THE TOTAL ACQUISITION NUMBER OF THE RANDOMLY WEIGHTED PATH

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1 Discussioes Mathematicae Graph Theory ) doi: /dmgt.1972 THE TOTAL ACQUISITION NUMBER OF THE RANDOMLY WEIGHTED PATH Aat Godbole Departmet of Mathematics ad Statistics East Teessee State Uiversity Elizabeth Kelley Departmet of Mathematics Uiversity of Miesota Emily Kurtz Departmet of Mathematics Wellesley College Pawe l Pra lat Departmet of Mathematics Ryerso Uiversity pralat@ryerso.ca ad Yiguag Zhag Departmet of Applied Mathematics ad Statistics The Johs Hopkis Uiversity yzha132@jhu.edu Abstract There exists a sigificat body of work o determiig the acquisitio umber a t G) of various graphs whe the vertices of those graphs are each iitially assiged a uit weight. We determie properties of the acquisitio Uautheticated Dowload Date 11/9/18 8:39 PM

2 920 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag umber of the path, star, complete, complete bipartite, cycle, ad wheel graphs for variatios o this iitial weightig scheme, with the majority of our work focusig o the expected acquisitio umber of radomly weighted graphs. I particular, we boud the expected acquisitio umber Ea t P )) of the -path whe distiguishable uits of itegral weight, or chips, are radomly distributed across its vertices betwee ad With computer support, we improve it by showig that Ea t P )) lies betwee ad We the use subadditivity to show that the limitig ratio limea t P ))/ exists, ad simulatios reveal more exactly what the limitig value equals. The Hoeffdig-Azuma iequality is used to prove that the acquisitio umber is tightly cocetrated aroud its expected value. Additioally, i a differet cotext, we offer a o-optimal acquisitio protocol algorithm for the radomly weighted path ad exactly compute the expected size of the resultat residual set. Keywords: total acquisitio umber, ssoizatio, dessoizatio, Maxwell-Boltzma ad Bose-Eistei allocatio Mathematics Subject Classificatio: 05C Itroductio I this paper, we cosider vertex-weighted graphs ad deote the weight of vertex v as wv). Let G be a graph with a iitial weight of 1 o each vertex. For adjacet v,u VG), weight ca be trasferred from v to u via a acquisitio move if the iitial weight o u is at least as great as the weight o v. Whe there are o more acquisitio moves possible, the set of vertices with o-zero weight forms a idepedet set referred to as the residual set. The miimal cardiality of this set, a t G), is the total acquisitio umber of G. A sequece of acquisitio moves that results i a residual set is referred to as a acquisitio protocol ad is optimal if the residual set has cardiality a t G). This cocept of acquisitio umber was first itroduced by [3] ad has subsequetly bee ivestigated i [4, 6]. Whe acquisitio moves are allowed to trasfer ay itegral amout of weight from a vertex, the miimum cardiality of the residual set is called the uit acquisitio umber, deoted a u G); see [6]. If acquisitio moves are allowed to trasfer ay o-zero amout of weight from a vertex, the miimum cardiality of the residual set is called the fractioal acquisitio umber, deoted a f G); see [5]. Because we cosider oly total acquisitio umber, ay istaces of the term acquisitio umber i this paper should be uderstood to mea total acquisitio umber. Although we offer a few mior results for graphs with the caoical weightig scheme where each vertex has iitial weight 1), we primarily cosider variats Uautheticated Dowload Date 11/9/18 8:39 PM

