Symmetric Range Assignment with Disjoint MST Constraints

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1 Symmetric Range Assignment with Disjoint MST Constraints Eric Schmtz Drexel University Philadelphia, Pa ,USA ABSTRACT If V is a set of n points in the nit sqare [0, 1] 2, and if R : V R + is an assignment of positive real nmbers (radii) to to those points, define a graph G(R) as follows: {v, w} is an ndirected edge if and only if the Eclidean distance d(v, w) is less than or eqal to min(r(v),r(w)). Given α 1 and k Z +, let Rk be the range assignment that minimizes the fnction J(R) = P R(v) α, sbject to the v V constraint that G(R) has at least k edge-disjoint spanning trees. For n random points in [0, 1] 2, the expected vale of the optimm, E(J(R k)), is asymptotically Θ(n 1 α 2 ). This is proved by analyzing a crde approximation algorithm that finds a range assignment R a k sch that the ratio J(Ra k ) J(R k ) is bonded. Categories and Sbject Descriptors F.2.m [Analysis of Algorithms and Problem Complexity]: Miscellaneos General Terms Algorithms, Performance, Reliability, Theory Keywords Range assignment, probabilistic analysis, approximation algorithm, spanning tree 1. INTRODUCTION If V is a finite set of points R 2, and R : V R + is an assignment of positive real nmbers (radii) to the points, define an ndirected graph G(R) as follows. The vertex set is V, and {, v} is an ndirected edge if and only if the Eclidean distance d(, v) is less than or eqal to min(r(), R(v)). This is one of the standard models for connections between nodes of a wireless network: R(v) represents the distance that node v can effectively transmit, and the presence of an Permission to make digital or hard copies of all or part of this work for personal or classroom se is granted withot fee provided that copies are not made or distribted for profit or commercial advantage and that copies bear this notice and the fll citation on the first page. To copy otherwise, to repblish, to post on servers or to redistribte to lists, reqires prior specific permission and/or a fee. DialM-POMC 08, Agst 22, 2008, Toronto, Ontario, Canada. Copyright 2008 ACM /08/08...$5.00. edge between two nodes indicates the availability of direct two-way commnication between those nodes. For a sitable vale of α, the fnction J(R) = P R(v) α has been v V sed as a measre of the energy spent by the nodes when transmitting. Hence it is a natral choice for an objective fnction to minimize. Some kind of connectivity constraint is needed to ensre that all pairs of nodes can commnicate (either directly via an edge, or indirectly by forwarding messages). In this paper, the constraint imposed is the existence of k edge-disjoint spanning trees. For positive integers k > 1, and real nmbers α 1, let Π k,α be the following optimization problem. Π k,α : Instance: V, a finite set of points in R 2. Objective: Choose R : V R + so as to minimize J(R) = v V R(v) α. Constraint: G(R) mst have at least k edge-disjoint spanning trees. Given an instance of Π k,α, let Rk denote an optimal choice of R. Ths J(Rk) is the minimm feasible energy. The main reslt in this paper is an asymptotic estimate for the expected vale of J(Rk). For the stochastic model in which a random instance is selected by choosing n points independently and niform randomly from [0,1] 2, we prove that E(J(Rk)) = Θ(n 1 α 2 ). More precisely,sppose that βα is the minimm spanning tree constant [3],that a < 2 βα and 5 that b > (2 + 2 α+2 kα+2 )β α. Then there is a positive constant N k,α sch that, for all n > N k,α, the expected vale of J(Rk) satisfies the ineqalities an 1 α 2 E(J(R k )) bn 1 α 2. One can regard Π k,α as the restriction of a problem Π, in which both the the positive integer k and the exponent α are given along with the points V as inpt. The problem Π is NP-hard, since it can be restricted to problems that are known to be NP-hard. In particlar, for k = 1, the existence of k spanning trees is eqivalent to connectivity. The problem of minimizing power, sbject to the graph being connected, is known to be NP-hard [7],[8].I believe Π k,α is NP-hard for general k and α, bt have not yet completed a proof. Several athors have considered range assignment problems with varios other connectivity constraints. Vertex connectivity is pariclarly important, and several athors

