5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica CTR: motivations the CTR: 2/12  Why studying the CTR problem is important? – In many situations, dynamically adjusting the node transmitting range is not feasible – for instance, because the wireless transceiver does not allow the transmit power to be adjusted  Characterizing the CTR helps the network designer to answer fundamental questions, such as: – Given n, which is the minimum value of the transmitting range that ensures connectivity? – Given a transmitter technology, how many nodes must be distributed in order to obtain a connected network? 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica The longest MST edge the CTR: 3/12  Solving the CTR problem is easy if node positions are know: the CTR is the longest edge of the Euclidean MST built on the nodes CTR 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica CTR: probabilistic approaches the CTR: 4/12  In many realistic scenarios, node positions are not known in advance (for instance, sensors spread from a moving vehicle)  Probabilistic approaches: nodes are distributed in R according to some distribution; which is the value of r which guarantees connectivity with high probability (w.h.p.)? Remark: In this context, w.h.p. means that the probability of connectivity converges to 1 as n grows 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica CTR: probabilistic tools the CTR: 5/12  The probabilistic characterizations of the CTR presented in the literature are based on one of the following applied probability theories: – Continuum percolation [MeesterRoy96] – Occupancy theory [Kolchin et al.78] – Geometric random graphs [Diaz et al.00]  The theory that is most suited to analyze the CTR problem is the theory of GRG 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica Geometric Random Graphs the CTR: 6/12  GRG: a set of n points are distributed in a d-dimensional region R according to some distribution, and some property of the resulting node placement is investigated Example: – length of the longest nearest neighbor link – length of the longest MST edge (CTR) – total cost of the MST  These results have been used in [PanchapakesanManjunath01] to prove the following result: – If nodes are distributed uniformly at random in [0,1] 2 , the CTR for connectivity w.h.p. is , for some constant c > 0 n n cr log = 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica Other probabilistic results the CTR: 7/12  The theory of GRG has been used also in [Bettstetter02a] to characterize the CTR for k-connectivity  The CTR for connectivity has been characterized also for the case of points uniformly distributed on a disk [GuptaKumar98]. In this case, we have where c(n) is an arbitrary function such that c(n) → ∞ for n → ∞ n ncn r ! )(log + = 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica The CTR for sparse networks the CTR: 8/12  The results presented so far refer to the case where the deployment region R is fixed, and n grows to infinity they can be applied only to dense networks  However, similar results have been proved also for the case of sparse networks. In this case, R=[0,l ] 2 , and connectivity is investigated for l → ∞  In case of sparse networks, the CTR for connectivity is of the form for some constant c > 0 [SantiBlough03] n l clr log = 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica CTR: more practical results the CTR: 9/12  Besides analytical characterization, the CTR for connectivity has been estimated through simulation 0,07651000 0,0894750 0,1082500 0,1533250 0,2353100 0,272075 0,325850 0,441525 0,656610 CTRn Table 1. (from [SantiBlough03]) Values of the CTR when n nodes are distributed uniformly in R = [0,1] 2 . Here, the CTR is defined as the minimum transmitting range that generates at least 99% of connected graphs 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica The COMPOW protocol the CTR: 10/12  In [Narayanaswamy et al.02], the authors introduce COMPOW, a protocol that attempts to determine the CTR for connectivity in a distributed way  Nodes maintain a routing table for each power level, and set as the common transmit power the minimum level such that the corresponding routing table contains all the nodes in the network  Setting the power to this minimum level achieves the three goals of: – maximizing network capacity, – reducing contention to access the wireless link – extending network lifetime with respect to the case of no TC  Drawbacks of the COMPOW protocol: – Considerable message overhead – Requires global knowledge (routing table) 5 th ACM MobiHoc – Tokyo, May 24, 2004 Istituto di Informatica e Telem atica The giant component the CTR: 11/12  Suppose all the nodes set their transmit power to 0, and start increasing their power simultaneously  W.h.p., connectivity occurs when the last isolated node disappears from the graph  In other words, a giant component is formed soon, and the remaining increase in the transmit power is needed to connect few isolated nodes  Thus, a lot of power is used to connect relatively few nodes  Giant component phenomenon supported by experimental data [SantiBlough03]: – reducing the transmitting range of about 40% with respect to CTR yields a graph in which 90% of the nodes are connected . ACM MobiHoc – Tokyo, May 24 , 20 04 Istituto di Informatica e Telem atica CTR: motivations the CTR: 2/ 12  Why studying the CTR problem is important? – In many situations, dynamically adjusting the. MobiHoc – Tokyo, May 24 , 20 04 Istituto di Informatica e Telem atica CTR: probabilistic approaches the CTR: 4/ 12  In many realistic scenarios, node positions are not known in advance (for instance,. simulation 0,07651000 0,0894750 0,10 825 00 0,153 325 0 0 ,23 53100 0 ,27 2075 0, 325 850 0,441 525 0,656610 CTRn Table 1. (from [SantiBlough03]) Values of the CTR when n nodes are distributed uniformly in R = [0,1] 2 . Here, the