Graph Contraction and Connectivity

Transcription

1 Chpter 14 Grph Contrtion n Connetivity So fr we hve mostly overe tehniques for solving problems on grphs tht were evelope in the ontext of sequentil lgorithms. Some of them re esy to prllelize while others re not. For exmple, we sw tht BFS hs some prllelism sine eh level n be explore in prllel, but there ws no prllelism in DFS 1 There ws lso limite prllelism in Dijkstr s lgorithm, but there ws plenty of prllelism in the Bellmn-For lgorithm. In this hpter we will over tehnique lle grph ontrtion tht ws speifilly esigne to be use in prllel lgorithms n llows us to get polylogrithmi spn for ertin grph problems. Also, so fr, we hve been only esribe lgorithms tht o not moify grph, but rther just trverse the grph, or upte vlues ssoite with the verties. As prt of grph ontrtion, in this hpter we will lso stuy tehniques for restruture grphs Preliminries We strt by reviewing n efining some new grph terminology. We will use the following grph s n exmple. Exmple We will use the following unirete grph s n exmple. b e f 1 In relity, there is prllelism in DFS when grphs re ense in prtiulr, lthough verties nee to visite sequentilly, with some re, the eges n be proesse in prllel. 213

2 214 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Rell tht in grph (either irete or unirete) vertex v is rehble from vertex u if there is pth from u to v. Also rell tht n unirete grph is onnete if ll verties re rehble from ll other verties. Our exmple is onnete. Exmple You n isonnet the grph by eleting two eges, for exmple (, f) n (b, e). b e f When working with grphs it is often useful to refer to prt of grph, whih we will ll subgrph. Question Any intuition bout how we my efine subgrph? A subgrph n be efine s ny subsets of eges n verties s long s the result is well efine grph, n in prtiulr: Definition 14.4 (Subgrph). Let G = (V, E) n H = (V, E ) be two grphs. H is subgrph of if V V n E E. There re mny subgrphs of grph. Question How mny subgrph woul grph with n verties n m eges hve? It is hr to ount the number of possible subgrphs sine we nnot just tke rbitrry subsets of verties n eges beuse the resulting subsets must efine grph. For exmple, we nnot hve n ege between two non-existing verties. One of the most stnr subgrphs of n unirete grph re the so lle onnete omponents of grph. Definition 14.6 ((Connete) Component). Let G = (V, E) be grph. A subgrph H of G is onnete omponent of G if it is mximl onnete subgrph of G.

3 14.1. PRELIMINARIES 215 In the efinition mximl mens we nnot ny more verties or eges from G without isonneting it. In generl when we sy n objet is mximl X, we men we nnot ny more to the objet without violting the property X. Question How mny onnete omponents o our two grphs hve? Our first exmple grph hs one onnete omponent (hene it is onnete), n the seon hs two. It is often useful to fin the onnete omponents of grph, whih les to the following problem: Definition 14.8 (The Grph Connetivity (GC) Problem). Given n unirete grph G = (V, E) return ll of its onnete omponents (mximl onnete subgrphs). When tlking bout subgrphs it is often not neessry to mention ll the verties n eges in the subgrph. For exmple for the grph onnetivity problem it is suffiient to speify the verties in eh omponent, n the eges re then implie they re simply ll eges inient on verties in eh omponent. This les to the importnt notion of inue subgrphs. Definition 14.9 (Vertex-Inue Subgrph). The subgrph of G = (V, E) inue by V V is the grph H = (V, E ) where E = {{u, v} E u V, v V }. Question In Exmple 14.1 wht woul be the subgrph inue by the verties {, b}? How bout {, b,, }? Using inue subgrphs llows us to speify the onnete omponents of grph by simply speifying the verties in eh omponent. The onnete omponents n therefore be efine s prtitioning of the verties. A prtitioning of set mens set of subsets where ll elements re in extly one of the subsets. Exmple Connete omponents on the grph in Exmple 14.2 returns the prtitioning {{, b,, }, {e, f}}. When stuying grph onnetivity sometimes we only re if the grph is onnete or not, or perhps how mny omponents it hs. In grph onnetivity there re no eges between the prtitions, by efinition. More generlly it n be useful to tlk bout prtitions of grph ( prtitioning of its verties) in whih there n be eges between prtitions. In this se some eges re internl eges within the inue subgrphs n some re ross eges between them.