3 The Total Acquisitio Number of the Radomly Weighted Path 921 o that weightig scheme where the iitial weights of vertices are allowed to assume ay itegral value. Ay -vertex graph with vertex labels {1,2,...,} ca be associated with a iteger sequece A = a 1,a 2,...,a ) where a i deotes the iitial weight give to vertex i. For such weight sequeces ad particular classes of graphs, we cosider i Sectio 2 the size a max G) of the largest possible residual set, whether a t G) chages or remais the same as the uit weight case, ad the existece of legal residual sets with sizes equal to every itegral value i the iterval [a t G),a max G)]. Our primary focus is, however, o the total acquisitio umber of graphs with radomly weighted vertices. Although there exists work o radom graphs whose vertices each begi with uit weight due to [1], we are ot aware of ay existig work o graphs with radomly-weighted vertices. I Sectio 2, we make some prelimiary remarks. I Sectio 3, we obtai bouds o the total expected acquisitio umber of the radomly weighted path, where vertex weightig is assiged accordig to both the sso ad Maxwell-Boltzma distributios i the latter case, the chips are thus cosidered to be distiguishable, ad obviously the Bose-Eistei distributio might yield completely differet results!). We also show that the limitig ratio limea t P ))/ exists, ad that the acquisitio umber is tightly cocetrated aroud its expected value. Additioally, i a differet cotext, we offer a o-optimal acquisitio protocol algorithm for the radomly weighted path ad exactly compute the expected size of the resultat residual set. 2. Basic Results I this sectio, we provide some basic results. The proofs of these results are ot very difficult, but the uderlyig logic is importat for Sectio a t G) = 1 I this subsectio, we cosider graphs i which the vertices ca have ay oegative iteger weight. Oe questio we could ask is what is the smallest maximum vertex weight ecessary to drive a t G) dow to 1? Deote such a value as smvg). We specifically cosider the complete graph K o vertices; the -cycle ad -path C,P ; ad the star, wheel, ad complete bipartite graphs deoted respectively by K 1,,W, ad K,m. It is clear that the smvk ) = smvw ) = smvk 1, ) = 1, because we ca have a special vertex the ceter vertex for W ad K 1, ; ay vertex for K ) absorb the weight of its eighbors first ad thus make it the largest weighted vertex i VG). For K,m, its smv value is 1 as well. Let A,B be the two vertex sets of K,m, where A B. Let the iitial weight of all vertices be 1. Let two vertices v A,v B Uautheticated Dowload Date 11/9/18 8:39 PM

4 922 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag be two arbitrary vertices from A ad B. At the first step, v A ca acquire all the weight from vertices i B\{v B } ad v B ca acquire all the weight from vertices i A\{v A }. Because A B, wv B ) wv A ) ad v B ca acquire the weight from v A. Thus, the smv value for complete bipartite graphs is also 1. We ext cosider paths ad cycles, for which the situatio is more uaced. Lemma 2.1. Let P = v 1 v 2 v be a path o 2 vertices, where v 1 ad v are the edpoits of the path. Let each vertex have weight at least oe. I order for v to be the oly vertex i the residual set, its iitial weight must be at least 2 2 ad i the extremal case) the path must correspod to the uique cofiguratio Proof. First ote that i order for v to be the oly vertex i the residual set, the aquisitio protocol must first move the weight from v 1 to v 2, the from v 2 to v 3, ad so o. Therefore each v i must start with weight at least the sum of the weights of v 1,v 2,...,v i 1. Now sice v 1 has weight at least 1, v 2 must start with weight at least 1, v 3 must start with weight at least = 2, ad the geeral statemet follows by iductio. The problem of fidig the smallest maximum vertex weight ecessary to drive a t G) dow to 1 for a path o vertices is equivalet to fidig the smallest iitial weight for the middle vertex to absorb all the weight i that path. By Lemma 2.1, a weight of 2 m is eeded to take all the weight o a m + 2)- path to a leaf vertex; applyig this fact to the two middle vertices, we see that a weight of 2 m suffices to move all the weight o P 2m+4 to these vertices ad the to oe of them. Solvig = 2m + 4 for m, we get m = /2 2 for eve. If is odd, we get m = /2 2 by the same reasoig. It follows immediately that smvc ) = 2 /2 2 as well. The smv value for a m grid graph is thus at most 2 /2 + m/2 4 ; here we use the strategy of movig all the weight i each row to the ceter vertex, ad the all the weight i the middle colum to the ceter of the grid. For the lower boud, let us ote that, regardless where the absorbig vertex v is, at least oe of the four corers is at distace m 1)/2 + 1)/2 from v. Hece, at the time whe the weight from this corer is pushed to the fial destiatio, the weight at v must be at least 2 m 1)/2 + 1)/2 1. However, perhaps some of this weight comes from the other three eighbours of v. As a result, we oly get that the iitial weight at v is at least 2 m 1)/2 + 1)/2 4, which is matchig the upper boud for m, both eve, ad is always by a multiplicative factor of at least 1/4 away from it Size of a residual set Let G = V,E) be a arbitrary graph. Note that the size of the maximum idepedet set is a atural upper boud of the size of a residual set. By choosig Uautheticated Dowload Date 11/9/18 8:39 PM

