Random Walks on Rooted Trees

Transcription

1 Random Walks on Rooted rees Felx Lazebnk, Wenbo L and Ryan Martn Department of Mathematcal Scences, Unversty of Delaware, Newark, DE Abstract For arbtrary postve ntegers h and m, we consder the famly of all rooted trees of heght h havng exactly m vertces at dstance h from the root. We refer to such trees as h, m-trees. For a tree from ths famly, we consder a smple random walk on whch starts at the root and termnates when t vsts one of the m vertces at dstance h from the root. Consder the problem of fndng a tree n the famly on whch the expected tme of a random walk s mnmal an extremal tree. In ths paper we present some propertes of extremal trees for arbtrary h and m, completely descrbe extremal 2, m- and 3, m-trees, descrbe a lower bound for the expected tme of any 4, m-tree, and refute the conjecture that the complete bnary tree s extremal n the class of all h, 2 h -trees wth the degree of the root at least 2. Introducton All mssng defntons related to probablty and Markov chans can be found n [2], and those related to graph theory can be found n [8]. Let the greatest dstance of any vertex from the root, the heght of, be h. A vertex at a dstance k, 0 k h, from the root s sad to be at ter k. An h, m-tree s a rooted tree of heght h havng exactly m vertces at ter h. Let Ω be a set of all walks W = v 0 v 1... v n n such that v 0 s the root, v n s at ter h, and v s at ter h mples = n. Let d u denote the degree of a vertex u n,.e., the number of edges of ncdent to u. We defne the probablty Pr{W } of a walk W Ω to be equal to the product of [d v] 1 for all vertces of a walk W, excludng the last. Let X be a random varable representng the length of a walk n Ω, and let E[X] be the expected value of X,.e., E[X] = W Ω lw Pr{W }, and we wll refer to t as the expected tme of a random walk. In ths paper, we are concerned wth the followng: Problem Gven two postve ntegers h and m, we wsh to fnd an h, m-tree on whch the expected tme of a random walk s mnmal. We refer to such a tree as beng extremal. he above was consdered by Lee [3] for the class of sphercally symmetrc trees, that s, trees n whch the degrees of all vertces at the same ter are the same. hs reduces the analyss of the problem to that of a Markov process on a path of length h. Let us descrbe the case n whch a complete soluton for sphercally symmetrc h, m-trees was obtaned. Suppose there exsts a Partally supported by the Undergraduate Research Program of the Unversty of Delaware. 1

2 postve nteger k such that m = k 1 2 k h 3. hen the unque extremal tree Lee tree has the degree of the root and of vertces at ter h equal to 1, the degree of the vertces at ters 1 and h 1 equal to k, and all other vertces of degree k + 1. If such k does not exst, the expected tme for the path stll can be calculated for all m and ths expected tme provdes a lower bound for the expected tme for all sphercally symmetrc trees. It was shown that ths bound s acheved f and only f m = k 1 2 k h 3 and k s an nteger. Problems related to random walks on varous famles of graphs were consdered by many authors. Here we menton just [1] and references theren. In ths paper we present some propertes of extremal trees for the class of all h, m-trees and arbtrary postve ntegers h and m Secton 1 as well as descrptons of the extremal tree for h = 2, 3 and all nteger values of m Secton 2. hen we dscuss the relatonshp between the expected tme of the Lee tree and the expected tme of any rooted tree of heght 4 Secton 2. In Secton 3 we provde a counterexample to a conjecture whch motvated our research see [3] and references theren. A partcular case of ths conjecture, whch we dsprove, can be stated as follows: Among all h, 2 h -trees wth the degree of the root at least two, the complete bnary tree s the unque extremal tree. We close by statng a new conjecture. 1 General Results We begn by exhbtng a smple operaton on a tree that strctly decreases the expected tme. Lemma 1.1 Let be an h, m-tree. Let the tree be constructed by addng a vertex v at ter h and jonng any vertex w at ter h 1 n to v. hen, s an h, m + 1-tree and E[X ] < E[X]. Proof. Let d = d w, then d + 1 = d w. Our proof s based on an dea known as couplng see, e.g., [5]. We ntroduce a tree that couples and. s created from by prunng v. he probabltes of movng from one vertex to another, however, wll not change. In other words, n, f u s a neghbor of w dstnct from v, then the probablty pw, u of movng from vertex w 1 to vertex u n one step s d+1. We would lke to pont out that a tree n the context of the couplng method s not just vewed as a graph but as a graph together wth a probablty dstrbuton on ts walks. Let W be a walk n that does not vst v. It must have the exact same probablty n because these transton probabltes have not changed. So, what about walks that do vst v n? Such walks vst v only once, when they reach ter h. Let W be such a walk n. In order to have W correspond to a walk n, we must dstrbute the remanng 1 d+1 among the transton probabltes of extng vertex w n. We can dstrbute them n any way we choose to create a new tree. As long as we ncrease the probablty pw, u where u s at ter h, then the expected tme n tree s less than the expected tme n tree. he valdty of ths pont can be seen by traversng W n and ts counterparts n smultaneously. he walk W n must splt nto walks n nstead of vstng v. Accordng to the dstrbuton of the transton probabltes n, some walks wll conclude wth one more step traversng some wu where u s at ter h nstead of traversng wv. hese walks wll not change the expected tme. But, some walks wll return to the parent of w. hs means that these walks wll take at least 3 steps to complete after vstng w. 2

