1. Local Computations in Large-Scale Networks Idit Keidar Technion

2. Material I. Keidar and A. Schuster: “ Want Scalable Computing? Speculate! ” SIGACT News Sep 2006. http://www.ee.technion.ac.il/people/idish/ftp/speculate.pdf http://www.ee.technion.ac.il/people/idish/ftp/speculate.pdf Y. Birk, I. Keidar, L. Liss, A. Schuster, and R. Wolff: “ Veracity Radius - Capturing the Locality of Distributed Computations ”. PODC'06. http://www.ee.technion.ac.il/people/idish/ftp/veracity_radius.pdf http://www.ee.technion.ac.il/people/idish/ftp/veracity_radius.pdf Y. Birk, I. Keidar, L. Liss, and A. Schuster: “ Efficient Dynamic Aggregation ”. DISC'06. http://www.ee.technion.ac.il/people/idish/ftp/eff_dyn_agg.pdf http://www.ee.technion.ac.il/people/idish/ftp/eff_dyn_agg.pdf E. Bortnikov, I. Cidon and I. Keidar: “ Scalable Load-Distance Balancing in Large Networks ”. DISC ’ 07. http://www.ee.technion.ac.il/people/idish/ftp/LD-Balancing.pdf http://www.ee.technion.ac.il/people/idish/ftp/LD-Balancing.pdf

3. Brave New Distributed Systems Large-scale Thousands of nodes and more.. Dynamic … coming and going at will... Computations … while actually computing something together. This is the new part.

4. Today ’ s Huge Dist. Systems Wireless sensor networks –Thousands of nodes, tens of thousands coming soon P2P systems –Reporting millions online (eMule) Computation grids –Harnessing thousands of machines (Condor) Publish-subscribe (pub-sub) infrastructures –Sending lots of stock data to lots of traders

5. Not Computing Together Yet Wireless sensor networks –Typically disseminate information to central location P2P & pub-sub systems –Simple file sharing, content distribution –Topology does not adapt to global considerations –Offline optimizations (e.g., clustering) Computation grids –“ Embarrassingly parallel ” computations

6. Emerging Dist. Systems – Examples Autonomous sensor networks –Computations inside the network, e.g., detecting trouble Wireless mesh network (WMN) management –Topology control –Assignment of users to gateways Adapting p2p overlays based on global considerations Data grids (information retrieval)

7. Autonomous Sensor Networks The data center is too hot! Let’s all reduce power Let’s turn on the sprinklers (need to backup first)

8. Autonomous Sensor Networks Complex autonomous decision making –Detection of over-heating in data-centers –Disaster alerts during earthquakes –Biological habitat monitoring Collaboratively computing functions –Does the number of sensors reporting a problem exceed a threshold? –Are the gaps between temperature reads too large?

9. Wireless Mesh Networks

10. Infrastructure (unlike MANET) City-wide coverage Supports wireless devices Connections to Mesh and out to the Internet –“ The last mile ” Cheap –Commodity wireless routers (hot spots) –Few Internet connections

11. Decisions, Decisions Assigning users to gateways –QoS for real-time media applications –Network distance is important –So is load Topology control –Which links to set up out of many “ radio link ” options –Which nodes connect to Internet (act as gateways) –Adapt to varying load

12. Centralized Solutions Don ’ t Cut It Load Communication costs Delays Fault-tolerance

13. Classical Dist. Solutions Don ’ t Cut It Global agreement / synchronization before any output Repeated invocations to continuously adapt to changes High latency, high load By the time synchronization is done, the input may have changed … the result is irrelevant Frequent changes -> computation based on inconsistent snapshot of system state Synchronizing invocations initiated at multiple locations typically relies on a common sequencer (leader) –difficult and costly to maintain

14. Locality to the Rescue! Nodes make local decisions based on communication (or synchronization) with some proximate nodes, rather than the entire network Infinitely scalable Fast, low overhead, low power, … L

