4.4 Shortest Paths in a Graph

Transcription

1 hortst Path Problm. hortst Paths i a Graph hortst path twork. Dirctd graph G = (V, E). ourc s, dstiatio t. Lgth = lgth of dg. hortst path problm: fid shortst dirctd path from s to t. cost of path = sum of dg costs i path shortst path from Pricto C dpartmt to Eisti's hous s Cost of path s----t = =. t Dijkstra's Algorithm Dijkstra's Algorithm Dijkstra's algorithm. Maitai a st of xplord ods for which w hav dtrmid th shortst path distac d(u) from s to u. Iitializ = { s }, d(s) = 0. Rpatdly choos uxplord od v which miimizs ( v) mi ( u, v) : u d( u), Dijkstra's algorithm. Maitai a st of xplord ods for which w hav dtrmid th shortst path distac d(u) from s to u. Iitializ = { s }, d(s) = 0. Rpatdly choos uxplord od v which miimizs ( v) mi ( u, v) : u d( u), add v to, ad st d(v) = (v). shortst path to som u i xplord part, followd by a sigl dg (u, v) add v to, ad st d(v) = (v). shortst path to som u i xplord part, followd by a sigl dg (u, v) d(u) u v d(u) u v s s

2 Dijkstra's Algorithm: Proof of Corrctss Dijkstra's Algorithm: Implmtatio Ivariat. For ach od u, d(u) is th lgth of th shortst s-u path. Pf. (by iductio o ) Bas cas: = is trivial. Iductiv hypothsis: Assum tru for = k. Lt v b xt od addd to, ad lt u-v b th chos dg. Th shortst s-u path plus (u, v) is a s-v path of lgth (v). Cosidr ay s-v path P. W'll s that it's o shortr tha (v). Lt x-y b th first dg i P that lavs, P ad lt P' b th subpath to x. P is alrady too log as soo as it lavs. P' x y (P) (P') + (x,y) d(x) + (x, y) (y) (v) ogativ wights iductiv hypothsis df of (y) s Dijkstra chos v istad of y u v For ach uxplord od, xplicitly maitai (v) mi d(u). (u,v):u Nxt od to xplor = od with miimum (v). Wh xplorig v, for ach icidt dg = (v, w), updat (w) mi { (w), (v) }. Efficit implmtatio. Maitai a priority quu of uxplord ods, prioritizd by (v). PQ Opratio Isrt ExtractMi ChagKy Dijkstra m Array Biary hap log log log Priority Quu d-way Hap d log d d log d log d Idividual ops ar amortizd bouds Fib hap IsEmpty log Total m log m log m/ m + log 9 0

3 Chaptr. Miimum paig Tr Grdy Algorithms lids by Kvi Way. Copyright 00 Parso-Addiso Wsly. All rights rsrvd. Miimum paig Tr Applicatios Miimum spaig tr. Giv a coctd graph G = (V, E) with ralvalud dg wights c, a MT is a subst of th dgs T E such that T is a spaig tr whos sum of dg wights is miimizd. MT is fudamtal problm with divrs applicatios. Ntwork dsig. tlpho, lctrical, hydraulic, TV cabl, computr, road Approximatio algorithms for NP-hard problms. travlig salsprso problm, tir tr 9 0 G = (V, E) T, T c = 0 9 Idirct applicatios. max bottlck paths LDPC cods for rror corrctio imag rgistratio with Ryi tropy larig salit faturs for ral-tim fac vrificatio rducig data storag i squcig amio acids i a proti modl locality of particl itractios i turbult fluid flows autocofig protocol for Ethrt bridgig to avoid cycls i a twork Cayly's Thorm. Thr ar - spaig trs of K. Clustr aalysis. ca't solv by brut forc

