ACSS 2006, T. Radzik1 Communication Algorithms for Ad-hoc Radio Networks Tomasz Radzik Kings Collage...

ACSS 2006, T. Radzik 1

Communication Algorithms for Ad-hoc Radio Networks

Tomasz Radzik

Kings Collage London


Radio Networks

• If a node v transmits, then the signal from v goes to all nodes within the range of v.

• If u listens, then it receives transmission from v if and only if v is the only transmitting node which has u in its range.

• If u is in the range of more than one transmitting node: collision, no data received (no collision detection).

• Unknown topology.

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u

z

y


Radio Networks



• If u is in the range of more than one transmitting node: collision, no data received (no collision detection).


v

u

ba


Radio Networks



• If u is in the range of more than one transmitting node: collision, no data received.


v

u

ba


Broadcasting

• Initially, a source node has a message M.

Msource


Broadcasting

• Initially, a source node has a message M.

• We want to distribute message M to all nodes in the network.

MM

M

M

M

MM

M

M

M

M

M

M

M

source


Gossiping

• Initially, each node i has its own message Mi.

M2M1

M4

M3

Mj

Mi

M5

Mn


Gossiping

• Initially, each node i has its own message Mi.

• We want to distribute all these messages to all nodes in the network.

M1,…, Mn

M1,…, Mn

M1,…, Mn

M1,…, Mn

M1,…, Mn

M1,…, Mn

M1,…, Mn

M1,…, Mn


Radio Networks – different variants

• Directed or undirected network

• Known network, unknown network, or partially known network (for example, each node knows its neighbours)

• No node labels (anonymous nodes), small node labels – from { 1, 2, …, O(n) }, or large node labels – from { 1, 2, …, O(N) } where N is an independent parameter.

• Randomized or deterministic protocol

• Bounded or unbounded messages

• Collision detection or no collision detection


Topics

1. Randomized broadcasting in unknown networks and deterministic broadcasting in known networks

2. Deterministic communication in unknown networks

a. Selectors or selective families of sets

b. Deterministic broadcasting in unknown networks

c. Deterministic gossiping in unknown networks.


Broadcasting

• Radius-2 networks– Randomized O(log2n) protocol – unknown net.

– Deterministic O(log2n) protocol – known net.

– Ω(log2n) lower bound

• General networks – Randomized O(D log(n/D) + log2n) protocol (optimal)

unknown network

– O(D + log3n) deterministic and O(D + log2n) randomized protocols – known networks


Broadcasting in radius-2 networks

source

L1:

L2:

• First round: the source sends the message to all nodes in L1

• Subsequent rounds: nodes from L1 try to send the message to the nodes in L2.


Randomized O(log2n) protocol• Repeat c log n times following phase

for i = 1 to log n do

each w in L1 transmits with prob. 2-i

• For a v in L2 with degree 2i-1 ≤ d(v ) < 2i:

P(v gets M in phase r)

≥ P(v gets M in iter. i of phase r )

= d(v) 2-i (1 – 2-i)d(v)-1 ≥ 1/8

• P(v doesn’t get M in c log n phases) ≤ 1/n2

• P(all v in L2 get M in c log n phases) ≥ 1-1/n


Deterministic O(log2n) protocol[Chlamtac, Weinstein, 1987]

• Known network

• De-randomize by conditional expectations

• Consider the first phase, iteration i

• X – { v in L2: 2i-1 ≤ d(v ) < 2i }• Y – nodes in X which get M in this iteration

• In randomized algorithm: E|Y| = ∑ { P(v gets M): v in X } ≥ 1/8 |X|

• In deterministic algorithm:select nodes from L1 for transmission such that

|Y| ≥ E|Y| ≥ 1/8 |X|


Deterministic O(log2n) protocol (cont.)• At the end of phase 1, the number of nodes in L2 without

M is at most (7/8) |L2|.