5 The Total Acquisitio Number of the Radomly Weighted Path 923 the weight of vertices i G strategically, how may differet sizes of residual sets ca we get for a give graph? I this subsectio, we focus o K,C,P,W, ad K,m. It is clear that the size of the residual set of K must be 1 for ay assigmet ofweights, becausethemaximumidepedetsetofk hassize1. Now, cosider C. Because the size of the maximum idepedet set is 2, the residual set ca be o larger tha 2. Ideed, by assigig the vertices i a largest idepedet set of C the first 2 largest weights, we ca obtai a residual set with 2 vertices. For example, /2 2 s ad /2 1 s ca do the job. Now, to obtai a residual set of size i, where 1 i 2, we eed to choose i vertices which form a idepedet set that are equally spaced to the extet possible, ad strategically assig the values of the i largest weights to these vertices so as to eable those vertices to acquire the weights of the other vertices. This ca be achieved because of the reasoig i Sectio 2.1 by usig weights of 1,1,2,...,2 /i 2 o each of the i paths that the cycle ca be thought of as beig comprised of. Next we see that the size of the residual set of W ca be ay iteger from 1 to 2 as well. We ca place the smallest weight o the ceter vertex to reduce the problem to the problem of C after the first move. With the same argumet, paths ca have residual set with size from 1 to 2, where the upper boud is the size of the largest idepedet set of P. The size of the residual set of K m, for m ca be ay iteger from 1 to m. By placig the m largest umbers of the sequece o the vertices of the largerside of the bipartite graph, we esure that o vertex o the smaller side ca acquire ay additioal chips, thus resultig i a residual set icludig all vertices of the larger side. After choosig appropriate weights of vertices, we ca esure that exactly oe vertex o the larger side is acquired by assigig the smallest umber to oe vertex o the larger side, the assigig the ext smallest umbers to the vertices of the smaller side. The, as log as the sum of the smallest umber ad the m th largest umber is smaller tha the m 1) th largest umber, we have a residual set of size m 1. We ca use a similar strategy to get ay other umber betwee 1 ad m Subadditivity Now, we cosider P with uit weights. Without makig use of the fact that a t P ) = 4, we ca show that: Lemma 2.2. The sequece {a t P )} =1 is subadditive. Proof. Cosider the graph P +m, with m, Z. If P +m is subdivided ito P ad P m ad distict acquisitio protocols are ru o each, the the sum of the resultig iduced residual sets is a t P m ) + a t P ). Because a t P +m ) is Uautheticated Dowload Date 11/9/18 8:39 PM

