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