4 216 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Exmple In Exmple 14.1 the prtitioning of the verties {{, b, }, {}, {e, f}} efines three inue subgrphs. The eges {, b}, {, }, n {e, f} re internl eges, n the eges {, }, {b, }, {b, e} n {, f} re ross eges Grph Contrtion We now return to the topi of the hpter, whih is grph ontrtion. Although grph ontrtion n be use to solve vriety of problems we will first look t how it n be use to solve the grph onnetivity problem esribe in the lst setion. Question Cn you think of wy of solving the grph-onnetivity problem? One wy to solve the grph onnetivity problem is to use grph serh. For exmple, we n strt t ny vertex n serh ll verties rehble from it to rete the first omponent, then move onto the next vertex n if it hs not lrey been serhe serh from it to rete the seon omponent. We then repet until ll verties hve been heke. Question Wht kins of lgorithms n you use to perform the serhes? Either BFS or DFS n be use for the iniviul serhes. Question Woul these pprohes yiel goo prllelism? Wht woul be the spn of the lgorithm? Using BFS n DFS le to perfetly sensible sequentil lgorithms for grph onnetivity, but they re not goo prllel lgorithms. Rell tht DFS hs liner spn. Question How bout BFS? Do you rell the spn of BFS? BFS tkes spn proportionl to the imeter of the grph. In the ontext of our lgorithm the spn woul be the imeter of omponent (the longest istne between two verties). Question How lrge n the imeter of omponent be? Cn you give n exmple? The imeter of omponent n be s lrge s n 1. A hin of n verties will hve imeter n 1

5 14.2. GRAPH CONTRACTION 217 Question How bout in ses when the imeter is smll, for exmple when the grph is just isonnete olletion of eges. Even if the imeter of eh omponent is smll, we might hve to iterte over the omponents one by one. Thus the spn in the worst se n be liner in the number of omponents. Contrtion hierrhies. We re intereste in n pproh tht n ientify the omponents in prllel. We lso wish to evelop n lgorithm whose spn is inepenent of the imeter, n ielly polylogrithmi in V. To o this let s give up on the ie of grph serh sine it seems to be inherently limite by the imeter of grph. Inste we will borrow some ies from our lgorithm for the sn opertion. In prtiulr we will use ontrtion. The ie grph ontrtion is to shrink the grph by onstnt frtion, while respeting the onnetivity of the grph. We n then solve the problem on the smller, ontrte grph n from tht result ompute the result for the tul grph. Exmple (A ontrtion hierrhy). The ruil point to remember is tht we wnt to buil the ontrtion hierrhy in wy tht respets n revels the onnetivity of the verties in hierrhil wy. Verties tht re not onnete shoul not beome onnete n vie vers. G log(n) G log(n)-1 b e G1 f This pproh is lle grph ontrtion. It is resonbly simple tehnique n n be

