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Constant Density Spanners for Wireless Ad hoc Networks Kishore Kothapalli (JHU) Melih Onus (ASU) Christian Scheideler (JHU) Andrea Richa (ASU) 1

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Ad hoc Networks ●Network created by wireless stations communicating over a wireless medium ●Two challenges –Lack of centralized infrastructure –Mobility 2

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Ad hoc Networks ●Network created by wireless stations communicating over a wireless medium ●Two challenges –Lack of centralized infrastructure –Mobility ●Need topology control in ad hoc networks –Local control strategies are needed –Support time and energy efficient routing ●How to model ad hoc networks? –Need models that are close to reality –But can still design algorithms using the model 2

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Modeling Wireless Networks ●Wireless communication very difficult to model accurately –Shape of transmission range –Interference –Mobility –Physical Carrier Sensing 3

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Outline Introduction → Models of Wireless networks ●Our model ●Our results ●Problem description –Running Example ●Conclusions 4

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Models of Wireless Networks ●Unit Disk Graph (UDG) –Given a transmission radius R, nodes u, v are connected if d(u,v) ≤ R –Too simple model 5 u R v u'

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●What is the problem? –Transmission range could be arbitrary shape 5 R R u R v u' u Models of Wireless Networks ●Unit Disk Graph (UDG) –Given a transmission radius R, nodes u, v are connected if d(u,v) ≤ R –Too simple model

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●Packet Radio Network (PRN) –Can handle arbitrary shapes –Widely used –Nodes u, v can communicate directly if they are within each other's transmission range, r t. 6 u v w v' Models of Wireless Networks

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What is the problem? ●Model for interference too simplistic 7 u v w v'

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●w can still interfere at u ●PRN model fails to address certain interference problems in practice v n-2 s t ≤rt ≤rt ≤ r t ≤ r i ≥ r t 7 What is the problem? u v w v'

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8 ●Transmission Range, Interference Range –Separate values for transmission range, interference range. –Interference range constant times bigger than transmission range. –Used in e.g., [Adler and Scheideler '98], [Kuhn et. al., '04] Models of Wireless Networks u rtrt v w u' riri

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8 ●Transmission Range, Interference Range –Separate values for transmission range, interference range. –Interference range constant times bigger than transmission range. –Used in e.g., [Adler and Scheideler '98], [Kuhn et. al., '04] ●What is the problem? –Extension of unit disk model to handle interference Models of Wireless Networks u rtrt v w u' riri

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9 Outline Introduction Models of Wireless Networks → Our Model ● Our results ●Problem description –Running Example ●Conclusions

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Our Model ●Transmission range, interference range via cost function ●Carrier sensing –Two types 1)Physical carrier sensing 2)Virtual carrier sensing 10

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Cost Function ●G r = (V, E r ), set of nodes V, Euclidean distance d(,) ●c is a cost function on nodes –symmetric: c(u,v) = c(v,u) − [0,1), depends on the environment –c(u,v) [(1- ) d(u,v), (1+ ) d(u,v)] ● Edge (u,v) E r if and only if c(u,v) ≤ r w u v a b 11

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Transmission and Interference Range ●Transmission range denoted r t (P), Interference range, r i (P) –If c(v,w) ≤ r i (P), node v can cause interference at node w. –If c(v,w) ≤r t (P) then v is guaranteed to receive the message from w provided no other node v' with c(v, v') ≤ r i (P) also transmits at the same time. 12 w rt(P)rt(P) v' ri(P)ri(P) u v c(v,w) r t (P) c(v, v') r i (P)

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Physical Carrier Sensing ●Provided by Clear Channel Assessment (CCA) circuit: –Monitor the medium as a function of Received Signal Strength Indicator (RSSI) –Energy Detection (ED) bit set to 1 if RSSI exceeds a certain threshold –Has a register to set the threshold in dB 13

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Physical Carrier Sensing ●Carrier sense transmission (CST) range, denoted r st (T, P) ●Carrier sense interference (CSI) range, denoted r si (T, P) ●Both the ranges grow monotonically in both T and P. w v r st (T,P) v' v'' 14 r si (T,P) c(w,v) r st (T, P) c(w, v') r si (T, P) c(w, v'') r si (T, P)

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Virtual Carrier Sensing 1. RTS s t 15 ●Done with the use of two control signals –Request To Send (RTS) –Clear To Send (CTS) ●DATA transmission begins after receipt of CTS

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Virtual Carrier Sensing ●Done with the use of two control signals –Request To Send (RTS) –Clear To Send (CTS) ●DATA transmission begins after receipt of CTS 15 2. CTS s t

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Virtual Carrier Sensing 15 3. DATA s t ●Done with the use of two control signals –Request To Send (RTS) –Clear To Send (CTS) ●DATA transmission begins after receipt of CTS

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16 Outline Introduction Models of Wireless Networks Our Model → Our Results ●Problem description –Running Example ●Conclusions

