Local Computations in Large-Scale Networks
Idit KeidarTechnion
Material
I. Keidar and A. Schuster: “Want Scalable Computing? Speculate!” SIGACT News Sep 2006. http://www.ee.technion.ac.il/people/idish/ftp/speculate.pdfY. 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.pdfY. 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
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.
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
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
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 considerationsData grids (information retrieval)
Autonomous Sensor NetworksThe data center is too hot!
Let’s all reduce power
Let’s turn on the sprinklers (need to
backup first)
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?
Wireless Mesh Networks
Wireless Mesh Networks
Infrastructure (unlike MANET)City-wide coverageSupports wireless devicesConnections to Mesh and out to the Internet – “The last mile”
Cheap – Commodity wireless routers (hot spots)– Few Internet connections
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
Centralized Solutions Don’t Cut It
LoadCommunication costsDelaysFault-tolerance
Classical Dist. Solutions Don’t Cut It
Global agreement / synchronization before any outputRepeated invocations to continuously adapt to changesHigh latency, high loadBy the time synchronization is done, the input may have changed … the result is irrelevant Frequent changes -> computation based on inconsistent snapshot of system stateSynchronizing invocations initiated at multiple locations typically relies on a common sequencer (leader) – difficult and costly to maintain
Locality to the Rescue!
Nodes make local decisions based on communication (or synchronization) with some proximate nodes, rather than the entire network
Infinitely scalableFast, low overhead, low power, …
L
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 SkipNetis 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…
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
approximationsAverage case view– Requires an a priori distribution of the inputs
To be continued…
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
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 )
Computations are Never “Done”
Speculative output may be over-ruledGood 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
Summary: Prerequisites for Speculation
Global synchronization is prohibitive Many instances amenable to local solutionsEventual 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
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
Reminder: Naïve Locality Definitions
Worst case view– Often applicable only to simplistic problems or
approximationsAverage case view– Requires an a priori distribution of the inputs
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]
“Per-Instance” Optimality Too Strong
Instance: assignment of inputs to nodesFor a given instance I, algorithm AI 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 ICan you beat that?
Need to measure optimality per-class not per-instanceChallenge: capture attainable locality
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: GG CCGIC : MA(I) const MLB(C)
A lower bound on a single instance is meaningless!
The Trick is in The Classification
Classification based on parameters– Peak load in WMN– Proximity to threshold in “voting”
Independent of system sizePractical solutions show clear relation between these parameters and costsParameters not always easy to pinpoint– Harder in more general problems– Like “general aggregation function”
Veracity Radius – Capturing the Locality of Distributed Computations
Yitzhak Birk, Idit Keidar, Liran Liss, Assaf Schuster, and Ran Wolf
Dynamic Aggregation
Continuous monitoring of aggregate value over changing inputsExamples:– More than 10% of sensors report of seismic activity– Maximum temperature in data center– Average load in computation grid
The Setting
Large graph (e.g., sensor network)– Direct communication only between neighbors
Each node has a changing inputInputs 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
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
Standard Aggregation Solution: Spanning Tree
7 black, 1 white
2 bl
ack
Global communication!
black!
1 bl
ack
black!20 black, 12 white
Spanning Tree: Value Change
Global communication!
6
black, 2 white
19 black, 13 white
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)
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%
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
Introducing Slack
Examine “neighborhood-like” environments that:– (1) include an (r)-neighborhood for some (r)<r– (2) are included in an r-neighborhood
Example: (r)=max{r-1,r/2}
Global result:VR
r = 2:wrong result
VR Yields a Class-Based Lower Bound
VR for both input assignments is rNode v cannot distinguish between I and I’ in fewer than r stepsLower bound of r on both output stabilization and quiescenceTrivially tight bound for output stabilization
n1 a’s
n2 b’s
only b’s
vr-1
I
n1 a’s
n2 b’s
only a’s
vr-1
I’
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…
Local Partition Hierarchy
Topology static– Input changes more frequently
Build structure to assist aggregation– Once per topology change– Spanning tree, but with locality properties
Minimal Slack LPH for Mesheswith (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:
Another View
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
I-LEAG’s Operation(Majority Voting)
Legend:Input:Output:Message:
!Tree edge:Conflict:
Initialization: Node’s output is its input
Startup: Communication AmongTree Neighbors
Legend:Input:Output:Message:
!Tree edge:Conflict:
Recall neighbor valueswill be used in all phases
Phase 0 Conflict Detection
Legend:Input:Output:Message:
!Conflict:!