4 Grdy Algorithms Grdy Algorithms Kruskal's algorithm. tart with T =. Cosidr dgs i ascdig ordr of cost. Isrt dg i T ulss doig so would crat a cycl. Rvrs-Dlt algorithm. tart with T = E. Cosidr dgs i dscdig ordr of cost. Dlt dg from T ulss doig so would discoct T. Prim's algorithm. tart with som root od s ad grdily grow a tr T from s outward. At ach stp, add th chapst dg to T that has xactly o dpoit i T. implifyig assumptio. All dg costs c ar distict. Cut proprty. Lt b ay subst of ods, ad lt b th mi cost dg with xactly o dpoit i. Th th MT cotais. Cycl proprty. Lt C b ay cycl, ad lt f b th max cost dg blogig to C. Th th MT dos ot cotai f. f C Rmark. All thr algorithms produc a MT. is i th MT f is ot i th MT Cycls ad Cuts Cycl-Cut Itrsctio Cycl. t of dgs th form a-b, b-c, c-d,, y-z, z-a. Claim. A cycl ad a cutst itrsct i a v umbr of dgs. Cycl C = -, -, -, -, -, - Cycl C = -, -, -, -, -, - Cutst D = -, -, -, -, - Itrsctio = -, - Cutst. A cut is a subst of ods. Th corrspodig cutst D is th subst of dgs with xactly o dpoit i. Cut = {,, } Cutst D = -, -, -, -, - Pf. (by pictur) C V -

5 Grdy Algorithms Grdy Algorithms implifyig assumptio. All dg costs c ar distict. Cut proprty. Lt b ay subst of ods, ad lt b th mi cost dg with xactly o dpoit i. Th th MT T* cotais. Pf. (xchag argumt) uppos dos ot blog to T*, ad lt's s what happs. Addig to T* crats a cycl C i T*. Edg is both i th cycl C ad i th cutst D corrspodig to thr xists aothr dg, say f, that is i both C ad D. T' = T* { } - { f } is also a spaig tr. ic c < c f, cost(t') < cost(t*). This is a cotradictio. f implifyig assumptio. All dg costs c ar distict. Cycl proprty. Lt C b ay cycl i G, ad lt f b th max cost dg blogig to C. Th th MT T* dos ot cotai f. Pf. (xchag argumt) uppos f blogs to T*, ad lt's s what happs. Dltig f from T* crats a cut i T*. Edg f is both i th cycl C ad i th cutst D corrspodig to thr xists aothr dg, say, that is i both C ad D. T' = T* { } - { f } is also a spaig tr. ic c < c f, cost(t') < cost(t*). This is a cotradictio. f T* 9 T* 0 Prim's Algorithm: Proof of Corrctss Implmtatio: Prim's Algorithm Prim's algorithm. [Jarík 90, Dijkstra 9, Prim 99] Iitializ = ay od. Apply cut proprty to. Add mi cost dg i cutst corrspodig to to T, ad add o w xplord od u to. Implmtatio. Us a priority quu ala Dijkstra. Maitai st of xplord ods. For ach uxplord od v, maitai attachmt cost a[v] = cost of chapst dg v to a od i. O( ) with a array; O(m log ) with a biary hap. Prim(G, c) { forach (v V) a[v] Iitializ a mpty priority quu Q forach (v V) isrt v oto Q Iitializ st of xplord ods } whil (Q is ot mpty) { u dlt mi lmt from Q {u} forach (dg = (u, v) icidt to u) if ((v ) ad (c < a[v])) dcras priority a[v] to c

6 Kruskal's Algorithm: Proof of Corrctss Implmtatio: Kruskal's Algorithm Kruskal's algorithm. [Kruskal, 9] Cosidr dgs i ascdig ordr of wight. Cas : If addig to T crats a cycl, discard accordig to cycl proprty. Cas : Othrwis, isrt = (u, v) ito T accordig to cut proprty whr = st of ods i u's coctd compot. Implmtatio. Us th uio-fid data structur. Build st T of dgs i th MT. Maitai st for ach coctd compot. O(m log ) for sortig ad O(m (m, )) for uio-fid. m log m is O(log ) sstially a costat Kruskal(G, c) { ort dgs wights so that c c... c m. T forach (u V) mak a st cotaiig siglto u Cas v u Cas } for i = to m ar u ad v i diffrt coctd compots? (u,v) = i if (u ad v ar i diffrt sts) { T T { i } mrg th sts cotaiig u ad v } mrg two compots rtur T Lxicographic Tibrakig To rmov th assumptio that all dg costs ar distict: prturb all dg costs by tiy amouts to brak ay tis.. Clustrig Impact. Kruskal ad Prim oly itract with costs via pairwis comparisos. If prturbatios ar sufficitly small, MT with prturbd costs is MT with origial costs..g., if all dg costs ar itgrs, prturbig cost of dg i by i / Implmtatio. Ca hadl arbitrarily small prturbatios implicitly by brakig tis lxicographically, accordig to idx. boola lss(i, j) { if (cost( i ) < cost( j )) rtur tru ls if (cost( i ) > cost( j )) rtur fals ls if (i < j) rtur tru ls rtur fals } Outbrak of cholra daths i Lodo i 0s. Rfrc: Nia Mishra, HP Labs