6 218 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY pplie to vriety of problems, beyon just onnetivity, inluing spnning trees n minimum spnning trees. As n nlogy, you n think of grph ontrtion s wy of viewing grph t ifferent levels of etil. For exmple, if you wnt to rive from Pittsburgh to Sn Frniso, you o not onern yourselves with ll the little towns on the ro n ll the ros in between them. Rther you think of highwys n the interesting ples tht you my wnt to visit. As you think of the grph t ifferent levels, however, you ertinly on t wnt to ignore onnetivity. For exmple, you on t wnt to fin yourself hopping on ferry to the UK on your wy to Sn Frniso. Contrting grph, however, is bit more omplite thn just piring up the o n even positione vlues s we i in the lgorithm for sn. Question Any ies bout how we might ontrt grph? When ontrting grph, we wnt to tke subgrphs of the grph n represent them with wht we ll superverties, justing the eges oringly. By seleting the subgrphs refully, we will mintin the onnetivity n t the sme time shrink the grph geometrilly (by onstnt ftor). We then tret the superverties s orinry verties n repet the proess until there re no more eges. The key question is wht kin of subgrphs shoul we ontrt n represent s single vertex. Let s first ignore effiieny for now n just onsier orretness. Question Cn we ontrt subgrph tht re isonnete? We surely shoul not tke isonnete subgrphs sine by repling eh subgrph with vertex, we re essentilly sying tht the verties in the subgrph re onnete. Therefore, the subgrphs tht we hoose shoul be onnete. Question Cn the subgrphs tht we hoose to ontrt overlp with eh other, tht is shre vertex or n ege. Although this might work, for our purposes, we will hoose isjoint subgrphs. We therefore will just onsier prtitions or our input grph where eh prtition is onnete.

7 14.2. GRAPH CONTRACTION 219 Figure 14.1: An illustrtion of grph ontrtion using mps. The ro to key west t three ifferent zoom levels. At first (top) there is just ro (n ege). We then see tht there is n isln (Mrthon) n in ft two highwys (1 n 5). Finlly, we see more of the islns n the ros in between.

8 220 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Exmple Prtitioning the grph in Exmple 14.1 might generte the prtitioning {{, b, }, {}, {e, f}} s inite by the following she regions: b b e ef f We nme the superverties b,, n ef. Note tht eh of the three prtitions is onnete by eges within the prtition. Prtitioning woul not return {{, b, f}, {, }, {e}}, for exmple, sine the subgrph {, b, f} is not onnete by eges within the omponent. One we hve prtitione the grph we n ontrt eh prtition into single vertex. We note, however, tht we now hve to o something with the eges sine their enpoints re no longer the originl verties, but inste re the new superverties. The internl eges within eh prtition n be thrown wy. For the ross eges we n relbel the enpoints to the new nmes of the superverties. Note, however, this n rete uplite eges (lso lle prllel eges), whih n be remove. Exmple For Exmple ontrting eh prtition n repling the eges. b b e ef f Prtition ientifie b ef b ef Contrte Duplite eges remove We re now rey to esribe grph-ontrtion bse on prtitioning. To simplify things we ssume tht prtitiongrph returns new set of superverties, one per prtition, long with tble tht mps eh of the originl verties to the supervertex to whih it belongs. Exmple For the prtitioning in Exmple 14.23, prtitiongrph returns the pir: ({b,, ef}, { b, b b, b,, e ef, f ef})

9 14.2. GRAPH CONTRACTION 221 b b f e ef Roun 1 Roun 3 b b bef ef Roun 2 ef bef b ef Figure 14.2: An exmple grph ontrtion. It ompletes when there re no more eges. We n now write the lgorithm for grph ontrtion s follows. Algorithm (Grph Contrtion). 1 funtion ontrtgrph(g = (V, E)) = 2 if E = 0 then V 3 else let 4 vl (V, P ) = prtitiongrph(v, E) 5 vl E = {(P [u], P [v]) : (u, v) E P [u] P [v]} 6 in 7 ontrtgrph(v, E ) 8 en This lgorithm returns one vertex per onnete omponent. It therefore llows us to ount the number of onnete omponents. An exmple is shown in Figure As in the verbl esription, ontrtgrph ontrts the grph in eh roun. Eh ontrtion on Line 4 returns the set of superverties V n tble P mpping every v V to v V. Line 5 uptes ll eges so tht the two enpoints re in V by looking them up in P : this is wht (P [u], P [v]) is. Seonly it removes ll self eges: this is wht the filter P [u] P [v] oes. One the eges re upte, the lgorithm reurses on the smller grph. The termintion onition is when there re no eges. At this point eh omponent hs shrunk own to singleton vertex. Nming superverties. In our exmple, we gve fresh nmes to superverties. It is often more onvenient to pik representtive from eh prtition s supervertex. We n then represent prtition s mpping from eh vertex to its representtive (supervertex). For exmple, we n return the prtition {{, b, }, {}, {e, f}} s the pir ({,, e}, {, b,,, e e, f e}).