! !
!
!
Phase 0 Conflict Resolution
Legend:Input:Output:Message:
!Tree edge:Conflict:
Updates sent by clusters that had
conflicts
Phase 1 Conflict Detection
Legend:Input:Output:Message:
!Tree edge:Conflict:
!
!
!!
No new Communication
Phase 1 Conflict Resolution
Legend:Input:Output:Message:
!Tree edge:Conflict:
Updates sent by clusters that had
conflicts
Phase 2 Conflict Detection
Legend:Input:Output:Message:
!Tree edge:Conflict:
Using information sent
at phase 0
No Communication
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
Simulation Study
VR also explains the locality of previous algorithms
Efficient Dynamic Aggregation
Yitzhak Birk, Idit Keidar, Liran Liss, andAssaf Schuster
Naïve Dynamic Aggregation
Periodically,– Each node samples input, initiates I-LEAG– Each instance I of I-LEAG takes O(VR(I)) time,
but sends (|V|) messagesSends messages even when no input changes– Costly in sensor networks
To save messages, must compromise freshness of result
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
Dynamic Lower Bound
Previous sample (instance) also plays a role– Example (majority voting):
Multi-shot lower bound: max{VRprev,VR}– On quiescence and output stabilization– Assumes sending zero messages when there are zero changes
I1 (VR2)
I2 (0 changes)
I3 (VR=0)
?!
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 timeEvery 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)
(~ VRprev)~(VR)
Matches max{VRprev,VR} lower boundwithin same factor as A
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
Multi-LEAG
The Veracity Level (VL) of node v is the highest LPH level in which v’s cluster has a conflict (VL<logVR+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
MultI-LEAG Operation
Physical nodes
Pivot nodes
Veracity Level
MultI-LEAG Operation
Case I: No changes
… no changes to report
… no conflicts… no conflicts
All is quiet…
Input Change
!
New veracity level
no conflicts, no communication
Abrupt Change Flips Outcome
Abrupt Change Flips Outcome
Clusters at VL recalculate, others forward up
Abrupt Change Flips Outcome
New Veracity level
no conflicts, no communication
MultI-LEAG Observations
O(max{VRprev,VR}) output stabilization and quiescenceMessage efficient:– Communication only in clusters with changes,
only when radius < max{VRprev,VR}
Sampling time is O(Diameter)– Good for cheap periodic aggregation– Can we do closer monitoring?
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 concurrentlyChallenges: – 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
Ruler Pipelining
Partial iterations, fewer in every levelChanges only communicated once
t
Sampling interval
Phase 2
Phase 1
Phase 0
Full iteration
Partial iteration
Memory usage: O(log(Diameter))
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
DynI-LEAG Operation
Phase below VLPhase above VL
0
2
1
“Previous VL” = 2
The influence of a conflict is
proportional to its level
t
Dynamic Aggregation: Conclusions
Local operation is possible – in dynamic systems – that solve inherently global problems
MultI-LEAG delivers periodic correct snapshots at minimal costDynI-LEAG responds immediately to input changes with a slightly higher message rate
Scalable Load-Distance Balancing
Edward Bortnikov, Israel Cidon, Idit Keidar
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
Distributed Setting
SynchronousDistributed 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)
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”?
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
Uniform Load
Skewed Load
Skewed Load
Load-Balance
Load-Balance
Skewed LoadLoad-Balance
Tree Clustering
As long as some cluster has improvable cost– Double it (merge with
hierarchy neighbor)
Clusters aligned at 2i indicesSimple, O(log N) convergence time
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 -)
Scalability: Cost
cost = Euclidian distance + linear load
Scalability: Locality
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