3 Wthout loss of generalty, we choose to create 1 by equally dstrbutng d+1 among the transton probabltes of extng w. So, f u s one of the d neghbors of w n, then pw, u = 1 d dd+1 = 1 d. herefore, f X s a random varable representng the length of a smple random walk n, E [X ] < E[X]. It s clear, though, that =. hen, E[X ] < E[X]. By repeated applcaton of ths lemma, we obtan the followng: Corollary 1.2 For a fxed h, let m be an extremal h, m -tree and X m the length of a walk n Ω m for = 1, 2. If m 1 > m 2, then E [X m1 ] < E [X m2 ]. We consder the class of all h, m-trees. A leaf n any graph s a vertex of degree 1, and an nternal leaf n a rooted tree s a leaf at a ter other than 0 or h. he followng lemma states that any random walk whch enters an nternal leaf s wastng tme. Lemma 1.3 An extremal tree has no nternal leaves. Proof. he orgnal proof of ths lemma see [6] dd not make use of the couplng method. As was suggested by Davd Wlson [7] we can agan utlze couplng. Let be a tree wth an nternal leaf v wth neghbor u and = \ {v}. Let X be the random varable representng the length of a random walk on and Y be the random varable representng the length of a random walk on. We wsh to couple X and Y approprately to show that E[X] > E[Y ]. Let W = w 0 w 1 w 2 w L be a walk n Ω and let W be a walk n Ω. We say W s coupled to W f the followng procedure results n W : Whenever w = v and w +1 = u then we delete both w and w +1. It s trval to observe that every walk n Ω s coupled to exactly one walk n Ω. So we create a jont probablty densty functon { Pr X,Y {W, W Pr } = X {W }, f W s coupled to W 0, otherwse. Note that W Ω W Ω Pr X,Y {W, W } = Pr Y {W } Pr X,Y {W, W 1 } = Pr Y {W } 1 deg u 1 ηw u = Pr X {W }, where η W u denotes the number of tmes that W vsts u. So, we use ths dea to compare the expectatons of X and Y E[X] = Pr X {W }lw = W Ω W Ω W Ω W Ω W Ω 3 Pr X,Y {W, W }lw Pr X,Y {W, W }lw

4 he nequalty occurs because f W s not coupled to W, Pr X,Y {W, W } = 0 and f W s coupled to W, lw lw. So we have E[X] W Ω Pr Y {W }lw = E[Y ]. And, n fact ths nequalty s strct because for each W there exsts at least one W wth postve probablty whch s coupled to W wth lw > lw. As a result, we shall from ths pont forward only consder trees wthout nternal leaves. Our goal now s to prove heorem 1.4. heorem 1.4 he root of an extremal tree has degree 1. Proof. Suppose there s an extremal tree where the root has degree d. can be decomposed nto d trees, 1 d, each of heght h and each wth a root of degree 1. Let X be the random varable measurng the length of a walk n. Wthout loss of generalty, these ndces can be chosen such that E[X 1 ] E[X 2 ]... E[X d ]. We defne p to be the probablty that a walk n Ω returns to the root. In the tree, let E 0 h [X ] denote the expected length of a walk gven that the walk does not return to the root, and E [X ] denote the expected length of the frst return to the root gven that the walk does ndeed return to the root. hen, by the law of total probablty, we have E[X ] = 1 p E 0 h [X ] + p E [X ] + E[X ] We now turn our attenton to the entre tree. By the law of total probablty accordng to the tree n whch the walk begns, we obtan 1 E[X] = 1 d d [ 1 p =1 E 0 h [X ] + p ] E [X ] + p E[X] 2 Solvng the above equaton for E[X] and usng 1, we get E[X] = d =1 [ 1 ] p E 0 h [X ] + p E [X ] d =1 1 p = d =1 [ 1 p E[X ] ] d =1 1 p, 3 whch s a weghted average of the E[X ] s. hs mples that the expected tme on one of the subtrees s at most that of the orgnal tree f the root of the orgnal tree s not a leaf. Addng vertces n the manner descrbed by Corollary 1 strctly lessens the tme. 4

