4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Applications

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1 Chaptr. Minimum panning Tr Grdy Algorithms lids by Kvin Wayn. Copyright 200 Parson-Addison Wsly. All rights rsrvd. Minimum panning Tr Applications Minimum spanning tr. Givn a connctd graph G = (V, E) with ralvalud dg wights c, an MT is a subst of th dgs T! E such that T is a spanning tr whos sum of dg wights is minimizd. MT is fundamntal problm with divrs applications.! Ntwork dsign. tlphon, lctrical, hydraulic, TV cabl, computr, road 2! Approximation algorithms for NP-hard problms. travling salsprson problm, tinr tr G = (V, E) T, " #T c = 0 9! Indirct applications. max bottlnck paths LDPC cods for rror corrction imag rgistration with Rnyi ntropy larning salint faturs for ral-tim fac vrification rducing data storag in squncing amino acids in a protin modl locality of particl intractions in turbulnt fluid flows autoconfig protocol for Ethrnt bridging to avoid cycls in a ntwork Cayly's Thorm. Thr ar n n-2 spanning trs of K n.! Clustr analysis. can't solv by brut forc

2 Grdy Algorithms Grdy Algorithms Kruskal's algorithm. tart with T = $. Considr dgs in ascnding ordr of cost. Insrt dg in T unlss doing so would crat a cycl. Rvrs-Dlt algorithm. tart with T = E. Considr dgs in dscnding ordr of cost. Dlt dg from T unlss doing so would disconnct T. Prim's algorithm. tart with som root nod s and grdily grow a tr T from s outward. At ach stp, add th chapst dg to T that has xactly on ndpoint in T. implifying assumption. All dg costs c ar distinct. Cut proprty. Lt b any subst of nods, and lt b th min cost dg with xactly on ndpoint in. Thn th MT contains. Cycl proprty. Lt C b any cycl, and lt f b th max cost dg blonging to C. Thn th MT dos not contain f. f C Rmark. All thr algorithms produc an MT. is in th MT f is not in th MT Cycls and Cuts Cycl-Cut Intrsction Cycl. t of dgs th form a-b, b-c, c-d,, y-z, z-a. Claim. A cycl and a cutst intrsct in an vn numbr of dgs. 2 Cycl C = -2, 2-, -, -, -, - 2 Cycl C = -2, 2-, -, -, -, - Cutst D = -, -, -, -, - Intrsction = -, - Cutst. A cut is a subst of nods. Th corrsponding cutst D is th subst of dgs with xactly on ndpoint in. 2 Cut = {,, Cutst D = -, -, -, -, - Pf. (by pictur) C V -

3 Grdy Algorithms Grdy Algorithms implifying assumption. All dg costs c ar distinct. implifying assumption. All dg costs c ar distinct. Cut proprty. Lt b any subst of nods, and lt b th min cost dg with xactly on ndpoint in. Thn th MT T* contains. Cycl proprty. Lt C b any cycl in G, and lt f b th max cost dg blonging to C. Thn th MT T* dos not contain f. Pf. (xchang argumnt)! uppos dos not blong to T*, and lt's s what happns.! Adding to T* crats a cycl C in T*.! Edg is both in th cycl C and in th cutst D corrsponding to % thr xists anothr dg, say f, that is in both C and D.! T' = T* & { - { f is also a spanning tr.! inc c < c f, cost(t') < cost(t*).! This is a contradiction.! f Pf. (xchang argumnt)! uppos f blongs to T*, and lt's s what happns.! Dlting f from T* crats a cut in T*.! Edg f is both in th cycl C and in th cutst D corrsponding to % thr xists anothr dg, say, that is in both C and D.! T' = T* & { - { f is also a spanning tr.! inc c < c f, cost(t') < cost(t*).! This is a contradiction.! f T* 9 T* 0 Prim's Algorithm: Proof of Corrctnss Implmntation: Prim's Algorithm Prim's algorithm. [Jarník 90, Dijkstra 9, Prim 99]! Initializ = any nod.! Apply cut proprty to.! Add min cost dg in cutst corrsponding to to T, and add on nw xplord nod u to. Implmntation. Us a priority quu ala Dijkstra.! Maintain st of xplord nods.! For ach unxplord nod v, maintain attachmnt cost a[v] = cost of chapst dg v to a nod in.! O(n 2 ) with an array; O(m log n) with a binary hap. Prim(G, c) { forach (v # V) a[v] ' ( Initializ an mpty priority quu Q forach (v # V) insrt v onto Q Initializ st of xplord nods ' $ whil (Q is not mpty) { u ' dlt min lmnt from Q ' & { u forach (dg = (u, v) incidnt to u) if ((v ) ) and (c < a[v])) dcras priority a[v] to c 2