2 have sed this as a constraint. See for example, Lloyd et. al., Hajiaghayi et al., [10]Kortsarz, Mirrokni,Ntov and Tsanko, [12] and the references therein, and Calinesc at.al [2],[5].The work of Blogh et.al.[4]is closely related to ors becase they estimated the expected vale of the optimm power in the case k = 1. We shold particlarly mention Π k,α = the symmetric range assignment problem with the constraint that G has a k-edge-connected spanning sbgraph. It is well-known [9]that any 2k-edge-connected graph has k disjoint spanning trees. Hence any approximation algorithm for Π 2k,α can, ipso facto, be sed as an approximation algorithm for Π k,α. See Calinesc at.al [2],[5],Lloyd et.al.[13], and the work of Kortsarz et.al.,[12] and references therein for algorithms that se edge-connectivity as a constraint for range assignments. It is important to note that a graph with k disjoint spanning trees need not be 2k-edge connected. Hence the optimm soltion for a Π k,α instance may have less power than the lowest power 2k-edge-connected range assignment for that instance. It follows that the worst case ratio of an algorithm for Π 2k,α does not apply when the algorithm is sed as an approximation algorithm for Π k,α. 2. MINIMUM SPANNING TREES In this section, some elementary ineqalities will be sed to bond the objective fnction J from above and below by less complicated fnctions related to minimm spanning trees. Like many athors, we will se minimm spanning tree(s) and related ineqalities to constrct and analyze an approximation algorithm. Let K c = K c(v ) denote the complete graph on V. For v, w V, define the cost of the edge ε = {v, w} to be W(ε) = d(v, w) α. Obviosly W depends on α, bt to limit notational cltter, we write W instead of W α. Throghot this paper, α is a fixed positive constant that is greater than or eqal to one. It is important to note that, for α > 1, W may not satisfy the triangle ineqality. More precisely, one can have vertices v 1, v 2, v 3 for which W({v 1, v 3}) > W({v 1, v 2}) + W({v 2, v 3}). For any spanning sbgraph H K c, define the cost of H to be W(H) = P W(ε), the sm of the costs of of its edges. ε H If H is a spanning sbgraph of K c, let = (H) be the maximm of the degrees of the vertices in the graph H, and define F(H) = P max d(w, v w v v)α, where the maximm is over all neighbors w that v has in H. Then we have Lemma 1. For any spanning sbgraph H of K c, 2 W(H) F(H) 2W(H). Proof. For any finite list of positive real nmbers, the maximm of the nmbers is trivially bonded from above by the sm of the nmbers. Therefore F(H) d(, v) α = 2W(H). v v For the lower bond, note that, for any finite list of real nmbers, the maximm of the nmbers is bonded below by the average of the nmbers. Ths, for each vertex v, max d(, 1 v v)α d(, v) α 1 d(, v) α. degree(v) v v Smming over all vertices v, we get the lower bond in Lemma 1. Let S k be a minimm cost nion of k disjoint spanning trees for G(R k). To apply Lemma 1, we also need the following lemma. Lemma 2. For all vertices v, Rk(v) = max d(, v), where the maximm is over vertices for which {, v} is an edge of S k. Proof. From the definition of S k (as a minimm cost nion of k disjoint spanning trees for G(R k)), it is clear that S k is a sbgraph of G(R k). This, and the definition of G(R k), imply that, for any edge {, v} of S k, we mst have d(, v) min(r k(), R k(v)). Ths R k(v) d(, v) for any vertex that is adjacent to v in S k,i.e. Rk(v) max d(, v). (1) We mst prove that the ineqality (1) cannot be strict. We sppose Rk(v) > max d(, v), and derive a contradiction. Define ( R R(w) = k(w), w v max d(v, ), w = v, where the maximm is over all vertices that are adjacent to v in S k. Note that R is feasible; G( R) has k disjoint spanning trees since it still contains S k. Bt J( R) < J(R k). This contradicts the minimality of J(R k) in the definition of R k Applying Lemma 1 to Lemma 2, we get Corollary 3. If = (S k) is the maximm vertex degree in S k, then 2 W(S k) J(R k) 2W(S k). Let S Kc be minimm cost nion of k spanning trees for K c. In general, W(S k) W(S Kc )) kw(t 1). It is well known that K c(v ) has a minimm weight spanning tree T 1 whose maximm degree less than or eqal to 5. (See, for example, Lemma 7.2 of [14]It is not tre for arbitrary weights, bt is tre for the weights W({, v}) = d(, v) α in this paper.) Hence Corollary 3 does yield a reasonable lower bond for E(J(R k)) in the special case when k = 1, namely E(J(R 1)) 2 5 E(W(T1)). The distribtion of W(T 1) has been intensively stdied by probability theorists[1],[3],[11],[17].for this paper, we need only the expected vale: there is a constant β α, called the minimm spanning tree constant, sch that E(W(T 1)) = β αn 1 α 2 (1 + o(1)). (2) Unfortnately, we do not have any pper bonds on = (S k) for k > 1, and we can no longer se this method to dedce a lower bond. However it is worth observing that R k R 1, so that we at least have these crde lower bonds: J(Rk)) 2 W(T1)) (3) 5