• Generally, each phase reduces the number of nodes in L2 without M at least by factor 7/8, so after O(log n) phases all nodes in L2 have M.

• Deciding nodes for transmission in iter. i :Π = { } // decisions made so farcalculate E(|Y|) = E(|Y| | Π)for each w in L1 do if E(|Y| | Π and w transmits) ≥ E(|Y| | Π) then Π ← Π U { “w transmits” } else Π ← Π U { “w doesn’t transmit” }// E(|Y| | Π) ≥ E(|Y|)


Deterministic O(log2n) protocol (cont.)

Calculate E(|Y| | decisions made so far)

• E(|Y|) = ∑ { d(v) 2-i (1 – 2-i)d(v)-1 : v in X }

• E(|Y| | w1, w2, w3 decided)

= ∑ { P(v gets M) : blue v in X }

+ ∑ { P(v gets M) : green v in X }

X:

w1 w2 w3

X:

X:v


Ω(log2n) lower bound [Alon, Bar-Noy, Linial, Peleg, 1991]

• L1 = { 1,2, … , n }• Network: H = {S1, S2, .. , Sm}, Si - subset of L1• Protocol: F = {R1, R2, .. , Rt}, Ri - subset of L1

• Fix a protocol F of length t = ε log2n and consider a random network H

• Show: Prob( F is good for H ) < exp{ - n log2n }

• There are ≤ exp{ n log2n } different protocols, so some fixed network H has no length t protocol.

L1:

L2:


Ω(log2n) lower bound (cont.)

• H = U Hq, where for q = 1, 2, … , log n, Hq = { S1, S2, … , Sm} – random network such

that for each 1 ≤ i ≤ n and 1 ≤ k ≤ m = n7,

Prob( i in Sk ) = 2-q, independently

• Sk – random subset of {1, … , n} of size ≈ n/2-q

• For a set R in protocol F, – if |R| ≈ 2q, |R ∩ Sk| = 1 with constant prob.

– if |R| << 2q, |R ∩ Sk| = 0 with high prob.– if |R| >> 2q, |R ∩ Sk| ≥ 2 with high prob.

• F needs Ω(log n) sets of size ≈ 2q, for each q. Or otherwise for some q, F is bad for Hq w.h.p.



• Combinatoria lemma:

For each family F of ε log2n subsets of {1,…,n}, there is an index q*, (1/4) log n ≤ q* ≤ (1/2) log n, such that

F = { A1, A2, … , Ax, B1, B2, … , By }, where

(i) |U Ai| ≤ 2q* log n

(ii) |Bj \ (U Ai)| ≥ 2q*

(iii) ∑ 2q*/ |Bj \ (U Ai)| ≤ log n

• F is not good for Hq*: for each set S in Hq*,

Prob( |S ∩ R| ≠ 1 for all R in F ) ≥ 1/n5

• Prob(F is good for Hq*) ≤ (1-1/n5) ↑ n7 ≤ exp{-n2}



• F – an arbitrary protocol of length ε log2n.• q* and F = { A1, A2, … , Ax, B1, B2, … , By }, as in the

lemma, and A = UAi.• S – a randon set in Hq*.

• Prob( |S ∩ Ai| = 0, for all Ai )

≥ (1 – 1/2q* )|A| ≥ 1/n2 (use (i))

• Prob( |S ∩ (Bi \ A)| ≥ 2 ), putting b = |Bi \ A|

≥ 1 – (1 – 1/2q*)b – (b/2q*)(1 – 1/2q*)b

≥ 1 – 0.9 ∙ 2q*/ b (use (ii))

• Prob( |S ∩ (Bi \ A)| ≥ 2, for all Bi )

≥ Π(1 – 0.9 ∙ 2q*/ |Bi \ A|) ≥ 1/n3 (use (iii))



• Prob( |S ∩ R| ≠ 1, for each set R in F )

≥ Prob( |S ∩ Ai| = 0, for all Ai and |S ∩ (Bi \ A)| ≥ 2, for all Bi )

= Prob( |S ∩ Ai| = 0, for all Ai ) ∙ Prob( |S ∩ (Bi \ A)| ≥ 2, for all Bi )

≥ (1/n2) ∙ (1/n3) = 1/n5.

