Abstract MAC Layers

1. Abstract MAC Layers Nancy Lynch MIT CSAIL February 24, 2010 Joint work with Fabian Kuhn and Cal Newport [DISC 2009]

2. For many years, we have been working on: • Algorithms, lower bounds, impossibility results, for problems to be solved in distributed systems. • Problems: Communication, consensus, data management, resource allocation, synchronization,… • Distributed systems: Mainly, traditional wired networks. • Modeling and analysis methods for distributed algorithms and systems. • I/O automata • Timed I/O Automata • Probabilistic I/O Automata • The theory is pretty well developed.

3. Distributed Algorithms for Wireless Networks • We would like the same kind of theory for wireless networks. • Problems, algorithms, analysis, lower bounds, impossibility results. • And they should all fit together in a common theory. • What kinds of problems? • Basic: Clock synchronization, localization. • Communication: Network-wide broadcast, point-to-point communication (message routing). • Building and maintaining network structures: Spanning trees, routes, clusters, leaders, dominating sets. • Auxiliary services: Consensus, group membership, distributed shared memory, resource allocation. • Applications: Data management, control (robots, vehicles).

4. Applications Auxiliary services Communication Network Structures Basic services Problems for Wireless Networks • Basic: Clock synchronization, localization. • Communication: Network-wide broadcast, point-to-point communication (message routing). • Building and maintaining network structures: Spanning trees, routes, clusters, leaders, dominating sets. • Auxiliary services: Consensus, group membership, distributed shared memory, resource allocation. • Applications: Data management, robot motion coordination, vehicle motion coordination. • We (and others) have results on many of these individual problems. • But no one has fit them all together.

5. Some individual results • Rambo reconfigurable distributed shared memory algorithm [Gilbert, Lynch, Shvartsman 02, 04] • Intruder tracking [Arora, Demirbas, Lynch, Nolte 04, 07] • Gradient clock synchronization [Fan, Lynch 06], [Kuhn, Locher, Oshman 09], [Kuhn, Oshman 09] • Virtual nodes [Brown, Chockler, Demirbas, Dolev, Gilbert, Lahiani, Lynch, Mitra, Newport, Nolte, Shvartsman, Spindel, Welch, 03-08] • Consensus [Chockler, Demirbas, Gilbert, Lynch, Newport, Nolte 05, 08] • Motion coordination [Gilbert, Lynch, Mitra, Nolte 05, 08] • Connectivity of robot swarms [Cornejo, Lynch 08], [Cornejo, Kuhn, Ley-Wild, Lynch 09]

6. What’s different about wireless networks? • Local broadcast communication: Message can reach many nodes. • Success of message delivery depends on distance between sender and receiver and on complex signal propagation patterns. • Contention: Different senders may send at the same time, leading to message collisions and loss. • Dynamic behavior: Nodes may fail, recover, join, leave. • Mobility: • A good theory for wireless networks must take these complications into account.

7. Signal propagation patterns • Can be complicated: • Unequal propagation in different directions. • Multipath effects. • Existing theory assumes idealized models: • Uniform propagation in all directions • Unit disk: • Quasi-unit-disk

8. Contention • Contention management strategies: • Binary Exponential Backoff. • Send repeatedly, adjusting probabilities. • ZigZag Decoding [Gollakota, Katabi 08]

9. Contention management in high-level algorithms • Algorithms for high-level tasks that are designed to run in collision-prone networks must manage contention, as well as performing high-level algorithmic work. • The combination can be very complicated. • Often most of the complexity involves contention management! • Example: Network-wide single-message broadcast [Bar-Yehuda, Goldreich, Itai 92] • Decay subroutine, decreases sending probabilities exponentially, repeats many times. • Main broadcast protocol: When a node first receives the message, it retransmits it using Decay.

10. Contention management in high-level algorithms • There has been considerable follow-on work. • Repetitious: • Different algorithms for network-wide broadcast and related problems perform similar work of managing contention. • Complicated: • Combination of high-level algorithm + contention management is pretty complicated. • Limits the complexity of algorithms that can be studied. • Unclear how results depend on assumptions: • Hard to trace dependence of algorithms on particular assumptions about signal patterns, collisions, failures,... • So must reprove results for different models.