15. The Locality Hype Locality plays a crucial role in real life large scale distributed systems C. Intanagonwiwat et.al, on sensor networks: “An important feature of directed diffusion is that … are determined by localized interactions...” N. Harvey et.al, on scalable DHTs: “The basic philosophy of SkipNet is to enable systems to preserve useful content and path locality…” John Kubiatowicz et.al, on global storage: “In a system as large as OceanStore, locality is of extreme importance…

16. What is Locality? Worst case view –O(1) in problem size [ Naor & Stockmeyer,1993 ] –Less than the graph diameter [ Linial, 1992 ] –Often applicable only to simplistic problems or approximations Average case view –Requires an a priori distribution of the inputs To be continued…

17. Interesting Problems Have Inherently Global Instances WMN gateway assignment: arbitrarily high load near one gateway –Need to offload as far as the end of the network Percentage of nodes whose input exceeds threshold in sensor networks: near-tie situation –All “ votes ” need to be counted Fortunately, they don’t happen too often

18. Speculation is the Key to Locality We want solutions to be “ as local as possible ” WMN gateway assignment example: –Fast decision and quiescence under even load –Computation time and communication adaptive to distance to which we need to offload A node cannot locally know whether the problem instance is local –Load may be at other end of the network Can speculate that it is (optimism )

19. Computations are Never “ Done ” Speculative output may be over-ruled Good for ever-changing inputs –Sensor readings, user loads, … Computing ever-changing outputs –User never knows if output will change due to bad speculation or unreflected input change –Reflecting changes faster is better If input changes cease, output will eventually be correct –With speculation same as without

20. Summary: Prerequisites for Speculation Global synchronization is prohibitive Many instances amenable to local solutions Eventual correctness acceptable –No meaningful notion of a “ correct answer ” at every point in time –When the system stabilizes for “ long enough ”, the output should converge to the correct one

21. The Challenge: Find a Meaningful Notion for Locality Many real world problems are trivially global in the worst case Yet, practical algorithms have been shown to be local most of the time ! The challenge: find a theoretical metric that captures this empirical behavior

22. Reminder: Na ï ve Locality Definitions Worst case view –Often applicable only to simplistic problems or approximations Average case view –Requires an a priori distribution of the inputs

23. Instance-Locality Formal instance-based locality: –Local fault mending [Kutten,Peleg95, Kutten,Patt-Shamir97] –Growth-restricted graphs [Kuhn, Moscibroda, Wattenhofer05] –MST [Elkin04] Empirical locality: voting in sensor networks –Although some instances require global computation, most can stabilize (and become quiescent) locally –In small neighborhood, independent of graph size –[Wolff,Schuster03, Liss,Birk,Wolf,Schuster04]

24. “ Per-Instance ” Optimality Too Strong Instance: assignment of inputs to nodes For a given instance I, algorithm A I does: – if (my input is as in I) output f( I) else send message with input to neighbor – Upon receiving message, flood it – Upon collecting info from the whole graph, output f( I ) Convergence and output stabilization in zero time on I Can you beat that? Need to measure optimality per-class not per-instance Challenge: capture attainable locality

25. Local Complexity [BKLSW ’ 06] Let –G be a family of graphs –P be a problem on G –M be a performance measure –Classification C G of inputs to P on a graph G into classes C –For class of inputs C, M LB (C) be a lower bound for computing P on all inputs in C Locality:  G  G  C  C G  I  C : M A (I)  const  M LB (C) A lower bound on a single instance is meaningless!