10 222 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Computing the omponents. Our previous oe just returns one vertex per omponent, n therefore llows us to ount the number of omponents. It turns out we n moify the oe slightly to ompute the omponents themselves inste of returning their ount. Rell thn in the ontrtion bse oe for sn we i work both on the wy own the reursion n on the wy bk up, when we e the results from the reursive ll to the originl elements to generte the o inexe vlues. A similr ie will work here. The ie is to use the lbels of the reursive ll on the superverties, to relbel ll verties. This is implemente by the following lgorithm. Algorithm (Contrtion-bse grph onnetivity). 1 funtion onnetecomponents(g = (V, E)) = 2 if E = 0 then (V, {v v : v V }) 3 else let 4 vl (V, P ) = prtitiongrph(v, E) 5 vl E = {(P [u], P [v]) : (u, v) E P [u] P [v]} 6 vl (V, P ) = onnetecomponents(v, E ) 7 in 8 (V, {v P [P [v]] : v V }) 9 en In ition to returning the set of omponent lbels s returne by ontrtgrph, this lgorithm returns mpping from eh of the initil verties to its omponent. Thus for our exmple grph it might return: ({}, {, b,,, e, f }) sine there is single omponent n ll verties will mp to tht omponent lbel. The only ifferene of this oe from ontrtgrph is tht on the wy bk up the reursion inste of simply returning the result (V ) we lso upte the representtives for V bse on the representtives from V. Consier our exmple grph. As before lets sy the first prtitiongrph returns V = {,, e} P = {, b,,, e e, f e} Sine the grph is onnete the reursive ll to omponents will mp ll verties in V to the sme vertex. Lets sy this vertex is giving: P = {,, e } Now wht Line 8 in the oe oes is for eh vertex v V, it looks for v in P to fin its representtive v V, n then looks for v in P to fin its onnete omponent (will be lbel from V ). This is implemente s P [P [v]]. For exmple vertex f fins e from P n then looks this up in P to fin. The bse se of the omponents lgorithm lbels every vertex with itself.

11 14.2. GRAPH CONTRACTION 223 Prtitioning the Grph. In our isussion so fr, we hve not speifie how to prtition the grph. There ifferent wys to o it, whih orrespons to ifferent forms of grph ontrtion. Question Wht properties re esirble in forming the grph prtitions to ontrt, i.e. the prtitions generte by grphprtition. There re hnful of properties we woul like when generting the prtitions, beyon the requirement tht eh prtition must be onnete. Firstly we woul like to be ble to form the prtitions without too muh work. Seonly we woul like to be ble to form them in prllel. After ll, one of the min gols of grph ontrtion is to prllelize vrious grph lgorithms. Finlly we woul like the number of prtitions to be signifintly smller thn the number of verties. Ielly it shoul be t lest onstnt frtion smller. This will llow us to ontrt the grph in O(log V ) rouns. Here we outline three methos for forming prtitions tht will be esribe in more etil in the following setions. Ege Contrtion: Eh prtition is either single vertex or or two verties with n ege between them. Eh ege will ontrt into single vertex. Str Contrtion: Eh prtition is str, whih onsists of enter vertex v n some number (possibly 0) stellite verties onnete by n ege to v. There n lso be eges mong the stellites. Tree Contrtion: For eh prtition we hve spnning tree Ege Contrtion In ege ontrtion, the ie is to prtition the grph into omponents onsisting of t most two verties n the ege between them. We then ontrt the eges to single vertex.

