorithm Grdy Algorithm 5-6 Dijktra Algorithm Analyi: Soundn Soundn For any nod v th path, P(P(P(v))), P(P(v)), P( v i a hortt v path Algorithm tay ahad Induction on S Ba ca: If S =, thn S = {}, and d() = 0 Induction ca: Lt P u dnot th path, P(P(P()), P(P(), P(, u Suppo th claim hold for S = k, that i for any u S th hortt path Lt v b addd on th nxt tp. Conidr any path P from to v othr than P v P u i Algorithm Grdy Algorithm 5-7 Algorithm Grdy Algorithm 5-8 Soundn (cntd) x y Thr i a point whr P lav S for th firt tim Lt it b arc (x,y) P u Th lngth of P i at lat u v th lngth of P x + th lngth of (x,y) + th lngth of y v Howvr, by th choic of v ln( Pv ) = ln( Pu ) u, v) ln( Px ) x, y) ln( P) Running Tim Lt th givn graph hav n nod and m arc n itration of th whil loop Straightforward implmntation rquir chcking up to m arc that giv O(mn) running tim Improvmnt: For ach nod v tor d' ( v) : = min = ( u, and updat it vry tim S chang Whn nod v i addd to S w nd to chang dg(v) valu m chang total O(m+n) `call Proprly implmntd thi giv O(m log n) Rcall hap and priority quu

4 Algorithm Spanning Tr 5-0 Th Minimum Spanning Tr Problm Lt G = (V,E) b a connctd undirctd graph A ubt T E i calld a panning tr of G if (V,T) i a tr Spanning Tr Dign and Analyi of Algorithm Andri Bulatov If vry dg of G ha a wight (poitiv) c tr alo ha aociatd wight c T Th Minimum Spanning Tr Problm Intanc Graph G with dg wight Objctiv Find a panning tr of minimum wight thn vry panning Algorithm Spanning Tr 5- Algorithm Spanning Tr 5- Prim Algorithm Input: graph G with wight c Output: a minimum panning tr of G choo a vrtx t S:={}, T:= pick a nod v not from S uch that th valu min= ( u, c i minimal t S:=S {v} and T:=T {} Krukal Algorithm Input: graph G with wight c Output: a minimum panning tr of G T:= whil T < V - do pick an dg with minimum wight uch that it i not from T and T {} do not contain cycl t T:=T {} Algorithm Spanning Tr 5- Algorithm Spanning Tr 5- Exampl Krukal Algorithm: Soundn a b Lmma (th Cut Proprty) Aum that all dg wight ar diffrnt. Lt S b a nonmpty ubt of vrtic, S V, and lt b th minimum wight dg conncting S and V S. Thn vry minimum panning tr contain c d U th xchang argumnt 5 f Lt T b a panning tr that do not contain W find an dg in T uch that rplacing with w obtain anothr panning tr that ha mallr wight

5 Algorithm Spanning Tr 5-5 Algorithm Spanning Tr 5-6 Lt = (v,w) Thr i a (uniqu) path P in T conncting v and w t q v f r u w Rplac in T dg with T = (T { }) {} T rmain a panning tr but lightr Lt u b th firt vrtx on thi path not in S, and lt = tu b th dg conncting S and V S. Algorithm Spanning Tr 5-7 Algorithm Spanning Tr 5-8 Krukal algorithm produc a minimum panning tr T i a panning tr It contain no cycl If (V,T) i not connctd thn thr i an dg uch that T {} contain no cycl. Th algorithm mut add th lightt uch dg (cntd) T ha minimum wight W how that vry dg addd by Krukal algorithm mut blong to vry minimum panning tr Conidr dg = (v,w) addd by th algorithm at om point, and lt S b th t of vrtic rachabl from v in (V,T), whr T i th t gnratd at th momnt Clarly v S, but w S Edg (v,w) i th lightt dg conncting S and V S Indd if thr i a lightr on, ay,, thn it i not in T, and hould b addd intad Algorithm Spanning Tr 5-9 Algorithm Spanning Tr 5-0 Prim Algorithm: Soundn (cntd) Prim algorithm produc a minimum panning tr : DIY Krukal Algorithm: Running Tim Suppo G ha n vrtic and m dg Straightforward: W nd to add n dg, and vry tim w hav to find th lightt dg that don t form a cycl Thi tak n m n n, that i O( mn ) Uing a good data tructur that tor connctd componnt of th tr bing contructd w can do it in O(m log n) tim