6 924 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag defiitioally the size of the miimal residual set, the fact that it is possible to obtai a set of size a t P )+a t P m ) gives the boud as desired. a t P +m ) a t P )+a t P m ), Corollary 1.For {a t P i )} a i=1, the limit lim tp ) exists ad is equal to if atp). Proof. This result follows directly from Lemma 2.2 ad Fekete s Lemma. Although it is ituitively obvious that this limit equals 1/4, the power of subadditivity will become clear i the ext sectio, where we use radom weights. 3. Total Acquisitio o Radomly Weighted Graphs 3.1. sso Distributio I this sectio, we cosider the total acquisitio umber of P whe each vertex begis with weight 1), i.e., the vertices have radom weights determied by a sequece of idepedet sso variables with uit mea. We deote this specific cofiguratio as P. It may be oted, however, that other distributios could have bee used by us, ad similar methods of proof could have bee used.) I geeral, the upper case letter A will be used for the acquisitio umber whe it is viewed as a radom variable. We ca begi by provig that the limit lim EA t P )) exists, ad provide upper ad lower bouds for EA t P )). We start with a few remarks. Remarks. First, let us ote that checkig whether a give weightig of P = v 1,v 2,...,v ) ca be used to move the total weight oto oe vertex ca be doe as follows. Startig from v 1, we push its weight to the right as much as possible, edig at v k for some k. The, idepedetly ad usig the iitial weightig), we start from v ad push its weight to the left as much as possible, edig at v l for some l 1. It is straightforward to see that if k l 1, the our task is possible; otherwise, it is ot. Next, fidig a t P ) for a give weightig) ca be easily doe as follows. Suppose that weights o the subpath v 1,v 2,...,v k ) ca be moved to oe vertex. If weights o the subpath v 1,v 2,...,v k,v k+1 ) ca also be moved to oe vertex, the this is at least as good strategy as movig oly weighs from v 1,v 2,...,v k ) ad the applyig the best strategy for the remaiig path cosider simple Uautheticated Dowload Date 11/9/18 8:39 PM

7 The Total Acquisitio Number of the Radomly Weighted Path 925 coupligbetweethetwostrategies). Asaresult, fidiga t P )forayweightig ca be doe with a easy greedy algorithm ad with the support of a computer). Fially, we get from the above remarks that EA t P )) is a icreasig fuctio of. Lemma 3.1. The sequece {E )) } =1 is subadditive. Proof. Same as the proof of Lemma 2.2 for each sample realizatio. We the take expectatios to get the result. Corollary 2. The limit lim EA tp )) exists ad is equal to if EAtP )). Proof. This result follows directly from Lemma 3.1 ad Fekete s Lemma. Theorem 3.2. The expected acquisitio umber of P is bouded as E )) Proof. Defie a islad as the clump of vertices to the left of the first zero weight; to the right of the last zero weight; or i betwee ay two successive zero weights. Islads are of o-egative size, ad thus each cosist of a possibly empty set of o-zero umbers. The islad size thus has a geometric distributio with success probability 1/e ad expected size e 1, yieldig a expected umber of /e+c;0 c 1, for the radom umber Λ of islads. Note that the expected umber of zeros is /e.) Theoretically, the expected total acquisitio umber could be calculated usig the coditioal probability expressio E A t P )) = Λ = j=1 )] ) E [a t P Λj = e +c E [ )] a t P Λ1 e +c ) j=0 E )) j P Λ1 = j), where the secod equality follows from Wald s Lemma. However, calculatig E [ A t Pj ) ] is difficult for arbitrary j. The probability that a islad is of size j equals 1 1 j 1 e) e). Thus, there is a roughly 84% probability that a islad has size three or less, ad a reasoable lower boud ca be obtaied by restrictig the calculatio to those cases. It is clear that E [ A t P0 ) ] = 0 ad that E [ A t P 1 ) ] = E [ A t P 2 ) ] = 1. For P 3, however, it is possible to have A t P 3 ) = 1 or A t P 3 ) = 2. Because A t P 3 ) = A t a b c) requires that wa) > wb) 1 ad wc) > wb) 1, we coditio o each of wa),wb), ad wc) Uautheticated Dowload Date 11/9/18 8:39 PM