26. The Trick is in The Classification Classification based on parameters –Peak load in WMN –Proximity to threshold in “ voting ” Independent of system size Practical solutions show clear relation between these parameters and costs Parameters not always easy to pinpoint –Harder in more general problems –Like “ general aggregation function ”

27. Veracity Radius – Capturing the Locality of Distributed Computations Yitzhak Birk, Idit Keidar, Liran Liss, Assaf Schuster, and Ran Wolf

28. Dynamic Aggregation Continuous monitoring of aggregate value over changing inputs Examples: –More than 10% of sensors report of seismic activity –Maximum temperature in data center –Average load in computation grid

29. The Setting Large graph (e.g., sensor network) –Direct communication only between neighbors Each node has a changing input Inputs change more frequently than topology –Consider topology as static Aggregate function f on multiplicity of inputs –Oblivious to locations Aggregate result computed at all nodes

30. Goals for Dynamic Aggregation Fast convergence –If from some time t onward inputs do not change … Output stabilization time from t Quiescence time from t Note: nodes do not know when stabilization and quiescence are achieved –If after stabilization input changes abruptly … Efficient communication –Zero communication when there are zero changes –Small changes  little communication

31. Standard Aggregation Solution: Spanning Tree 7 black, 1 white 2 black Global communication! black! 1 black black! 20 black, 12 white

32. Spanning Tree: Value Change Global communication! 6 black, 2 white 19 black, 13 white

33. The Bad News Virtually every aggregation function has instances that cannot be computed without communicating with the whole graph –E.g., majority voting when close to the threshold “ every vote counts ” Worst case analysis: convergence, quiescence times are  (diameter)

34. Local Aggregation – Intuition Example – Majority Voting: Consider a partition in which every set has the same aggregate result (e.g., >50% of the votes are for ‘ 1 ’ ) Obviously, this result is also the global one! 51% 59% 84% 88% 80% 98% 76% 57% 91% 73% 93%

35. Veracity Radius (VR) for One-Shot Aggregation [BKLSW,PODC ’ 06] Roughly speaking: the min radius r 0 such that " r> r 0 : all r-neighborhoods have same result Example: majority Radius 1: wrong result Radius 2: correct result VR=2

36. Introducing Slack Examine “ neighborhood-like ” environments that: –(1) include an a (r)-neighborhood for some a (r)<r –(2) are included in an r-neighborhood Example: a (r)=max{r-1,r/2} Global result: VR a=3 r = 2: wrong result

37. VR Yields a Class-Based Lower Bound VR for both input assignments is  r Node v cannot distinguish between I and I ’ in fewer than r steps Lower bound of r on both output stabilization and quiescence Trivially tight bound for output stabilization n1 a’s n2 b’s only b’s v r-1 I n1 a’s n2 b’s only a’s v r-1 I’

38. Veracity Radius Captures the Locality of One-Shot Aggregation [BKLSW,PODC ’ 06] I-LEAG ( I nstance -L ocal E fficient A ggregation on G raphs) –Quiescence and output stabilization proportional to VR –Per-class within a factor of optimal – Local: depends on VR, not graph size! Note: nodes do not know VR or when stabilization and quiescence are achieved –Can ’ t expect to know you ’ re “ done ” in dynamic aggregation …

39. Local Partition Hierarchy Topology static –Input changes more frequently Build structure to assist aggregation –Once per topology change –Spanning tree, but with locality properties

40. Minimal Slack LPH for Meshes with a (r)=max(r-1,r/2) Level 0 pivot: Level 2 pivot: Level 1 pivot: Level 0 edge: Level 2 edge: Level 1 edge: Mesh edge:

41. Another View

42. The I-LEAG Algorithm Phases correspond to LPH levels Communication occurs within a cluster only if there are nodes with conflicting outputs –All of the cluster ’ s nodes hold the same output when the phase completes –All clusters ’ neighbors know the cluster ’ s output Conflicts are detected without communication –I-LEAG reaches quiescence once the last conflict is detected

43. I-LEAG ’ s Operation (Majority Voting) Legend: Input: Output: Message: ! Tree edge: Conflict: Initialization: Node’s output is its input

44. Startup: Communication Among Tree Neighbors Legend: Input: Output: Message: ! Tree edge: Conflict: Recall neighbor values will be used in all phases