4 Kruskal's Algorithm: Proof of Corrctnss Implmntation: Kruskal's Algorithm Kruskal's algorithm. [Kruskal, 9]! Considr dgs in ascnding ordr of wight.! Cas : If adding to T crats a cycl, discard according to cycl proprty.! Cas 2: Othrwis, insrt = (u, v) into T according to cut proprty whr = st of nods in u's connctd componnt. Implmntation. Us th union-find data structur.! Build st T of dgs in th MT.! Maintain st for ach connctd componnt.! O(m log n) for sorting and O(m + (m, n)) for union-find. m * n 2 % log m is O(log n) ssntially a constant Kruskal(G, c) { ort dgs wights so that c * c 2 *... * c m. T ' $ forach (u # V) mak a st containing singlton u Cas v u Cas 2 for i = to m ar u and v in diffrnt connctd componnts? (u,v) = i if (u and v ar in diffrnt sts) { T ' T & { i mrg th sts containing u and v mrg two componnts rturn T Lxicographic Tibraking To rmov th assumption that all dg costs ar distinct: prturb all dg costs by tiny amounts to brak any tis.. Clustring Impact. Kruskal and Prim only intract with costs via pairwis comparisons. If prturbations ar sufficintly small, MT with prturbd costs is MT with original costs..g., if all dg costs ar intgrs, prturbing cost of dg i by i / n 2 Implmntation. Can handl arbitrarily small prturbations implicitly by braking tis lxicographically, according to indx. boolan lss(i, j) { if (cost( i ) < cost( j )) rturn tru ls if (cost( i ) > cost( j )) rturn fals ls if (i < j) rturn tru ls rturn fals Outbrak of cholra daths in London in 0s. Rfrnc: Nina Mishra, HP Labs

5 Clustring Clustring of Maximum pacing Clustring. Givn a st U of n objcts labld p,, p n, classify into cohrnt groups. photos, documnts. micro-organisms Distanc function. Numric valu spcifying "closnss" of two objcts. numbr of corrsponding pixls whos intnsitis diffr by som thrshold k-clustring. Divid objcts into k non-mpty groups. Distanc function. Assum it satisfis svral natural proprtis.! d(p i ) = 0 iff p i = p j (idntity of indiscrnibls)! d(p i ), 0 (nonngativity)! d(p i ) = d(p j, p i ) (symmtry) pacing. Min distanc btwn any pair of points in diffrnt clustrs. Fundamntal problm. Divid into clustrs so that points in diffrnt clustrs ar far apart.! Routing in mobil ad hoc ntworks.! Idntify pattrns in gn xprssion.! Documnt catgorization for wb sarch.! imilarity sarching in mdical imag databass! kycat: clustr 0 9 sky objcts into stars, quasars, galaxis. Clustring of maximum spacing. Givn an intgr k, find a k-clustring of maximum spacing. spacing k = Grdy Clustring Algorithm Grdy Clustring Algorithm: Analysis ingl-link k-clustring algorithm.! Form a graph on th vrtx st U, corrsponding to n clustrs.! Find th closst pair of objcts such that ach objct is in a diffrnt clustr, and add an dg btwn thm.! Rpat n-k tims until thr ar xactly k clustrs. Ky obsrvation. This procdur is prcisly Kruskal's algorithm (xcpt w stop whn thr ar k connctd componnts). Rmark. Equivalnt to finding an MT and dlting th k- most xpnsiv dgs. Thorm. Lt C* dnot th clustring C*,, C* k formd by dlting th k- most xpnsiv dgs of a MT. C* is a k-clustring of max spacing. Pf. Lt C dnot som othr clustring C,, C k.! Th spacing of C* is th lngth d* of th (k-) st most xpnsiv dg.! Lt p i b in th sam clustr in C*, say C* r, but diffrnt clustrs in C, say C s and C t.! om dg (p, q) on p i -p j path in C* r spans two diffrnt clustrs in C.! All dgs on p i -p j path hav lngth * d* sinc Kruskal chos thm.! pacing of C is * d* sinc p and q ar in diffrnt clustrs.! C* r C s C t p i p q p j 9 20

6 MT Algorithms: Thory Extra lids Dtrministic comparison basd algorithms.! O(m log n) [Jarník, Prim, Dijkstra, Kruskal, Boruvka]! O(m log log n). [Chriton-Tarjan 9, Yao 9]! O(m -(m, n)). [Frdman-Tarjan 9]! O(m log -(m, n)). [Gabow-Galil-pncr-Tarjan 9]! O(m + (m, n)). [Chazll 2000] Holy grail. O(m). Notabl.! O(m) randomizd. [Kargr-Klin-Tarjan 99]! O(m) vrification. [Dixon-Rauch-Tarjan 992] Euclidan.! 2-d: O(n log n). comput MT of dgs in Dlaunay! k-d: O(k n 2 ). dns Prim 22 Dndrogram Dndrogram of Cancrs in Human Dndrogram. cintific visualization of hypothtical squnc of volutionary vnts.! Lavs = gns.! Intrnal nods = hypothtical ancstors. Tumors in similar tissus clustr togthr. Gn Gn n Rfrnc: 2 Rfrnc: Botstin & Brown group gn xprssd gn not xprssd 2