12 224 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Exmple The figure below shows n exmple ege ontrtion. e Roun 1 b f e e In the bove exmple, we were ble to reue the size of the grph by ftor of two by ontrting long the selete eges. Remrk Fining set of isjoint eges is lso lle vertex mthing, sine it tries to mth every vertex with nother vertex (monogmously), n is stnr problem in grph theory. In ft fining the vertex mthing with the mximum number of eges is well known problem. Here we re not onerne if it hs mximum number of eges. Question Cn you esribe n lgorithm for fining vertex mthing? One wy to fin vertex mthing is to strt t vertex n pir in up with one of its neighbors. This n be ontinue until there re no longer ny verties to pir up, i.e., ll eges re lrey inient on selete pir. The pirs long with ny unpire verties form the prtition. The problem with this pproh is tht it is sequentil. Question Cn you think of wy to fin vertex mthing in prllel? An lgorithm for forming the pirs in prllel will nee to be ble to mke lol eisions t eh vertex. One possibility is in for eh vertex in prllel to pik one of its neighbors to pir up with.

13 14.2. GRAPH CONTRACTION 225 Question Wht is the problem with this pproh? The problem with this pproh is tht it is not possible to ensure isjointness. We nee wy to brek the symmetry tht rises when two verties try to pir up with the sme vertex. Question Cn you think of wy to use rnomiztion to brek this symmetry? We n use rnomiztion to brek the symmetry. One pproh is to flip oin for eh ege n pik the ege if the ege (u, v) flips hes n ll the eges inient on u n v flip tils. Question Cn you see how mny verties we woul pir up in our yle grph exmple? Lets nlyze this pproh on grph tht is yle. Let R e be n initor rnom vrible enoting whether e is selete or not, tht is R e = 1 if e is selete n 0 otherwise. The expettion of initor rnom vribles is the sme s the probbility it hs vlue 1 (true). Sine the oins re flippe inepenently t rnom, the probbility tht vertex piks hes n its two neighboring eges pik tils is = 1. Therefore we hve E[R e] = 1/8. Thus summing over ll eges, we onlue tht expete number of eges elete is m 8 (note, m = n). In the hpter on rnomize lgorithms Setion 8.3 we rgue tht if eh roun of n lgorithm shrinks the size by onstnt frtion in expettion, n if the rnom hoies in the rouns re inepenent, then the lgorithm will finish in O(log n) rouns with high probbility. Rell tht ll we neee to o is multiply the expete frtion tht remin ross rouns n then use Mrkov s inequlity to show tht fter some k log n rouns the probbility tht the problem size is lest 1 is very smll. For yle grph, this tehnique les to n lgorithm for grph ontrtion with liner work n O(log 2 n) spn. Question Cn you think of wy to improve the expete number of eges ontrte? We n improve the probbility tht we remove n ege by letting eh ege pik rnom number in some rnge n then selet n ege if it is the lol mximum, i.e., it pike the highest number mong ll the eges inient on its en points. This inreses the expete number of eges ontrte in yle to m 3. Question So fr in our exmple, we hve onsiere simple yle grph. Do you think this tehnique woul work s effetively for rbitrry grphs?

14 226 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY v Ege ontrtion oes not work with generl grphs. The problem is tht if the vertex tht n ege is inient on hs high egree, then it is highly unlikely for the vertex to be pike. In ft, mong ll the eges inient n vertex only one n be pike for ontrtion. Thus in generl, using ege ontrtion, we n shrink the grph only by smll mount. As n exmple, onsier str grph: Definition (Str). A str grph G = (V, E) is n unirete grph with enter vertex v V, n set of eges E = {{v, u} : u V \ {v}}. In wors, str grph is grph me up of single vertex v in the mile (lle the enter) n ll the other verties hnging off of it; these verties re onnete only to the enter. It is not iffiult to onvine ourselves tht on str grph with n + 1 verties 1 enter n n stellites ny ege ontrtion lgorithm will tke Ω(n) rouns. To fix this problem we nee to be ble to form prtitions tht re more thn just eges Str Contrtion We now onsier more ggressive form of ontrtion. The ie is to prtition the grph into set of str grphs, onsisting of enter n stellites, n ontrt eh str into vertex in one roun. Exmple In the grph below (left), we n fin 2 isjoint strs (right). The enters re olore blue n the neighbors re green. The question is how to ientify the isjoint strs to form the prtitioning.