45. Phase 0 Conflict Detection Legend: Input: Output: Message: ! Conflict: ! !! ! !

46. Phase 0 Conflict Resolution Legend: Input: Output: Message: ! Tree edge: Conflict: Updates sent by clusters that had conflicts

47. Phase 1 Conflict Detection Legend: Input: Output: Message: ! Tree edge: Conflict: ! ! ! ! No new Communication

48. Phase 1 Conflict Resolution Legend: Input: Output: Message: ! Tree edge: Conflict: Updates sent by clusters that had conflicts

49. Phase 2 Conflict Detection Legend: Input: Output: Message: ! Tree edge: Conflict: Using information sent at phase 0 No Communication

50. Phase 2 Conflict Resolution Legend: Input: Output: Message: ! Tree edge: Conflict: No conflicts found, no need for resolution This region has been idle since phase 0

51. Simulation Study VR also explains the locality of previous algorithms

52. Efficient Dynamic Aggregation Yitzhak Birk, Idit Keidar, Liran Liss, and Assaf Schuster

53. Na ï ve Dynamic Aggregation Periodically, –Each node samples input, initiates I-LEAG –Each instance I of I-LEAG takes O(VR( I )) time, but sends  (|V|) messages Sends messages even when no input changes –Costly in sensor networks  To save messages, must compromise freshness of result 

54. Dynamic Aggregation at Two Timescales Efficient multi-shot aggregation algorithm (MultI- LEAG) –Converges to correct result before sampling the inputs again –Sampling time may be proportional to graph size Efficient dynamic aggregation algorithm (DynI- LEAG) –Sampling time is independent of graph size –Algorithm tracks global result as close as possible

55. Dynamic Lower Bound Previous sample (instance) also plays a role –Example (majority voting): Multi-shot lower bound: max{VR prev,VR} –On quiescence and output stabilization –Assumes sending zero messages when there are zero changes I 1 (VR  2) I 2 (0 changes) I 3 (VR=0) ? !

56. Dynamic Aggregation: Take II Initially, run local one-shot algorithm A –Store distance information travels in this instance, dist Let D = A ’ s worst-case convergence time Every D time, run a new iteration (MULTI-A) –If input did not change, do nothing –If input changed, run full information protocol up to dist –If new instance ’ s VR isn ’ t reached, invoke A anew –Update dist (~VR) (~ VR prev ) (~VR) Matches max{VR prev,VR} lower bound within same factor as A

57. A is for I-LEAG I-LEAG uses a pre-computed partition hierarchy –LPH: Local Partition Hierarchy – cluster sizes bounded both from above and from below (doubling sizes) –Spanning tree in each cluster, rooted at pivot –Computed once per topology I-LEAG phases correspond to LPH levels –Active phase: full-information from cluster  pivot –Phase result communicated to cluster and its neighbors –Phase active only if there is a conflict in the previous level –Conflicts detected without new communication

58. Multi-LEAG The Veracity Level (VL) of node v is the highest LPH level in which v ’ s cluster has a conflict (VL< log VR+1) A multi-LEAG iteration ’ s phases correspond to LPH levels: –Phase level < VL: propagate changes (if any) to pivot active only if there are changes –Phase level  VL: fall back to I-LEAG active only if new VR is larger than previous –Cache partial aggregate results in pivot nodes allows conflict detection between active and passive clusters

59. MultI-LEAG Operation Physical nodes Pivot nodes Veracity Level

60. MultI-LEAG Operation Case I: No changes … no changes to report … no conflicts All is quiet…

61. Input Change ! New veracity level no conflicts, no communication

62. Abrupt Change Flips Outcome

63. Clusters at VL recalculate, others forward up

64. Abrupt Change Flips Outcome New Veracity level no conflicts, no communication

65. Final Outcome

66. MultI-LEAG Observations O(max{VR prev,VR}) output stabilization and quiescence Message efficient: –Communication only in clusters with changes, only when radius < max{VR prev,VR} Sampling time is O(Diameter) –Good for cheap periodic aggregation –Can we do closer monitoring?