8 926 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag beig o-egative, ad, settig Wb) = k, we get P [A t P 3 ) = 2] = e e 1 k! k 1 j k = e 1) 3 k! = e 2 e 1) 3 Thus E[A t P 3 )] ca be calculated as k 1 k=1 1 k! j k+1 1 e e 1 ) j! 1 j! 2 Γk +1,1) Γk +1) 2 ) ) E[A t P 3 )] = P [A t P 3 ) = 1]1)+P [A t P 3 ) = 2]2) Usig 1) ad the mootoicity of EA t P )), we ca ow calculate a lower boud for E )) as E )) ) e +c 1 1 ) 1 e e ) 2 1 e e ) ) 3 e To obtai a upper boud, we use the fact that a t P j ) j+1 2 for ay j. Returig to our coditioal probability expressio, this allows us to costruct a upper boud for EA t P )) as E )) ) 3 = e +c t P j ))P Λ 1 = j=1ea j)+ EA t P j ))P Λ 1 = j) j 4 ) 3 e +c j= e = e ) EA t P j ))P Λ 1 = j)+ e +c 1 1 e 2e 1 1 e 2e j=4 j 1 1 j e) e 1) 3 3+e) e , which gives us the desired bouds for E )). j 4 j +1 2 P Λ 1 = j) Uautheticated Dowload Date 11/9/18 8:39 PM

9 The Total Acquisitio Number of the Radomly Weighted Path 927 The above bouds ca certaily be improved, but we do ot do so here rather, we poit out methods that might lead to a tighteig. First we ca compute PA t P 4 ) = 2) or eve more higher order terms so as to improve the lower boud. For the upper boud, oe may do a more careful calculatio by usig the fact that A t P j ) j 2, ad separatig the argumet for j 4 ito the eve ad odd cases. However, these methods are likely to yield oly icremetal improvemets, ad so we ext report o the results of simulatios which yield theoretical bouds that are vastly better tha the oes above, ad also suggest the value of the limitig costat Simulatios As we already remarked i Subsectio 3.1, with the support of a computer, it is easy to fid a t P j ) for a give iitial weightig. By cosiderig all possible k j cofiguratios of weights at most k = kj), we ca easily estimate EA t P j )) from below ad above. We cosidered all paths o at most 21 vertices to obtai the followig bouds for more details, see [7]). j k = kj) lower boud upper boud Based o that we get the followig. Corollary 3. The expected acquisitio umber of P is bouded as E )) Uautheticated Dowload Date 11/9/18 8:39 PM

10 928 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag Moreover, we performed a umber of experimets o paths of legth = 100,000,000,000. Simulatios suggest that E )) agai, for more details, see [7]) dessoized model Although cosiderig a sso model for weight distributio makes it sigificatly easier to boud E )), there is typically more iterest i models where a fixed amout of weight is distributed o P. I particular, we will let A t P) ad A t P) x be the total acquisitio umber whe or x tokes are radomly placed o P. A t P) will be referred to as the dessoized model. I order to traslate our result for the ssoized model to this dessoized model, we begi by establishig two lemmas. Lemma 3.3. For P, assigig a iitial weight of 1) chips to each vertex is equivalet to cosiderig the model i which we geerate the total umber of chips accordig to a ) distributio, ad the distribute them idepedetly ad uiformly o the vertices. Proof. Oe half of the proof follows from the fact that the sum of idepedet 1) variables has a ) distributio. Next, cosider a radom distributio of ) chips o P as i the statemet of the lemma. The probability that two particular vertices, u, v, receive x, y chips respectively the same argumet holds for ay umber of vertices) is give by P [wu) = x,wv) = y] [ ] [ ] = P wv i ) = r P wu) = x,wv) = y wv i ) = r = r=0 r=x+y i=1 e r r! r x,y ) 1 ) x+y 1 2 i=1 ) r x y = e x!y! r=x+y 2) r x y r x y)! = e 2 x!y! which we recogize as the product of the probability that wu) = x; wv) = y if the iitial weights o the vertices are determied by a idepedet 1) process, as desired. The followig lemma is critical ad valid oly for special graphs such as P. Lemma 3.4. Chagig the iitial weight o a sigle vertex ca chage a t P ) by at most 1. Proof. The proof is a easy cosequece of the remarks at the begiig of Sectio 3.1. Ideed, after applyig the greedy algorithm metioed there, we Uautheticated Dowload Date 11/9/18 8:39 PM

