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Local Computations in Large-Scale Networks

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  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 • 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 • 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 • 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

  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 turn on the sprinklers (need to backup first) Let’s all reduce power

  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. Wireless Mesh Networks • 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! L • 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, …

  15. The Locality Hype • Locality plays a crucial role in real life large scale distributed systems John Kubiatowicz et.al, on global storage: “In a system as large as OceanStore, locality is of extreme importance… 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…”

  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 aMeaningful 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: Naïve 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 AIdoes: • 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 notper-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 CG of inputs to P on a graph G into classes C • For class of inputs C, MLB(C) be a lower bound for computing P on all inputs in C • Locality: GGCCGIC : MA(I)  const  MLB(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 20 black, 12 white Global communication! black! 7 black, 1 white black! 2 black 1 black

  32. Spanning Tree: Value Change 19 black, 13 white Global communication! 6 black, 2 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% 73% 98% 57% 84% 91% 88% 93% 76% 59% 80%

  35. Veracity Radius (VR) for One-Shot Aggregation [BKLSW,PODC’06] • Roughly speaking: the min radius r0 such that"r> r0: 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} r = 2: wrong result Global result: VRa=3

  37. I only b’s I’ only a’s n1 a’s v r-1 n1 a’s v r-1 n2 b’s n2 b’s 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

  38. Veracity Radius Captures the Locality of One-Shot Aggregation [BKLSW,PODC’06] • I-LEAG (Instance-Local Efficient Aggregation on Graphs) • 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. Mesh edge: Level 0 edge: Level 1 edge: Level 2 edge: Level 0 pivot: Level 1 pivot: Level 2 pivot: Minimal Slack LPH for Mesheswith a(r)=max(r-1,r/2)

  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 AmongTree 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 Updates sent by clusters that had conflicts • Legend: Input: Output: Message: Tree edge: ! Conflict:

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

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

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

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