3 and E(J(R k)) E(J(R 1)) 2 5 βαn1 α 2 (1 + o(1)). (4) 3. APPROIMATION ALGORITHM This section presents a method for selecting a (sboptimal) set of k disjoint spanning trees for K c. Then, in the following section, we estimate the expected vale of the sm of the weights of the set of trees that it selects, and then se this nmber to derive an pper bond the expected optimm vale E(J(R k)). Let T 1 be an MST for K c having maximm degree less than 6. Fix a vertex v 1 V as the root, and define the level of each vertex v as the nmber of edges on the niqe path from v to v 1 in the tree T 1. (Level 0 consists of the root vertex v 1.) Let h be the maximm of the levels, and for i = 0,1, 2,..., h, let L i consists of those vertices at level i, i.e. the vertices v for which the graph distance is i. Let f : V/{v 1} V be the following sccessor fnction: f(v) = the first vertex after v on the path from v to v 1. If f (t) denotes f composed with itself t times, then the domain of f (t) is S L i. If f (t) (v) is a vertex in L j, then v is a i t vertex in L t+j. Ths {v, f (t) (v)} is an edge of K c that joins vertices whose levels differ by exactly t. We se f to define k 1 forests as follows. For t = 2,..., k, let F t consist of all edges of the form {v, f (t) (v)} where v is a vertex in level t or higher. Since the edges of F t join vertices that are separated by exactly t levels,it is clear that the k 1 forests are edgedisjoint and also share no edge with T 1. We need to modify them slightly so that they are in fact spanning trees rather than forests. For each t, let ω t be the nmber of components that the forest F t has. Let τ t,j,1 j ω t be the component trees of F t, and let V t,j be the vertex set of τ t,j. Also let ρ t,j be the root (lowest level vertex) of τ t,j. The idea will be to hook all these roots p with low level vertices in the largest of the τ t,j s. Choose tmax,jmax sch that V tmax,jmax has maximm cardinality, and let I = (t, j) : 1 j ω t and 1 t k and (t, j) (tmax,jmax). Let Z consist of those vertices of V tmax,jmax whose level is less than or eqal to 5 k. Later we will verify that Z > I. We can therefore choose a 1 1 fnction φ from I to Z. Now define T t to be the tree that is obtained from F t by adjoining the edges {ρ t,j, φ(t, j)} with 1 j ω t, and (t, j) (tmax,jmax). The validity of or constrction depended pon or assmption that I < Z. To verify this fact, begin with the following observation. For each l, there are at most 5 l vertices in level l (becase the a maximm degree of T 1 is strictly less than 6). On the other hand, each root ρ t,j is at some level less than t. Hence ω t t 1 P I = k P t=2 ω t < 5k Ths l=0 5 l < 5t 4. and I < 5 k. (5) The size of the largest component is at least as large as the average component size, so V tmax,jmax > 4n. If 5 k n ω tmax n > 25k 4 V tmax,jmax > 5 k. (6) Combining (6) and (5), we get the desired reslt: Z > I. 4. ANALYSIS Let S a k be the nion of k spanning trees that was constrcted in the previos section. Define R a k(v) = max d(, v), where the maximm is over all vertices for which {, v} is an edge of S a k. We know that R a k is feasible becase G(R a k) contains S a k. Therefore, by the definition of R k, we mst have By Lemma 1, It follows from (7) and (8) that J(R k) J(R a k). (7) J(R a k) 2W(S a k) (8) E(J(R k)) 2E(W(S a k)). (9) Ths we mst estimate the expected cost of the spanning trees that are constrcted sing method in the previos section. The choice of spanning trees in the preceding section involved three phases: The first phase, in which the MST T 1 is fond. The second phase, in which the forests F t are chosen. The third phases, in which the forests trees roots are linked to vertices in V tmax,jmax Let C 1,k, C 2,k, C 3,k respectively be the costs of the edges added in these phases. Then by (9), we have E(J(R k)) 2E(C 1,k ) + 2E(C 2,k ) + 2E(C 3,k ). (10) We already cited an estimate for the first term (see eqation 2),so or goal in this section is to prove pper bonds for the second and third terms in (10). First we estimate E(C 2,k ). Sppose c = {w, z} is an edge of F t and e = {x, y} is an edge of the MST T 1. We say c covers e, and write either c e or e c, if the niqe path in T 1 from w to z incldes the edge e. Note that every edge c in F t covers exactly t edges e of T 1. In the special case α = 1 we can se the triangle ineqality and write W(c) W(e) e c = e T 1,c e W(e) t e T 1 W(e). Then, by smming on t and then averaging, we get and then E(C 2,k ) k 2 C 2,k k 2 E(W(T 1)) = k 2 W(T 1), (11) β 1n 1/2 (1 + o(1)). (12) Unfortnately, for α > 1, the triangle ineqality does not hold and we need a crder argment. Sppose c = {w, z} is an edge of F t and that w = x 0, x 1, x 2,..., x t 1, x t = z are