11 The Total Acquisitio Number of the Radomly Weighted Path 929 decompose P ito a t P ) subpaths; each subpath has the total acquisitio of 1 with the iitial weightig iduced o correspodig vertices). Now, chagig the iitial weight o a sigle vertex ca icrease the total acquisitio of the correspodig path by 1. Hece, globally, a t P ) ca icrease by at most 1. This fiishes the proof as it also implies that it caot decrease by more tha 1 if it decreases by more tha that, the after switchig back to the origial weightig, the parameter icreases by more tha oe). Eve though the mai itet of the use of Lemma 3.4 is dessoizatio, it also quickly gives a very sharp cocetratio of A t P ) aroud E )). ) Theorem 3.5. For determied by a series of radom uit sso trials, X 1,...,X ad ay φ), Pr [ A t P ) )) E At P > 2φ)] 0 ) as ad is therefore tightly cocetrated i a iterval of width φ) ) O aroud E )) = Θ). Proof. From Lemma 3.4, we kow that for every i ad ay two sequeces of possible outcomes x 1,...,x ad x 1,...,x i 1,x i,x i+1,...,x, A t P ) ) X1 = x 1,...,X = x A t P ) Xj = x j j i),x i = x i It follows from the Hoeffdig-Azuma iequality that [ ) )) Pr E At P > 2φ)] 2e φ) 0, as desired. ) 1. We will let EA t P )) ad EA t P x )) respectively deote the expected total acquisitio umber whe ad x tokes are radomly placed o P. Lemma 3.6. For all x [ φ),+φ) ], where φ) is arbitrary, EA t P )) φ) EA t P x )) EA t P ))+φ). Proof. This follows immediately from Lemma 3.4, ad, moreover, holds for the radom variable A t P ) as well, before expectatios are take. Together, these lemmas ca be used to show that, whe is sufficietly large, the bouds from Theorem 3.2 also apply to the dessoized model. Uautheticated Dowload Date 11/9/18 8:39 PM

12 930 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag Theorem 3.7. For the dessoized chip distributio process o P, EA t P lim )) E )) = lim. That is, the limit both exists ad is idetical to the limit for the ssoized model. Proof. By Chebychev s iequality, with probability at most 1 1/φ 2 ) we have x φ) if x ), so that by Lemma 3.6 A t P ) A t P x ) φ). O the other had, if x > φ), the trivially A t P ) A t P x ). Combiig the above two facts, we see that for ay fixed φ) = o ) such that φ) as, ad with B = [ φ),+φ) ] E )) = e x EA t P)) x x! x 0 = e x EA t P)) x + e x EA t P x 2) )) x! x! x B x B EA tp)) + φ) + 1 EAt P ) )) φ 2 +1 ) = 1+o1)) EA tp)) 3), as E )) = Ω). Likewise by just icludig the first term i 2), we get that 4) E )) EAt P 1 o1)) )) φ) ) = 1 o1)) EA tp)). Iequalities 3) ad 4) prove the result Uiform distributio So far, our discussio has focused primarily o optimal acquisitio protocols. For small examples or particularly simple graphs, it is ofte possible to defiitively determie the optimal acquisitio protocol. I larger or more complicated cases, however, doig so becomes laborious ad complexity issues become more relevat. I order to sidestep this issue, we shift i this sectio to cosiderig the size of the Uautheticated Dowload Date 11/9/18 8:39 PM