67. Dynamic Aggregation Take III: DynI-LEAG Sample inputs every O(1) link delays –Close monitoring, rapidly converges to correct result Run multiple MultI-LEAG iterations concurrently Challenges: –Pipelining phases with different (doubling) durations –Intricate interaction among concurrent instances E.g., which phase 4 updates are used in a given phase 5.. –Avoiding state explosion for multiple concurrent instances

68. Ruler Pipelining Partial iterations, fewer in every level Changes only communicated once t Sampling interval Phase 2 Phase 1 Phase 0 Full iteration Partial iteration Memory usage: O(log(Diameter))

69. VL and Output Estimation Problem: correct output and VL of an iteration is guaranteed only after O(Diameter) time –cannot wait that long … Solution: choose iteration with highest VL according to most recent information –Use this VL for new iterations and its output as MultI-LEAG ’ s current output estimation Eventual convergence and correctness guaranteed

70. DynI-LEAG Operation Phase below VL Phase above VL 0 2 1 “Previous VL” = 2 The influence of a conflict is proportional to its level t

71. Dynamic Aggregation: Conclusions Local operation is possible –in dynamic systems –that solve inherently global problems MultI-LEAG delivers periodic correct snapshots at minimal cost DynI-LEAG responds immediately to input changes with a slightly higher message rate

72. DynI-LEAG Observations An outdated conflict at a certain level i cannot influence future iterations for more than const *i time –This defines an influence envelope relative to the last change –DynI-LEAG converges extremely fast when recent VRs are low t 0 (Last change) VR t Diameter r Convergence time is bounded by const*r Among clusters with radius < recent VRs, communication occurs only in those that experienced change

73. Related Work Work on data aggregation methods (Intanagonwiwat00, Kempe03, Bawa03, Considine04) –Does not deal with locality Work on Instance-based locality –Self-stabilization (Kutten&Pelleg95, Kutten&Patt-Shamir97), location services (Li00), MST (Elkin04) Initial work on instance-local aggregation (Wolff&Schuster03, Liss04) –Empirical results only

74. Scalable Load-Distance Balancing Edward Bortnikov, Israel Cidon, Idit Keidar

75. Load-Distance Balancing Two sources of service delay –Network delay (depends on distance to server) –Congestion delay (depends on server load) –Total = Network + Congestion Input –Network distances and congestion functions Optimization goal –Minimize the maximum total delay NP-complete, 2-approximation exists

76. Distributed Setting Synchronous Distributed assignment computation –Initially, users report location to the closest servers –Servers communicate and compute the assignment Requirements: –Eventual quiescence –Eventual stability of assignment –Constant approximation of the optimal cost (parameter)

77. Impact of Locality Extreme global solution –Collect all data and compute assignment centrally –Guarantees optimal cost –Excessive communication/network latency Extreme local solution –Nearest-Server assignment –No communication –No approximation guarantee (can ’ t handle crowds) No “ one-size-fits-all ” ?

78. Workload-Sensitive Locality The cost function is distance-sensitive –Most assignments go to the near servers –… except for dissipating congestion peaks Key to distributed solution –Start from the Nearest-Server assignment –Load-balance congestion among near servers Communication locality is workload-sensitive –Go as far as needed … –… to achieve the required approximation

79. Uniform Load

80. Skewed Load

81. Load-Balance

82. Skewed Load Load-Balance

83. Tree Clustering As long as some cluster has improvable cost –Double it (merge with hierarchy neighbor) Clusters aligned at 2i indices Simple, O ( log N ) convergence time

84. Ripple Clustering Adaptive merging –Better cost in practice As long as some cluster is improvable –Merge with smaller-cost neighbors Conflicts possible –A  B  C –A  B  C –Random tie-breaking to resolve –Many race conditions (we love it - )

85. Scalability: Cost cost = Euclidian distance + linear load

86. Scalability: Locality