7 Clustrig Clustrig of Maximum pacig Clustrig. Giv a st U of objcts labld p,, p, classify ito cohrt groups. photos, documts. micro-orgaisms Distac fuctio. Numric valu spcifyig "closss" of two objcts. umbr of corrspodig pixls whos itsitis diffr by som thrshold k-clustrig. Divid objcts ito k o-mpty groups. Distac fuctio. Assum it satisfis svral atural proprtis. d(p i, p j ) = 0 iff p i = p j (idtity of idiscribls) d(p i, p j ) 0 (ogativity) d(p i, p j ) = d(p j, p i ) (symmtry) pacig. Mi distac btw ay pair of poits i diffrt clustrs. Fudamtal problm. Divid ito clustrs so that poits i diffrt clustrs ar far apart. Routig i mobil ad hoc tworks. Idtify pattrs i g xprssio. Documt catgorizatio for wb sarch. imilarity sarchig i mdical imag databass kycat: clustr 0 9 sky objcts ito stars, quasars, galaxis. Clustrig of maximum spacig. Giv a itgr k, fid a k-clustrig of maximum spacig. spacig k = Grdy Clustrig Algorithm Grdy Clustrig Algorithm: Aalysis igl-lik k-clustrig algorithm. Form a graph o th vrtx st U, corrspodig to clustrs. Fid th closst pair of objcts such that ach objct is i a diffrt clustr, ad add a dg btw thm. Rpat -k tims util thr ar xactly k clustrs. Ky obsrvatio. This procdur is prcisly Kruskal's algorithm (xcpt w stop wh thr ar k coctd compots). Rmark. Equivalt to fidig a MT ad dltig th k- most xpsiv dgs. Thorm. Lt C* dot th clustrig C*,, C* k formd by dltig th k- most xpsiv dgs of a MT. C* is a k-clustrig of max spacig. Pf. Lt C dot som othr clustrig C,, C k. Th spacig of C* is th lgth d* of th (k-) st most xpsiv dg. Lt p i, p j b i th sam clustr i C*, say C* r, but diffrt clustrs i C, say C s ad C t. om dg (p, q) o p i -p j path i C* r spas two diffrt clustrs i C. All dgs o p i -p j path hav lgth d* sic Kruskal chos thm. pacig of C is d* sic p ad q C s C t ar i diffrt clustrs. C* r p i p q p j 9 0

8 MT Algorithms: Thory Extra lids Dtrmiistic compariso basd algorithms. O(m log ) [Jarík, Prim, Dijkstra, Kruskal, Boruvka] O(m log log ). [Chrito-Tarja 9, Yao 9] O(m (m, )). [Frdma-Tarja 9] O(m log (m, )). [Gabow-Galil-pcr-Tarja 9] O(m (m, )). [Chazll 000] Holy grail. O(m). Notabl. O(m) radomizd. [Kargr-Kli-Tarja 99] O(m) vrificatio. [Dixo-Rauch-Tarja 99] Euclida. -d: O( log ). comput MT of dgs i Dlauay k-d: O(k ). ds Prim Ddrogram Ddrogram of Cacrs i Huma Ddrogram. citific visualizatio of hypothtical squc of volutioary vts. Lavs = gs. Itral ods = hypothtical acstors. Tumors i similar tissus clustr togthr. G G Rfrc: Rfrc: Botsti & Brow group g xprssd g ot xprssd