4 the vertices on the path in T 1 from w to z. Since d(, ) does satisfy the triangle ineqality, we have Therefore t 1 d(w,z) d(x i, x i+1) t max d(xi, xi+1). 0 i<t i=0 d(w, z) α t α max d(xi, t 1 0 i<t xi+1)α t α d(x i, x i+1) α. As before, we have W(c) = t α e T 1,c e t α i=0 W(e) e c W(e) t α+1 e T 1 W(e). By smming on t and averaging, we get (for any α 1), E(C 2,k ) kα+2 kα+2 α E(W(T1)) = α + 2 α + 2 βαn1 2 (1 + o(1)). (13) The estimate for C 3,k can be qite crde becase only O(1) edges are added in phase three. Let M = max d(, v), where the maximm is over all edges {, v} of T 1.For each edge {r, v} that is added in phase 3, there is a path from r to v in T 1, consisting of at most 5 k + k edges of T 1. By the triangle ineqality (for d), we have d(r,v) (5 k +k)m, and conseqently Combining (5) with (14), we get W({r,v}) (5 k + k) α M α. (14) C 3,k 5 k (5 k + k) α M α. (15) From the reslts in Penrose [15]we know that, with asymptotic probability one, M < ( log n ) 2 1 n. Hence E(C 3,k ) = O(( log n n ) α 2 ) = o(n 1 α 2 ). (16) Ptting (2),(13), and (16) into (10), we get the main reslt: Theorem 4. If b > (2 + 2 α+2 kα+2 )β α, then for all sfficiently large n, E(J(R k)) < bn 1 α WORST CASE RATIO In this section, a slight modification of the preceding argments will be sed to bond the ratio J(Ra k ) of the energy J(R ) k compted by the approximation algorithm to the optimal energy. The idea is that both the nmerator and denominator are Θ(W(T 1)). For the denominator, we have (3), namely J(R k) J(R 1) 2 5 W(T1). For the nmerator, recall that where J(R a k) C 1,k + C 2,k + C 3,k (17) C 1,k = W(T 1) (18) and C 2,k k 2 W(T 1). (19) Recall from (15) that C 3,k 5 k (5 k + k) α M α. where M is the length of the longest edge of the MST T 1. Obviosly no edge of T 1 can be longer than 2, so a ridiclosly crde bond for M α is Hence M α = M α 1 M 2 α 1 M 2 α 1 W(T 1). C 3,k 5 k (5 k + k) α 2 α 1 W(T 1). (20) Define a constant B (depending on k and α, bt not on n) by B = 1 + `k k (5 k + k) α 2 α 1. 2/5 Ptting (18),(19), and (20) into (17), and then sing (3), we get Theorem 5. 1 J(Ra k ) J(R k ) < B. 6. DISCUSSION The method for selecting spanning trees in section 3 is constrctive, bt it clearly inferior to known algorithms for finding a minimm weight set of k disjoint spanning trees [6][16].I chose this method as an analytical device becase it simplified the estimation of E(Jk) = the average vale of the optimm power Jk. However, I was not able to evalate the limit lim E(J k)n α 2 1. Presmably the limit exists for n all α 1, bt even that fact has not been proved. 7. REFERENCES [1] D. Aldos and J. M. Steele. Asymptotics for Eclidean minimal spanning trees on random points. Probab. Theory Related Fields, 92(2): , [2] E. Althas, G. Calinesc, I. I. Mandoi, S. Prasad, N. Tchervenski, and A. Zelikovsky. Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks. Wirel. Netw., 12(3): , [3] F. Avram and D. Bertsimas. The minimm spanning tree constant in geometrical probability and nder the independent model: a nified approach. Ann. Appl. Probab., 2(1): , [4] D. M. Blogh, M. Leoncini, G. Resta, and P. Santi. On the symmetric range assignment problem in wireless ad hoc networks. In TCS 02: Proceedings of the IFIP 17th World Compter Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Compter Science, pages 71 82, Deventer, The Netherlands, The Netherlands, Klwer, B.V. [5] G. Calinesc and P. jn Wan. Range assignment for biconnectivity and k-edge connectivity in wireless ad hoc networks. Mob. Netw. Appl., 11(2): , [6] J. Clasen and L. A. Hansen. Finding k edge-disjoint spanning trees of minimm total weight in a network: an application of matroid theory. Math. Programming Std., (13):88 101, Combinatorial optimization, II (Proc. Conf., Univ. East Anglia, Norwich, 1979).

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