• Prob( for each S in H, exists R in F: |S ∩ R| = 1)

≤ (1 – 1/n5) ↑ n7 ≤ exp{ – n2 }

• There are ≤ exp{ n log2n } different protocols of length

ε log2n.

• Hence there is a radius-2 network H with n8 nodes

without a protocol of length ε log2n.


Broadcasting in general network• Network with diameter D

• O(D log2n) protocol

• O(D log n + log2n) protocol [Bar-Yehuda, Goldreich, Itai, 1992]:

Processors with M repeat (synchronized) phases:

for i = 1 to log n do transmits with prob. 2-i source v

• With constant probability, in one phase, message M is send to the next node.

• Large D: expected D phases → O(D) phases w.h.p.


Unknown undirected network, randomized alg.

• Ω(D log (n/D) + log2n) lower bound[Kushilevitz, Mansour, 1998]

• O(D log (n/D) + log2n) algorithm [Czumaj, Rytter, 2003]

source vShortest path:

• Average node degree: O(n/D).

• If each node degree is O(n/D), then the transmission probabilities < D/n not needed, so only log(n/D) iterations in one phase.

• General case: keep steps with transmission probabilities < D/n, but make them less frequent.


Known undirected network, deterministic alg.[Gasieniec, Peleg, Xin, 2005]

• BFS tree: source

• Rank the nodes from the leaves: increase the rank of the parent, if two children have same max rank.

1 1 1 1 11 1

3

1 13 1 2

11 113 2

1 2 111 2

11 11122

1

( ≤ log n )


Deterministic algorithm (cont.)

source

• Can find a BFS tree so that no two egdes (r,r) between layers i and i+1 have a cross edge.

1 1 1 1 11 1

3

1 13 1 2

11 113 2

1 2 111 2

11 11122

r r

no cross edge

simultaneoustransmissionspossible

rr 1



source


1 1 1 1 11 1

3

1 13 1 2

11 113 2

1 2 111 2

11 11122

r r

no cross edge


rr 1



source


1 1 1 1 11 1

3

1 23 1 2

11 213 2

1 2 211 2

11 11122

r r

no cross edge


rr 1


Pipelining

rmax

11 321 2 2 …

11 322 2 3 …

11 321 2 2 …2

• A node at layer i with rank q transmits at step

i + q + k rmax , for k = 0, 1, 2, …


Pipelining

rmax

11 321 2 2 …

11 322 2 3 …

11 321 2 2 …2


i + q + k rmax , for k = 0, 1, 2, …


Pipelining

rmax

11 321 2 2 …

11 322 2 3 …

11 321 2 2 …2


i + q + k rmax , for k = 0, 1, 2, …


Pipelining

rmax

11 321 2 2 …

11 322 2 3 …

3211 1 2 2 …2


i + q + k rmax , for k = 0, 1, 2, …

4 4

11 1



• Separate transmissions from consecutive layers, so that only one in every three consecutive layers transmits.

• If M is at the first node of the same-rank length d path at step t, then M is send to the end of this path in O(log n) + d steps.

• How can we pass messages between node of different ranks?

• For each pair of consecutive layers, repeatedly run the protocol for radius-2 networks. Interleave this with the pipeline.

source

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• Number of steps required:

fast (green) transmissions:

D + O(log n) ∙ O(log n)

slow (red) transmissions:

O(log n) ∙ O(log2n), if deterministic alg.

O(log2n) w.h.p, if randomized alg.

• Total running time:

D + O(log3n), deterministic alg.

D + O(log2n), randomized alg.

source

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ACSS 2006, T. Radzik1 Communication Algorithms for Ad-hoc Radio Networks Tomasz Radzik Kings Collage...

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