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Constant Density Spanners for Wireless Ad-Hoc Networks

Discrete Mathematics and Algorithms Seminar

Melih Onus

April 5 2005

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Ad-Hoc Networks Mobile Devices communicating via radio Network without centralized control The wireless units, or nodes, are represented by a graph,

and two nodes are connected by an edge if and only if they are within transmission range of each other

Transmissions of messages interfere at a node if at least two of its neighbors transmit a message at the same time.

A node can only receive a message if it does not interfere with any other message.

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Unit Disk Graph Model In theory, its assumed that nodes form a unit disk graph

Two nodes can communicate if they are within Euclidean distance 1 (equal transmission ranges)

Problems: In reality

Signal propagation of real antennas not clear-cut disk

Thus, algorithms designed for unit disk graph model may not work well in practice

The transmission range of a message is not the same as its interference range

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Our communication model The transmission range of a message is not the

same as its interference range The transmission and interference areas of a node

are not necessarily disk-shaped Provides a realistic model for physical carrier sensing

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Our communication modelA set V of nodes are distributed in an arbitrary way in a 2-dimensional Euclidean space

For a given cost function c and given transmission range rt, transmission area of u is { vV | c(u,v) rt }

For given interference range ri, interference area of v is { uV | c(u,v) ri }

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w. .

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Transmission & Interference Area

Node u is guaranteed receive a message from a node v in its transmission area as long as there is no other node w V in its interference area that transmits a message at the same time

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Transmission Range

.u rt/(1+)

rt/(1-)

Nodes can communicate if distance rt/(1+ )

Nodes cannot communicate if distance > rt/(1- )

In range (rt/(1- ), rt/(1+ )), it is unspecified whether massage arrives

Cost Function:

c(v,w) [(1- )d(v,w), (1+ )d(v,w)]

[0,1), fixed constant

d(v,w): the Euclidean distance between v and w

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w. .

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Physical carrier sensing

Nodes cannot only send and receive messages but they can also perform physical carrier sensing

Nodes can set their sensing threshold T

Sensing range grows monotonically with T

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Carrier Sense Transmission & Interference Areas

For a given carrier sensing threshold T, carrier sensing transmission area of u is { vV | c(u,v) rst(T) }

For a given carrier sensing threshold T, carrier sensing interference area of u is { vV | c(u,v) rsi(T) }

rst(T): carrier sensing transmission(CST) range

rsi(T): carrier sensing interference(CSI) range

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If node v transmits a message and v is in the CST range of node u, then u senses the message transmission

If node u senses a message transmission, then there is at least one node w in the CSI area of u that transmitted a message

Carrier Sensing

v

.u.

.w

ok!

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Dominating set

A dominating set (DS) is a subset of nodes such that either a node is in DS or has a neighbor in DS.

A minimum dominating set (MDS) is a DS with smallest possible number of nodes

A B

C

D

E

F

GGB D

E

F

A

C

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Our Results

The nodes do not know the total number of nodes The dominating set protocol generates a constant

approximation of a MDS in O(log4 n) communication rounds, with high probability

If physical carrier sensing is not available and the nodes have no estimate of the size of the network, then (n) are necessary for obtaining a constant approximation of MDS. (Jurdzinski, Stachowiak 2002)

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Preliminary Scenario

rst=rt, so CST area is equal to transmission area

rsi=ri, so CSI area is equal to interference area

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Preliminary DS Algorithm

Nodes can either be active or inactive The active nodes are the candidates for the

dominating set Algorithm:

– If v is active, then v sends out an ACTIVE signal. If v is inactive and v did not sense any ACTIVE signal, it becomes active again.

– If v is active, then v sends out a LEADER signal with probability ½. If v decides not to send out a LEADER signal, but senses a LEADER signal from at least one other node, then v becomes inactive.

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Example I

A

B

C

D

E

A

B

C

D

E

Active

Inactive

Active signal

Leader signal

Interference range

Transmission range

Dominating Set {B, C}

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Example II

A

B

C

D

E

A

B

C

D

E

Active

Inactive

Active signal

Leader signal

Interference range

Transmission range

C will sense leader signal of B

{B} is not a dominating set

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Ideas

There may be active nodes within range rt at the end of the algorithm, but at most constant number of them

Distributed Coloring: Each node divides the time into time frames of k slots for a given constant k

There is no active node with same time slot within range ri of an active node

Two different sensing threshold

k is number of active nodes in CSI area of a node

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Sensing Thresholds

The nodes use two different sensing thresholds, Ta and Ti, depending on their state

– The sensing threshold Ta has a CSI range of rt

– The sensing threshold Ti has a CST range of ri

rs

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DS Algorithm

Time Step I:– If v is active and in its active slot, then v sends out an

ACTIVE signal– If v is inactive and v did not sense any ACTIVE signal for the

last k slots using a sensing threshold of Ta, v senses with threshold Ti, and if it does not sense anything, it becomes active and declares the current slot number as its active slot

– If v did sense some ACTIVE signal in one of the last k slots, it just performs sensing with threshold Ta and records the outcome

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DS Algorithm Time Step II

– If v is active and is in its active slot, then v sends out a LEADER message containing its ID with some fixed probability p

– If v decides not to send out a LEADER message but it either senses a LEADER message with threshold Ta or receives a LEADER message, v becomes inactive.

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Why k slots?

If an inactive node v sensed an active signal, there is at least one active node u in its carrier sense interference area

There is at most constant number of active nodes in carrier sense interference area of a node, say k’

Choose k as k > k’ Then, if there is an active node in carrier sense

interference area of u, but there is no active node in its transmission area, then v will be active at this slot

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Analysis

If there is no active node in transmission area of an active node u, then u will stay as active forever, since inactive nodes cannot be active in its slot.

If u become active after v, then c(u, v) > rs, since u will sense all k slots before becoming active.

rs is the CST range when CSI range is equal to rt

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Analysis

A node u is called leader if it is active and there is no other active node v of same color with c(u,v) rt

Lemma: Every connected component of active nodes needs a most O( log n) steps, w.h.p., until every node in it either becomes inactive or becomes a leader

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Analysis

Lemma: At any time, if active, nonleading nodes cover an area A=(log3n), the number of leaders emerging from these nodes is (A/log2n), w.h.p.

Theorem: If all nodes are initially inactive, then after O(log4 n) rounds of the algorithm, the leaders form a static dominating set of constant density, with high probability.

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Assumptions Fixed identification numbers of any form are not

required The nodes do not know the total number of nodes We only require that the mobile hosts can

synchronize up to some reasonably small time difference, which can be done, for example, with the help of GPS signals

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Constant density spanner Constant density spanner: Given a graph G find subgraph G’

of G such that distance of two nodes in G’ is less than a constant factor of original distance– Dominating Set– Distributed Coloring– Gateway Selection

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Conclusion More realistic transmission and interference model New communication model that considers physical

sensing Polylogarithmic constant approximation DS algorithm

under the realistic wireless model

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References

K. Kothapalli, C Scheideler, M. Onus, A. Richa. Constant Density Spanners For Wireless Ad-hoc Networks, submitted to SPAA 05

T. Jurdzinski, G. G. Stachowiak. Probabilistic algorithms for the wakeup problem in single hop radio networks, ISAAC 535-549, 2002

Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer, Initializing Newly Deployed Ad Hoc and Sensor Networks, MOBICOM, Philadelphia, USA, September 2004.