11. Applications Auxiliary services Communication Network Structures Basic services Contention management in high-level algorithms • Repetitious. • Complicated. • Unclear how results depend on assumptions. • So, not an adequate approach for a general theory for wireless algorithms. • We need something better…

12. We need good abstractions! • In particular, at the lowest level, to mask some of the complexities of the physical networks, such as signal patterns and collisions. • [Walter, Welch, Vaidya]: • Dynamic graph model. • Ignore contention; assume reliable, fast local communication. • Study many high-level problems. • Simple, but unrealistically optimistic. • [Gilbert, Newport,…]: • Use abstract collision detection, contention management services • More realistic, but complicated. • We need abstractions that are realistic, yet not too complicated.

13. bcast abort rcv ack bcast abort rcv ack Abstract MAC Need Abstractions • Realistic, but not too complicated… • New approach: Abstract MAC service • Reliable local message broadcast communication, with time bounds that depend on the amount of local contention. • Expressed in terms of delay functions,which are bounds on: • Time for a message to reach a particular neighboring node, • Time for sender to know a message has been received by all neighbors, and • Time for a receiver to receive some message when some neighbor is sending.

14. bcast abort rcv ack bcast abort rcv ack Abstract MAC Abstract MAC Service • Reliable local message broadcast communication, with time bounds that depend on the amount of local contention. • Delay functions, bounds on: • Time for message to reach a particular neighboring node, • Time for sender to know message has been received by all nbrs, • Time for receiver to receive some message when some neighbor is sending. • Stated as simple worst-case bounds, but sometimes consider that they may be achieved with high probability. • Usual contention-management strategies are designed to achieve such guarantees.

15. Multi-Message Broadcast Algorithms using Abstract MAC Layers • Studied problem of Multi-Message Broadcast, broadcasting many messages throughout a network. • Algorithm 1: Simple greedy algorithm over Abstract MAC layer, global delivery time for message m bounded by O( (D + k) Fprog + (k-1) Fack ), where: • D = network diameter • k bounds the number of messages whose broadcast overlaps m, • Fprog bounds receiver’s time to receive something, • Fack bounds sender’s time to complete a local broadcast. • Algorithm 2: Uses regional leaders, O( D + k + bFprog + Fack), • b bounds the length of a node id. • Extends to dynamic networks.

16. Work in progress • We are: • Demonstrating how our Abstract MAC Layer can be implemented using existing methods of probabilistic transmission and network coding.  • Showing how one can use such layers to simplify the analysis of existing algorithms, by splitting the algorithms and their analysis into smaller pieces.  • Showing how our layer can be used to solve other problems, such as Neighbor Discovery. • And there is much more to be done…

17. Talk Outline • Introduction  • Abstract MAC layers • Using Abstract MAC layers for broadcast in static networks • Using Abstract MAC layers for broadcast in dynamic networks • Current work: • Decomposing the analysis of complex broadcast algorithms • Neighbor discovery • Zig-zag decoding implementations • Conclusions and future work

18. Abstract MAC Layers

19. bcast abort rcv ack bcast abort rcv ack Abstract MAC Abstract MAC Service • Reliable local message broadcast communication, with time bounds that depend on the amount of local contention. • Delay functions, bounds on: • Time for message to reach a particular neighboring node, • Time for sender to know message has been received by all nbrs, • Time for receiver to receive some message when some neighbor is sending. • Stated as simple worst-case bounds, but sometimes consider that they may be achieved with high probability. • Usual contention-management strategies are designed to achieve such guarantees.

20. bcast abort rcv ack bcast abort rcv ack Abstract MAC Abstract MAC layer interface • bcast(m): Send message m. • rcv(m): Deliver message m to a particular node. • ack(m): Tell sender of m that “everyone who should” has received the message. • abort(m): Stop processing on behalf of currently-active message m.

21. Delivery and ack guarantees • Message delivery and ack guarantees depend on the underlying physical network. • We specify this abstractly in terms of two dynamic (directed or undirected) graphs, G and G. • G: Messages mustbe delivered, guaranteed communication • G: Messages may be delivered, possible communication or interference. • G and G defined in terms of node locations and signal patterns. • Dynamic: As nodes move, or the physical environment changes, the graphs G and G change accordingly. G = (V, E), G = (V, E) E  E

22. Delivery and ack guarantees • Local broadcast (to all neighbors). • A message reachesall G-neighbors and no non-G-neighbors. • It might reach some G-neighbors that aren’t G-neighbors. • If G and G change, requirements become weaker: • Required to reach only those nodes connected in G for the entire broadcast interval. • Allowed to reach only those nodes connected in G at some point in the broadcast interval.