15 14.3. STAR CONTRACTION 227 Question How n we fin isjoint strs? Similr to ege ontrtion, it is possible to to onstrut strs sequentilly pik n rbitrry vertex, tth ll its neighbors to the str, remove the str from the grph, n repet. However, gin, we wnt prllel lgorithm tht mkes lol eisions. As in ege ontrtion when mking lol eisions, we nee wy to brek the symmetry between two verties tht wnt to beome the enter of the str. Question Cn you think of rnomize pproh for seleting strs (enters n stellites)? As with ege ontrtion we n use rnomiztion to ientify strs. An lgorithm n use oin flips to eie whih verties will be enters n whih ones will be stellites, n n then eie how to pir up eh stellite with enter. To etermine the enters, the lgorithm n flip oin for eh vertex. If it omes up hes, tht vertex is str enter, n if it omes up tils, then it is potentil stellite it is only potentil stellite beuse quite possibly, none of its neighbors flippe he so it hs no enter to hook to. At this point, we hve etermine every vertex s potentil role, but we ren t one: for eh stellite vertex, we still nee to eie whih enter it will join. For our purposes, we re only intereste in ensuring tht the strs re isjoint, so it oesn t mtter whih enter stellite joins. We will mke eh stellite rbitrrily hoose ny enter mong its neighbors. Exmple An exmple str prtition. Coin flips turne up s inite in the figure. H T T H T T H T T H T H T H T b e b e b e oin flips (hes(v,i)) fin potentil enters (TH) ompute "hook" eges (P) Before esribing the lgorithm for prtitioning grph into strs, we nee to sy ouple wors bout the soure of rnomness. Wht we will ssume is tht eh vertex is given (potentilly infinite) sequene of rnom n inepenent oin flips. The i th element of the sequene n be esse hes(v, i) : vertex int bool.

16 228 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY Algorithm (Str Prtition). 1 % requires: n unirete grph G = (V, E) n roun number i 2 % returns: V = remining verties fter ontrtion, 3 % P = mpping from V to V 4 funtion strprtition(g = (V, E), i) = 5 let 6 % selet eges tht go from til to he 7 vl TH = {(u, v) E hes(u, i) hes(v, i)} 8 % mke mpping from tils to hes, removing uplites 9 vl P = (u,v) TH {u v} 10 % remove verties tht hve been remppe 11 vl V = V \ omin(p ) 12 % Mp remining verties to themselves 13 vl P = {u u : u V } P 14 in (V, P ) en The funtion returns true if the i th flip on vertex v is hes n flse otherwise. Sine most mhines on t hve true soures of rnomness, in prtie this n be implemente with pseuornom number genertor or even with goo hsh funtion. The lgorithm for str ontrtion is given in Algorithm In our exmple grph, the funtion hes(v, i) on roun i gives oin flip for eh vertex, whih re shown on the left. Line 7 selets the eges tht go from til to he, whih re shown in the mile s rrows n orrespon to the set TH = {(, ), (, b), (e, b)}. Notie tht some potentil stellites (verties tht flippe tils) re jent to multiple enters (verties tht flippe hes). For exmple, vertex is jent to verties n b, both of whih got hes. A vertex like this will hve to hoose whih enter to join. This is sometimes lle hooking n is eie on Line 13, whih removes uplites for til using union, giving P = { b, e b}. In this exmple, is hooke up with b, leving enter without ny stellite. Line 11 tkes the verties V n removes from them ll the verties in the omin of P, i.e. those tht hve been remppe. In our exmple omin(p ) = {, e} so we re left with V = {, b, }. In generl, V is the set of verties whose oin flippe hes or whose oin flippe tils but in t hve neighboring enter. Finlly we mp ll verties in V to themselves n union this in with the hooks giving P = {, b b, b,, e b}. Anlysis of Str Contrtion. When we ontrt these strs foun by strcontrt, eh str beomes one vertex, so the number of verties remove is the size of P. In expettion, how big is P? The following lemm shows tht on grph with n non-isolte verties, the size of P or the number of verties remove in one roun of str ontrtion is t lest n/4.