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Our Results ●More general model for ad hoc wireless networks ●Constant density topological spanner for the original network –Local-control –Self-stabilizing [Dijkstra '74] –No knowledge of size or topology of network, including estimate of size –Nodes do not need globally distinct labels –Constant storage and constant size messages 17

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18 Outline Introduction Models of Wireless Networks Our Model Our Results → Problem Description –Running Example ●Conclusions

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Topological Spanners ●Definition: Given a graph G = (V,E), find a sub- graph H = (V, E') such that d H (u,v) ≤ t d G (u,v) –H is also called a t-spanner. ●Previous Work –[Alzoubi et. al., '03] 5-spanner –[Dubhashi et. al., '03] log n – spanner 19

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Our Approach 20 ●Dominating set: Given a graph G = (V,E) a subset U such that all nodes are either in U or have a neighbor in U. –Density of U is the maximum number of neighbors that any node has in U. Dominator Density = 3

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●Seek a connected dominating set of constant density. 20 Our Approach ●Dominating set: Given a graph G = (V,E) a subset U such that all nodes are either in U or have a neighbor in U. –Density of U is the maximum number of neighbors that any node has in U. Dominator Density = 3

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Our Approach ●Each round has time slots reserved for each phase of the protocol 21 Time One round Phase IPhase II Phase III Phase IPhase IIPhase III ●Three phase protocol 1.Phase I: Dominating set 2.Phase II: Distributed Coloring 3.Phase III: Gateway Discovery

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22 Outline Introduction Models of Wireless Networks Our Model Our Results Problem Description → Running Example ●Conclusions

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Phase I: Constant Density Dominating Set ●Observation: In G r = (V, E r ) any Maximal Independent Set (MIS) is also a dominating set of constant density ●Maximal Independent Set –well studied starting from [Luby '85], [Dubhashi et. al., '03], [Kuhn et. al., '04], [Gandhi and Parthasarathy '04] ●Our solution –Uncertainties in our model make it harder –Without knowing size of network, have to use physical carrier sensing –Randomized protocol that runs in O(log 4 n) w.h.p. 23

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●Nodes choose a state {Active, Inactive} ●Two different sensing thresholds: –active nodes use CSI range = r t –inactive nodes use CST range = r i 24 Inactive Active Phase I: Constant Density Dominating Set

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●If an active node sends a LEADER signal, the nodes that receive/sense the leader signal stay inactive Changed from Active Inactive ●Nodes choose a state {Active, Inactive} ●Two different sensing thresholds: – active nodes use CSI range = r t – inactive nodes use CST range = r i 24 Inactive Active and sent LEADER Phase I: Constant Density Dominating Set

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Dominators Inactive nodes ●Nodes choose a state {Active, Inactive} ●Two different sensing thresholds: –active nodes use CSI range = r t –inactive nodes use CST range = r i ●If an active node sends a LEADER signal, the nodes that receive/sense the leader signal stay inactive ●If no active node available, inactive nodes become active again 24

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Phase II: Distributed Coloring ●U := output of phase I, also called active nodes ●Active nodes choose colors (equiv. time rounds) such that nodes that are not r i r i apart have different colors. 25 riri riri u v w u v w

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Output of Phase II ●Time arranged as rounds ●Each round has time slots reserved for communication in each phase ●Transmission of active node during the corresponding round is free of interference! 26 One round Phase IPhase II Phase III Phase IPhase IIPhase III Time

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Phase III: Gateway Discovery ●Dominating set U may not be a connected dominating set –Extend U by gateway nodes. ●Observation: Each node in U needs O(1) gateways. ●Uses coloring achieved in Phase II to minimize interference problems ●Approach similar to that used in [Wang and Li, '03] 27

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Phase III Gateway Discovery u l l' CLIENT(u) v ● If CLIENT messages interfere at active node, l, then the active nodes sends a COLLISION signal 28

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Phase III Gateway Discovery ACK ● Eventually only one inactive node sends a CLIENT message to an active node 28 u l l' v

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Phase III Gateway Discovery 28 ●After receiving ACK from l, u advertises presence of l via ADV message. ADV(u,l) (( () )) u l l' v

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Phase III Gateway Discovery Phase III 28 GATEWAY(l,u,v,l') (( () )) u l l' v u l v Active node Inactive node Gateway node Gateway edge Other edges

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3-Spanner ●Our construction achieves a 3-spanner of constant density for the original network. ●Total runtime = O(log 4 n + (D log D) log n) whp. –D is the density of the original network 29 u l l' v s t Active node Inactive node Gateway node Gateway edge Other edges

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30 Outline Introduction Models of Wireless Networks Our Model Our Results Problem Description Running Example → Conclusions

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Conclusions and Future Work ●More realistic model for wireless networks ●Still possible to design efficient algorithms –Constant density 3-spanner –Algorithms are simple and use constant sized messages, constant storage at nodes ●Further applications –Higher communication primitives e.g., broadcasting, gathering –Handling mobility 31

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