5 2 Results for h = 2, 3, 4 In ths secton we collect our results on the problem n cases when h = 2, 3, 4. heorem 2.1 he extremal tree for h = 2 and any m s a star wth m + 2 vertces, wth the center of the star at ter 1. he expected tme for ths tree s m. Let h = 3 and m = k 2 + α, where 0 α 2k. Let t be the number of vertces at ter 2. If 0 α k 1, E[X] acheves ts mnmum only when t = k. If k + 1 α 2k, E[X] acheves ts mnmum only when t = k + 1. If α = k, E[X] acheves ts mnmum f and only f t = k or t = k + 1. he mnmum s E[X] = 0 α k k + 2 k α1+ k 2 k α 2k. k 3 +3k 2 +k+α k + 2 k α1+ 2 k k 3 +2k 2 +α Proof. he proof of s a much shorter verson than that of whch we present below. By heorem 1.4, the extremal tree has one vertex at ter 1. Its degree s denoted to be t + 1. If v s a vertex at ter 2, 1 t, then let the degree of v be d. herefore, t =1 d = m + t. Let E 0 be the expected tme of the walk ntated from the root of the tree. Let E 1 be the expected tme of the walk ntated from the vertex at ter 1. Let E 2, be the expected tme of the walk ntated from v at ter 2, 1 t. We have the followng system of equatons: E 0 = 1 + E 1 E 1 = t+1 E t+1 E 2, = d E 1. t E 2, =1 Solvng these equatons gves us the followng expresson for E 0 = E[X]: t + 1 E[X] = t t 1. =1 d 4 For a fxed t, the sum t =1 d s fxed. Snce each d s a postve nteger, by Jensen s nequalty, t=1 1 d s a mnmum f and only f max {d } mn {d } 1 and that ths set of values of d s s 1 t 1 t unque. If we let s = m t, then we obtan that an extremal graph contans ts + t m vertces of degree s + 1 and m ts vertces of degree s + 2 at ter 2. hus the mnmum E[X] s of the form E[X] = t+1s+1s+2 m+tss+1, whch for a fxed m s a functon of t only. It can be shown that the mnmum for E[X] occurs when t = k f 0 α k 1, when t = k + 1 f k + 1 α 2k, and when t = k or t = k + 1 f α = k. he proof s elementary. See Chapter 3 n [6] for all detals. In the case of h = 4 and arbtrary m we fal to descrbe extremal 4, m-trees, but f m = k 1 2 k, where k s a postve nteger, t can be shown that the Lee tree defned n Secton 1 s the only extremal tree. Its expected tme s 4 + 6k 4. It can also be shown that the functon of k 1 2 k n the expresson of the expected tme on Lee tree serves as a lower bound for the expected tme 5

6 on an arbtrary 4, k 1 2 k-tree. hs s acheved usng elementary methods such as Jensen s nequalty and Lagrange multplers. he detals can be found n [6]. 3 Counterexample and Conjecture he counterexample to the conjecture descrbed at the end of Introducton was constructed n the followng three steps. 1. Let H 1 and H 2 be two dentcal h, 2 h 1 -trees wth the degree of the root beng 1. Let H be a tree obtaned from H 1 and H 2 by dentfcaton of ther roots. hen H s an h, 2 h -tree wth the root of degree 2. If p, = 1, 2, denotes the probablty that a walk n ΩH returns to the root, then p 1 = p 2, E[H 1 ] = E[H 2 ], and 2 mples E[H] = 1 p 1 E[H 1 ] + 1 p 2 E[H 2 ] 1 p p 2 = E[H 1 ]. 2. Consder an 8, 2 7 -tree B 0 n whch the root has degree 1 and each vertex at ters 1 to 7 has exactly two chldren. By dentfyng roots of two copes of such tree, we obtan the complete bnary 8, 2 8 -tree B. From step 1, E[B] = E[B 0 ]. 3. Let C 0 be an 8, 2 7 -tree gven by Fgure 1, and C be the tree obtaned from two copes of C 0 by dentfyng ther roots. From step 1, E[C] = E[C 0 ]. By drect computaton, we have E[B 0 ] = and E[C 0 ] = , whch mples E[C] < E[B]. Hence C s a counterexample. he followng new conjecture was motvated by the dscusson n the last secton, numercal evdence and [3]. Conjecture 3.1 If k s a postve real root of the polynomal fx = x 1 2 x h 3 m, then E[X] k + 1 k 1 h + 4 2k k 1 2, and ths bound s acheved f and only f k s an nteger. 6

7 Fgure 1: he tree C 0 References [1] Doyle, Peter and Snell, J. Laure. Random Walks and Electrcal Networks. he Mathematcal Assocaton of Amerca: Washngton, DC, [2] Isaacson, Dean L. and Madsen, Rchard W. Markov Chans, heory and Applcatons. John Wley & Sons, Inc.: New York, [3] Lee, Susan. Optmal Drft on the Unt Interval. Dssertaton, Cornell Unversty, [4] Lee, Susan. Optmal Drft on [0,1]. ransactons of the AMS, 346: [5] Lndvall, orgny. Lectures on the Couplng Method. John Wley & Sons, Inc.: New York, [6] Martn, Ryan. Mnmum Expected me of Random Walks on Rooted rees. Undergraduate Honors hess, Unversty of Delaware, [7] Wlson, Davd. Prvate Communcaton. [8] Wlson, Robn. Introducton to Graph heory. John Wley & Sons, Inc.: New York,