17 14.3. STAR CONTRACTION 229 Lemm For grph G with n non-isolte verties, let X n be the rnom vrible initing the number of verties remove by strcontrt(g, r). Then, E [X n ] n/4. Proof. Consier ny non-isolte vertex v V (G). Let H v be the event tht vertex v omes up hes, T v tht it omes up tils, n R v tht v omin(p ) (i.e, it is remove). By efinition, we know tht non-isolte vertex v hs t lest one neighbor u. So, we hve tht T v H u implies R v sine if v is til n u is he v must either join u s str or some other str. Therefore, Pr [R v ] Pr [T v ] Pr [H u ] = 1/4. By the linerity of expettion, we hve tht the number of remove verties is [ ] E I {R v } = E [I {R v }] n/4 v:v non-isolte sine we hve n verties tht re non-isolte. v:v non-isolte Exerise Wht is the probbility tht vertex with egree is remove. Cost Speifition (Str Contrtion). Using ArrySequene n STArrySequene, we n implement strcontrt resonbly effiiently in O(n + m) work n O(log n) spn for grph with n verties n m eges Returning to Connetivity Now lets nlyze the ost of the lgorithm for ounting the number of onnete omponents we esribe erlier when using str ontrtion for ontrt. Let n be the number of nonisolte verties. Notie tht one vertex beomes isolte (ue to ontrtion), it stys isolte until the finl roun (ontrtion only removes eges). Therefore, we hve the following spn reurrene (we ll look t work lter): S(n) = S(n ) + O(log n) where n = n X n n X n is the number of verties remove (s efine erlier in the lemm bout strcontrt). But E [X n ] = n/4 so E [n ] = 3n/4. This is fmilir reurrene, whih we know solves to O(log 2 n). As for work, ielly, we woul like to show tht the overll work is liner sine we might hope tht the size is going own by onstnt frtion on eh roun. Unfortuntely, this is not the se. Although we hve shown tht we n remove onstnt frtion of the non-isolte verties on one str ontrt roun, we hve not shown nything bout the number of eges. We n rgue tht the number of eges remove is t lest equl to the number of verties sine

18 230 CHAPTER 14. GRAPH CONTRACTION AND CONNECTIVITY removing vertex lso removes the ege tht tthes it to its str s enter. But this oes not help symptotilly boun the number of eges remove. Consier the following sequene of rouns: roun verties eges 1 n m 2 n/2 m n/2 3 n/4 m 3n/4 4 n/8 m 7n/8 In this exmple, it is ler tht the number of eges oes not rop below m n, so if there re m > 2n eges to strt with, the overll work will be O(m log n). Inee, this is the best boun we n show symptotilly. Hene, we hve the following work reurrene: W (n, m) W (n, m) + O(n + m), where n is the remining number of non-isolte verties s efine in the spn reurrene. This solves to E [W (n, m)] = O(n + m log n). Altogether, this gives us the following theorem: Theorem For grph G = (V, E), numcomponents using strcontrt grph ontrtion with n rry sequene works in O( V + E log V ) work n O(log 2 V ) spn Tree Contrtion Tree ontrtion tkes set of isjoint trees n ontrts them. When ontrting tree, we n use str ontrtion s before but we n show tighter boun on work beuse the number of eges erese geometrilly (the number of eges in tree is boune by the number of verties). Therefore the number of eges must go own geometrilly from step to step, n the overll ost of tree ontrtion is O(m) work n O(log 2 n) spn using n rry sequene. When ontrting grph, we n lso use prtition funtion tht selet isjoint trees (str ontrtion is speil se). As in str ontrtion, we woul pik the subgrph inue by the verties in eh tree n ontrt them.