23. Delivery and ack guarantees • Termination: If the sender doesn’t abort a message, the MAC will eventually ack it. • Ack correctness: An ackis preceded by rcvs at all nodes that remain G-connected to sender. • Well-formedness: Each sender must wait until its previous message is acked before bcasting another one. • Time bound guarantees: Time for local broadcast depends on local contention.

24. How do we measure contention? • Contention c at a node i is the number of G-neighbors with active local broadcasts. Node i Contention = 2

25. bcast abort rcv ack bcast abort rcv ack Abstract MAC Receive bound • frcv(c) • If a node i does a bcast(m) that is received by j, then the time until rcv(m) at j is at most frcv(c), where c is the contention at the receiver j. • frcv(c) must be at least linear in c.

26. bcast abort rcv ack bcast abort rcv ack Abstract MAC Acknowledgement bound • fack(c) • If a node i does a bcast(m) that results in an ack(m), then the time until ack(m) is at most fack(c), where c is the total contention at the sender and all of its G-neighbors. • fack(c) is even larger than frcv(c).

27. Progress bound • frcv, fack: Guaranteed bounds for all messages. • Most messages take a long time, so frcv and fack are large. • Some messages will be much faster. • Progress bound: Guarantees some messages are fast.

28. bcast abort rcv ack bcast abort rcv ack Abstract MAC Progress bound • fprog(c) • If a node j has at least one G-neighbor with an active local bcast, then the time until j receives some message from some G-neighbor is at most fprog(c), where c is the contention at j. • frcv(c) can be much smaller than frcv and fack, e.g., poly-logarithmic in c.

29. Why is a small progress bound interesting? • Consider global broadcast of a message m. • At some point, some nodes have received m, others haven’t. • Receiving m from any neighbor is good enough!

30. bcast abort rcv ack bcast abort rcv ack Abstract MAC Abstract MAC time bounds (summary) • frcv(c): Upper bound on time from sending to receipt of a message, when the contention at the receiver is bounded by c. • fack(c): Upper bound on time from sending to ack at sender, when the total contention at the sender and all its receivers is bounded by c. • fprog(c): Upper bound on time for a receiver to receive some message from among a collection of messages by contending senders, when contention at the receiver is bounded by c.

31. bcast abort rcv ack bcast abort rcv ack Abstract MAC What Abstract MAC does not provide: • Neighborhood knowledge: • Guarantees nearby nodes receive a message, distant nodes don’t. • But the sender doesn’t know who is nearby, nor who receives the message. • To gain such information, nodes must do more, e.g., run a Neighbor Discovery protocol over the MAC layer. • Unicast/ack • Can implement this over the MAC layer.

32. What can we do with Abstract MAC layers? • Solve higher-level problems over the layers. • Describe and analyze algorithms to implement the layers over physical network models. • Combine the analyses to get overall bounds for high-level algorithms over physical network models.

33. Using Abstract MAC Layers for Broadcast, in Static Networks

34. j i Single-Message Broadcast • Assume static, undirected graphs G and G. • Theorem: Broadcasting a message m to all nodes takes time at most D  Fprog, where: • D = diameter of G, and • Fprog = f prog(+1), where  is the maximum degree in G. • Proof: Easy (because we don’t have to analyze contention). • Consider source i, path of length d from i to receiver j. • Time for m to progress one hop along the path is at most Fprog.

35. Multi-Message Broadcast (MMB) • Messages originate anywhere, at any times. • Everyone should receive every message. • Basic Multi-Message Broadcast (BMMB) algorithm: • Simple greedy algorithm. • Each process maintains a local queue of messages it has received. • Sends messages from its queue in order, using Abstract MAC, waiting for an ack for the previous message before sending the next. • Handles each message only once. • Messages can delay other messages all along the way. • Nevertheless, we obtain:…

36. Multi-Message Broadcast using BMMB • Assume G = G. • Theorem: Broadcasting a message m to all nodes takes time at most O((D + k) Fprog + (k-1) Fack), where: • D, , Fprog as before, • k  number of messages whose broadcast overlaps m, • Fack = f ack((+1)2). • Diameter is multiplied only by (small) Fprog, not by Fack. • Captures the effect of pipelining. • Proof: Not so easy, but still tractable (because we don’t worry about contention). • Double induction, on the number k of messages, and on the distance d to a receiver. • Although messages can block other messages, some messages always make progress. • “Either m gets through quickly, or a lot of other messages do.”

