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

2. ACSS 2006, T. Radzik2 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 x u z y

3. ACSS 2006, T. Radzik3 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 b a

4. ACSS 2006, T. Radzik4 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 b a

5. ACSS 2006, T. Radzik5 Broadcasting Initially, a source node has a message M. M source

6. ACSS 2006, T. Radzik6 Broadcasting Initially, a source node has a message M. We want to distribute message M to all nodes in the network. M M M M M M M M M M M M M M source

7. ACSS 2006, T. Radzik7 Gossiping Initially, each node i has its own message Mi. M2 M1 M4 M3 Mj Mi M5 Mn

8. ACSS 2006, T. Radzik8 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

9. ACSS 2006, T. Radzik9 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

10. ACSS 2006, T. Radzik10 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.

11. ACSS 2006, T. Radzik11 Broadcasting Radius-2 networks – Randomized O(log 2 n) protocol – unknown net. – Deterministic O(log 2 n) protocol – known net. – Ω(log 2 n) lower bound General networks – Randomized O(D log(n/D) + log 2 n) protocol (optimal) unknown network – O(D + log 3 n) deterministic and O(D + log 2 n) randomized protocols – known networks

12. ACSS 2006, T. Radzik12 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.

13. ACSS 2006, T. Radzik13 Randomized O(log 2 n) 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 2 i-1 ≤ d(v ) < 2 i : 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/n 2 P(all v in L2 get M in c log n phases) ≥ 1-1/n

14. ACSS 2006, T. Radzik14 Deterministic O(log 2 n) protocol [Chlamtac, Weinstein, 1987] Known network De-randomize by conditional expectations Consider the first phase, iteration i X – { v in L2: 2 i-1 ≤ d(v ) < 2 i } 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|

15. ACSS 2006, T. Radzik15 Deterministic O(log 2 n) 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 far calculate 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|)

16. 16 Deterministic O(log 2 n) 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: w1w2w3 X: v

17. 17 Ω(log 2 n) 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 log 2 n and consider a random network H Show: Prob( F is good for H ) < exp{ - n log 2 n } There are ≤ exp{ n log 2 n } different protocols, so some fixed network H has no length t protocol. L1: L2:

18. ACSS 2006, T. Radzik18 Ω(log 2 n) 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 = n 7, 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| ≈ 2 q, |R ∩ Sk| = 1 with constant prob. – if |R| << 2 q, |R ∩ Sk| = 0 with high prob. – if |R| >> 2 q, |R ∩ Sk| ≥ 2 with high prob. F needs Ω(log n) sets of size ≈ 2 q, for each q. Or otherwise for some q, F is bad for Hq w.h.p.

19. ACSS 2006, T. Radzik19 Ω(log 2 n) lower bound (cont.) Combinatoria lemma: For each family F of ε log 2 n 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| ≤ 2 q* log n (ii) |Bj \ (U Ai)| ≥ 2 q* (iii) ∑ 2 q* / |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/n 5 Prob(F is good for Hq*) ≤ (1-1/n 5 ) ↑ n 7 ≤ exp{-n 2 }

20. ACSS 2006, T. Radzik20 Ω(log 2 n) lower bound (cont.) F – an arbitrary protocol of length ε log 2 n. 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 – 1/2 q* ) |A| ≥ 1/n 2 (use (i)) Prob( |S ∩ (Bi \ A)| ≥ 2 ), putting b = |Bi \ A| ≥ 1 – (1 – 1/2 q* ) b – (b/2 q* )(1 – 1/2 q* ) b ≥ 1 – 0.9 ∙ 2 q* / b (use (ii)) Prob( |S ∩ (Bi \ A)| ≥ 2, for all Bi ) ≥ Π(1 – 0.9 ∙ 2 q* / |Bi \ A|) ≥ 1/n 3 (use (iii))

21. ACSS 2006, T. Radzik21 Ω(log 2 n) 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/n 2 ) ∙ (1/n 3 ) = 1/n 5. Prob( for each S in H, exists R in F: |S ∩ R| = 1) ≤ (1 – 1/n 5 ) ↑ n 7 ≤ exp{ – n 2 } There are ≤ exp{ n log 2 n } different protocols of length ε log 2 n. Hence there is a radius-2 network H with n 8 nodes without a protocol of length ε log 2 n.

22. 22 Broadcasting in general network Network with diameter D O(D log 2 n) protocol O(D log n + log 2 n) 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.

23. ACSS 2006, T. Radzik23 Unknown undirected network, randomized alg. Ω(D log (n/D) + log 2 n) lower bound [Kushilevitz, Mansour, 1998] O(D log (n/D) + log 2 n) algorithm [Czumaj, Rytter, 2003] source v Shortest 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.

24. 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. 1111111 3 11312 111132 12111 2 1111122 1 ( ≤ log n )

25. 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. 1111111 3 11312 111132 12111 2 1111122 r r no cross edge simultaneous transmissions possible rr 1

26. 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. 1111111 3 11312 111132 12111 2 1111122 r r no cross edge simultaneous transmissions possible rr 1

27. 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. 1111111 3 12312 112132 12211 2 1111122 r r no cross edge simultaneous transmissions possible rr 1

28. 28 Pipelining r max 1132122 … 1132223 … 1132 12 2 … 2 A node at layer i with rank q transmits at step i + q + k r max, for k = 0, 1, 2, …

29. 29 Pipelining r max 1132122 … 1132223 … 1132 12 2 … 2 A node at layer i with rank q transmits at step i + q + k r max, for k = 0, 1, 2, …

30. 30 Pipelining r max 1132122 … 1132223 … 1132 12 2 … 2 A node at layer i with rank q transmits at step i + q + k r max, for k = 0, 1, 2, …

31. 31 Pipelining r max 1132122 … 1132223 … 32 11 12 2 … 2 A node at layer i with rank q transmits at step i + q + k r max, for k = 0, 1, 2, … 44 11 1

32. ACSS 2006, T. Radzik32 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 v r r r r’ r”

33. ACSS 2006, T. Radzik33 Deterministic algorithm (cont.) Number of steps required: fast (green) transmissions: D + O(log n) ∙ O(log n) slow (red) transmissions: O(log n) ∙ O(log 2 n), if deterministic alg. O(log 2 n) w.h.p, if randomized alg. Total running time: D + O(log 3 n), deterministic alg. D + O(log 2 n), randomized alg. source v r r r r’ r”