13 The Total Acquisitio Number of the Radomly Weighted Path 931 residual set produced by a algorithmic acquisitio protocol. We will cosider the process which is defied i Theorem 3.9, which offers a algorithmic acquisitio protocol for istaces of the radomly weighted path where each vertex has a uique weight. I order to motivate the use of this algorithm, however, we first state below oe of the mai results of [2], amely that if a total of t 5 chips are distributed o P usig a uiform radom process, the probability that two or more vertices have the same iitial weight goes to zero as t. If t 5 this probability is tedig to 1 as.) This result exhibits a sceario uder which the coditios of Theorem 3.9 are satisfied. Lemma 3.8. Let a total of t chips, where t 5, be distributed o P usig a uiform radom process ad let X = X) deote the umber of pairs of vertices that receive the same umber of chips. The, lim Pr[X 1] = 0. I what follows, we assume that the path is weighted such that vertex weights are distict. Accordig to Lemma 3.8, this may be realized with high probability by placig, e.g., t = t 5 tokes radomly o P but we do ot use this fact explicitly or implicitly. Theorem 3.9. Let P be weighted by usig a radom permutatio of distict itegers w 1,...,w. If each acquisitio move cosists of the vertex with the highest weight receivig the weight of its immediate eighbors, the the expected acquisitio umber A td satisfies E[A td P )] lim = e2 1 2e 2. Proof. Let the chips be radomly distributed o P. Let w k i), for 1 i, deote the weight of vertex i after the kth step of the algorithm. Iitially, each vertex is equally likely to have the largest weight. Without loss of geerality, suppose w 0 j) > w 0 i) for all i < j ad i > j. If j has two eighbors, the at the ed of the first step w 1 j) = w 0 j 1)+w 0 j)+w 0 j+1). If j has oe eighbor, the w 1 j) = w 0 j)+w 0 j 1) or w 1 j) = w 0 j)+w 0 j+1). After the first step, vertex j caot acquire the weight of ay other vertices i the path because all its eighbors have zero weight. Therefore, calculatig the acquisitio umber of P reduces to calculatig the acquisitio umbers of the resultig smaller path or paths, as show i Figure 1, sice a td P ) = a td P )+a td P )+1. We therefore obtai the idetity E[A td P )] = ) E[A td P i )]+E[A td P 2 i )] = E[A td P i )]. i=0 i=0 Uautheticated Dowload Date 11/9/18 8:39 PM

14 932 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag P P P 1 2 j-2 j-1 j j+1j j-2 j-1 j j+1j+2-1 Figure1. AillustratioofP beforeleft)adafterright)thefirststepofthealgorithm. If vertex j has the highest weight, the calculatig the acquisitio umber of P reduces to calculatig the acquisitio umbers of P ad P. Let Gx) = i 0 E[A tdp i )]x i be the geeratig fuctio for E[A td P )]. I order to obtai a closed form expressio for Gx), we first ote that 2 E[A td P )] = +2 E[A td P i )], ad, multiplyig through by x ad summig over = 0,1,... a chage i the order of summatio is eeded for the third term) we see that xg x) = i=0 x x2 +2Gx) 1 x) 2 1 x. Solvig the resultig differetial equatio, we fid Gx) = 1+2Ce 2x 2 4x+2x 2 = Ce 2x x 1) x 1) 2, where C is a ukow costat. We would like to fid a explicit formula for E[A td P )]. To do so, we use kow geeratig series represetatios to ote that Ce 2x ) x 1) 2 = C 2) i ) x i j +1)x j, i! i=0 i=0 ad equate the x coefficiets to obtai E[A td P )] = C +1 i) 2)i i! i= We the use the fact that A td P 2 ) = 1 to determie that C = 1/2. Therefore, E[A td P )] = 1 +1 i) 2)i i! 2 ad This fiishes the proof of the result. i=0 E[A td P )] lim = e2 1 2e 2. Uautheticated Dowload Date 11/9/18 8:39 PM