37. j i Multi-Message Broadcast using BMMB • Theorem: Broadcasting a message m to all nodes takes time at most O( (D + k) Fprog + (k-1) Fack ). • Proof: • Double induction, on the number k of messages, and on the distance d to a receiver. • Key idea: Once a particular set M of k-1 messages is completely finished along a G-path from i to j, and i has another message not in M, then some message not in M reaches j quickly, in time dist(i,j)  Fprog.

38. Can we do better? • Idea: • Use geographical information. • Divide the plane into regions. • Elect leaders in regions and let them handle the message transmission.

39. Can we do better? • Idea: • Use geographical information. • Divide the plane into regions. • Elect leaders in regions and let them handle the message transmission.

40. Regional Leader Election • Elect a leader in each region. • Assume regions are all occupied. • Basic algorithm: Within each region, everyone sends their identifiers, elect max. • Time complexity: O(Fack) • Better algorithm: Selects leader based on bit representations of node identifiers. • Time complexity: O( b Fprog), where b = number of bits in a node identifier.

41. Region-Based Multi-Message Broadcast • Algorithm: Broadcast as in BMMB, but on the network of leaders. • Good enough, because other nodes hear messages from their region’s leader. • Uses a Collect sub-algorithm to transfer arriving messages from individual nodes to their leaders. • Assumes a bound on arrival rates sufficient to avoid queue overflow (1 / Fack).

42. Region-Based Multi-Message Broadcast • Theorem: Broadcasting a message m to all nodes takes time O(D + k + b Fprog + Fack). • Which is better than the previous bound, O((D + k) Fprog + (k-1) Fack). • Savings arises because, when only leaders broadcast, the contention is constant-bounded. • Interesting, not-so-simple broadcast algorithm. • Tractable analysis, because we don’t worry about contention. • Can extend to G  G.

43. Using Abstract MAC layers for Broadcast, in Dynamic Networks

44. Multi-Message Broadcast in Dynamic Networks • Now suppose the nodes can move in the plane. • We can adapt the region-based broadcast algorithm. • Dynamic algorithm consists of: • Dynamic Leader Election • Dynamic Collect • Dynamic Broadcast on Leader Network • Ensures that messages are delivered everywhere even though nodes move between regions. • Assumption: When a process enters a region, it records a region exit time bound, and promises to remain in the region at least until this time.

45. Dynamic Leader Election • Leader election runs concurrently with Collect and Broadcast. • Maintains regional leaders as nodes move, choosing new leaders when necessary. • Algorithm: • Runs static leader election algorithm repeatedly. • Each time: • If there is a current leader, and its exit bound is not too soon, then retain the leader. • Otherwise, choose the node with the largest exit bound.

46. Dynamic Leader Election • Leader election runs concurrently with Collect and Broadcast. • Maintains regional leaders as nodes move, choosing new leaders when necessary. • Algorithm: • Runs static leader election algorithm repeatedly. • Each time: • If there is a current leader, and its exit bound is not too soon, then retain the leader. • Otherwise, choose the node with the largest exit bound.

47. Dynamic Leader Election • Leader election runs concurrently with Collect and Broadcast. • Maintains regional leaders as nodes move, choosing new leaders when necessary. • Algorithm: • Runs static leader election algorithm repeatedly. • Each time: • If there is a current leader, and its exit bound is not too soon, then retain the leader. • Otherwise, choose the node with the largest exit bound.

48. Dynamic Collect • Transfers arriving messages from the front of a non-leader node i’s arrival queue to its leader j’s broadcast queue, provided that: • i and j remain in their region, and • j remains a leader, for sufficiently long after the message arrival. • Algorithm: Uses a broadcast from i to j and an ack from j to i, both sent over the MAC layer.

49. Dynamic Broadcast on Leader Network • Propagates message m from broadcast queue of a leader i in a region r to broadcast queues of leaders in all adjacent regions. • Algorithm: • Broadcasts on the leader network as in BMMB. • If a leader becomes a non-leader, it empties its broadcast queue into its arrival queue.

50. Dynamic Broadcast on Leader Network • Propagates message m from broadcast queue of a leader i in a region r to broadcast queues of leaders in all adjacent regions. • Algorithm: • Broadcasts on the leader network as in BMMB. • If a leader becomes a non-leader, it empties its broadcast queue into its arrival queue.