ACSS 2006, T. Radzik 1
Communication Algorithms for Ad-hoc Radio Networks
Tomasz Radzik
Kings Collage London
ACSS 2006, T. Radzik 2
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.
vx
u
z
y
ACSS 2006, T. Radzik 3
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.
v
u
ba
ACSS 2006, T. Radzik 4
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.
• Unknown topology.
v
u
ba
ACSS 2006, T. Radzik 5
Broadcasting
• Initially, a source node has a message M.
Msource
ACSS 2006, T. Radzik 6
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
ACSS 2006, T. Radzik 7
Gossiping
• Initially, each node i has its own message Mi.
M2M1
M4
M3
Mj
Mi
M5
Mn
ACSS 2006, T. Radzik 8
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
ACSS 2006, T. Radzik 9
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
ACSS 2006, T. Radzik 10
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.
ACSS 2006, T. Radzik 11
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
ACSS 2006, T. Radzik 12
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.
ACSS 2006, T. Radzik 13
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
ACSS 2006, T. Radzik 14
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|
ACSS 2006, T. Radzik 15
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|)
ACSS 2006, T. Radzik 16
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
ACSS 2006, T. Radzik 17
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:
ACSS 2006, T. Radzik 18
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.
ACSS 2006, T. Radzik 19
Ω(log2n) lower bound (cont.)
• 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}
ACSS 2006, T. Radzik 20
Ω(log2n) lower bound (cont.)
• 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))
ACSS 2006, T. Radzik 21
Ω(log2n) lower bound (cont.)
• 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.
ACSS 2006, T. Radzik 22
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.
ACSS 2006, T. Radzik 23
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.
ACSS 2006, T. Radzik 24
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 )
ACSS 2006, T. Radzik 25
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
ACSS 2006, T. Radzik 26
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
ACSS 2006, T. Radzik 27
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 23 1 2
11 213 2
1 2 211 2
11 11122
r r
no cross edge
simultaneoustransmissionspossible
rr 1
ACSS 2006, T. Radzik 28
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, …
ACSS 2006, T. Radzik 29
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, …
ACSS 2006, T. Radzik 30
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, …
ACSS 2006, T. Radzik 31
Pipelining
rmax
11 321 2 2 …
11 322 2 3 …
3211 1 2 2 …2
• A node at layer i with rank q transmits at step
i + q + k rmax , for k = 0, 1, 2, …
4 4
11 1
ACSS 2006, T. Radzik 32
Deterministic algorithm (cont.)
• 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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ACSS 2006, T. Radzik 33
Deterministic algorithm (cont.)
• 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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