15 The Total Acquisitio Number of the Radomly Weighted Path 933 Assumig that each vertex has uique weight, we ca similarly apply this algorithm to cycles, stars, wheels, ad complete bipartite graphs. After a sigle iteratio of the algorithm, the cycle graph C reduces to P 3 ad therefore E[A td C )] = 1+E[A td P 3 )]. For the star graph S, there are two possible cases. Let v c deote the ceter vertex of S. If v c has the highest weight, the durig the first iteratio it will acquire the weight of all other vertices ad the acquisitio umber will trivially be 1. If v c is ot the vertex with highest weight, the durig the first iteratio oe of the leaves will acquire the weight o v c, leavig 1 ucoected vertices ad givig the graph the acquisitio umber 1. Thus, we have E[A td S )] = 1)2 + 1 = 2 2. For the wheel graph W, there are agai two cases. Agai, let v c deote the ceter vertex of W. If v c has the highest weight, the it will similarly acquire the weight of all other vertices, producig a acquisitio umber of 1. If v c is ot the vertex with highest weight, the after the first iteratio W will reduce to C. It follows that we have E[A td W )] = E[A tdp 3 )]). Fially, let us cosider the complete bipartite graph K,m = U,V,E). Let v deote the vertex with highest weight. If v U, the durig the first iteratio of the algorithm v will acquire the weight of all vertices i V, leavig vertices i U totally discoected ad producig a acquisitio umber of. If v V, the durig the first iteratio v will similarly acquire the weight of all vertices i U, producig a acquisitio umber of m. Thus, we have E[A td K,m )] = 2 +m 2 +m. Similarly, the multipartite graph K 1,..., k has E[A td K 1,..., k )] = k i=1 2 i k i=1. i 4. Ope Questios Fially, we ca coclude by offerig some ope questios that arose durig our ivestigatio of radomly-weighted graphs. Uautheticated Dowload Date 11/9/18 8:39 PM

16 934 A. Godbole, E. Kelley, E. Kurtz, P. Pra lat ad Y. Zhag Questio 4.1. If t chips are radomly distributed, what are the expected total acquisitio umbers of K,, K,m, K 1,m, L, ad G m,? Questio 4.2 Diameter two graph). For ay graph G with diameter two, it is kow that a t G) 32lll [4] ad cojectured that a t G) c where perhaps c = 2). If we istead radomly distribute chips o a graph with diameter two, ca a t G) be similarly bouded? What if we radomly distribute t chips? Questio 4.3. For the caoical acquisitio problem, where each vertex of G begis with weight 1, there exists a acquisitio game variat where two players, Max ad Mi, make alterate acquisitio moves i a attempt to, respectively, maximize ad miimize the size of the residual set. The game acquisitio umber, a g G), is defied as the size of the residual set uder optimal play. Similar ivestigatios could be doe o the game acquisitio umber of radomly weighted graphs. Refereces [1] D. Bal, P. Beett, A. Dudek ad P. Pra lat, The total acquisitio umber of radom graphs 2016), arxiv: [2] E. Czabarka, M. Marsili ad L.A. Székely, Threshold fuctios for distict parts: revisitig Erdős-Leher, i: Iformatio Theory, Combiatorics, ad Search Theory i Memory of Rudolph Ahlswede), H. Aydiia, F. Cicalese ad C. Deppe Eds.), Spriger-Verlag, Lecture Notes i Comput. Sci ) doi: / [3] D.E. Lampert ad P.J. Slater, The acquisitio umber of a graph, Cogr. Numer ) [4] T. LeSaulier, N. Price, P. Weger, D. West ad P. Worah, Total acquisitio i graphs, SIAM J. Discrete Math ) doi: / [5] P. Weger, Fractioal acquisitio i graphs, Discrete Appl. Math ) doi: /j.dam [6] D. West, N. Price ad P. Weger, Uit acquisitio i graphs, preprit. [7] P. Pra lat, C++ program ad results ca be foud at pralat/. Received 14 August 2015 Revised 15 July 2016 Accepted 15 July 2016 Uautheticated Dowload Date 11/9/18 8:39 PM