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## Broadcasting in UDG Radio Networks with Unknown Topology

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**Broadcasting in UDG Radio Networks with Unknown Topology**WeizmannLiverpoolWeizmannQuébecWeizmannLiverpool Yuval Emek, Leszek Gąsieniec, Erez Kantor, Andrzej Pelc,David Peleg, Chang Su,**stations = points in**UDG radio networks • transmitting range = 1 • unit disk graph – UDG • (nodes, edges, paths, …) • distributed synchronous model • in every round: transmit or receive • message heard iff exactly one neighbor transmits • else: silence or collision (same effect)**distributed synchronous model**(1) no transmission (silence) (2) single transmission (3) multiple transmission v can receive the message from u v u w vcannotreceive the message**distributed synchronous model**(1) no transmission (silence) (3) multiple transmission • collisions cannot be distinguished from silence v u w vcannotreceive the message**Unknown topology (ad hoc)**• a unique coordinate system • each node knows its own coordinates • does not know the: • coordinates of any other node • the number of nodes • the diameter**Unknown topology (ad hoc)**• known granularity g = • inverse of minimum Euclidean distance • , for every pair of nodes • typically:d is much smaller then 1 and g is much larger than 1**Broadcasting**• a distinguished source node • source’s message should be heard by all nodes • remote nodes – use graph’s paths • connected graphs**Broadcasting**• two models are considered: • conditional wake up: - nodes are initially idle • wakes up upon hearing a message • spontaneous wake up: • – all nodes are awake from the beginning • execution time = • #rounds until all nodes hear the source’s message**Deterministic model**• decisions of a node on round tdepends only on: • own coordinates • t itself • messages heard so far**= diameter of the UDG network (in hops)**• = granularity: inverse of min Euclidean distance • s • v This work • execution time depends on two parameters: • not Euclidean diameter**This work**• upper bound • lower bound • conditionalwake up • spontaneouswake up**Previous results**• roughly divided into 2 subareas: • centralized: complete knowledge, designing fast schedulers • distributed: local knowledge, designing fast protocols (this work)**from**• to • Alon, Bar-Noy, Linial, Peleg ’91: constant D Centralized model • Chlamtac, Kutten ’85: formulating the model of radio networks • Chlamtac, Weinstein ‘91 • Gaber, Mansour ‘95 • Elkin, Kortsarz ‘05 • Gasieniec, Peleg, Xin ‘05 • Kowalski, Pelc (to appear)**Bar-Yehuda, Goldreich, Itai ’92:**• Kushilevitz, Mansour ’98: • Czumaj, Rytter ’03: Distributed model • unknown topology, no labels, randomized: • first to study distributed broadcasting (also deterministic) • (tight!)**Chlebus, Gasieniec, Gibbons, Pelc, Rytter ’02:**• Kowalski, Pelc ’05: Distributed model • unknown topology, knowing own labels, spontaneous wake up, deterministic: • Kowalski, Pelc ’05: unknown topology, knowing own labels, conditional wake up, deterministic**Spontaneous wake up – lower bound**• Theorem. deterministic broadcasting algorithm A, and choice of parameters D,g, UDG network N of diameter D and granularity g s.t. A requires • rounds to broadcast in N under the spontaneous wake up model.**clusters**• k consists of • cells Chain networks • each cell may be occupied with a node or empty • each cluster contain at least one occupied cell • source cell (always occupied) in source cluster 0**clusters**• form a clique Chain networks • there is no edge between any and any for |k-i|>1**the message go from directly to**Chain networks • from to when only one node from transmit the message**the message go from directly to**Chain networks • from to when only one node from transmit the message**Chain networks**• if there exists a node in that heard the message • then all the nodes of must being heard the source message**The broadcasting algorithm A**• knows the coordinates of the cells • does not know which cells are occupied and which are empty (except the source) • knows that there is at least one occupied cell in every cluster • a typical instruction: “transmit if occupied” • St = cells scheduled to transmit on round t by A**decisions are made separately for every k and online**based on The adversary • goal: slow down the broadcasting algorithm • decides for every cell whether occupied or empty**= number of occupied cells in**Game between the algorithm and the adversary • u • St schedule to transmit • adversary decide: • (1) single transmission • (2) silence / collision • algorithm can learn? • what u can learn?**u**Game between the algorithm and the adversary • adversary: • (1) reveal these cells (occupied/empty) • (2) report silence / collision • must be consists with previous reports**u**Game between the algorithm and the adversary • v • algorithm knows v • St schedule to transmit by the algorithm • algorithm can learn whether: • (1) • (u hear v) • (2) • (u did not hear v)**u**Game between the algorithm and the adversary • adversary: • (1) reveal these cells • (2) report silence / collision • (2) report that collision occur • must be consists with previous reports**Lower bound**• ti = first round on which the nodes of ireceive the message • , number of round for delivering the message from ito i+1**execution time:**Lower bound • adversary guarantees : • for ti<cg2 • , for i<cg2/log (g)**execution time:**• rounds • rounds • rounds • rounds Conditional wake up – lower bound • chain network • N1 • N2 • N3 • ND/2 • diameter 2**Conditional wake up – lower bound**• Theorem. deterministic broadcasting algorithm A, and g, UDG network N of diameter 2 and granularity g s.t. A requires • rounds to broadcast in N under the conditional wake up model.**blocks**• in each block: • 1> • auxiliary cells • opposite each block: • a target cell • 1> The network N • exactly 1 target cell is occupied • g auxiliary cells • target**The network N**• target cell is outside of the • transmitting range of any other blocks • there is at least oneoccupied cell in the block that opposite to the occupied target cell • the network is connected • auxiliary • target**Adversary**• can no longer guarantee that no messages are being heard • distinguish silence from collision (stronger model)**Game between the algorithm and the adversary**• st • Adversary: • (1) reveal some cells • (2) report: collision occur • (3) report: silence / collision**execution continues for**• rounds Adversarial policy • dead blocks– all cells are revealed, target cell is empty • on every round we “kill” at most 1 block and reveal at most 1 cell in each “live” block**The concatenate network**• the auxiliary cells of Niis outside the transmitting range of the next source node si+1 • the target cell of Niis inside of the transmitting range of the next source node si+1**The concatenate network**• the message must be delivered via target nodes and auxiliary nodes**The concatenate network**• execution time:**Summary**• upper bound • lower bound • conditionalwake up • spontaneouswake up**END**• Thank You!!!