dynamic load balancing in distributed hash tables

IPTPS 2005IPTPS 2005 Mirek Korzeniowski:

International Graduate School of Dynamic Intelligent Systems

HEINZ NIXDORF INSTITUTUniversity of Paderborn

Algorithms and Complexity

Dynamic Load Balancing in DHTsDynamic Load Balancing in DHTs

Dynamic Load Balancing in Distributed Hash Tables

Marcin Bieńkowski

Mirosław Korzeniowski

Friedhelm Meyer auf der Heide






Ring / Hypercube Topologies

• Chord, Tapestry, Pastry, de Bruijn• Space: ring [0,1]• Arithmetics mod 1• Links to:

– predecessor and successor– shortcuts

• Degree: log n• Diameter: log n (or log n / log log n)• Balance: (log n) • Smoothness (ratio of the longest and shortest interval): (n¢log n)

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Balls into Bins

If we throw n balls into n bins uniformly at random, the fullest bin will have (log n / log log n) balls.

If we throw n¢ log n balls into n bins, each bin will have (log n) balls

If we throw n balls sequentially into n bins and each ball chooses the emptiest of log n randomly chosen bins, each bin will have (1) balls






Previous Results

Recent publications concerning balancing in P2P:– Moni Naor, Udi Wieder „Novel Architectures for P2P

Applications: the Continous-Discrete Approach” SPAA 2003– Micah Adler, Eran Halperin, Richard Karp, Vijay Vazirani „A

stochastic process on the hypercube with appications to peer-to-peer networks” STOC 2003

• Both optimal • Both sequential






Previous Results

Recent publications concerning balancing in P2P:– David Karger, Matthias Ruhl, Simple Efficient Load

Balancing Algorithms for Peer-to-Peer Systems, SPAA 2004• each join / leave operation makes log log n nodes

migrate– Gurmeet Singh Manku, Balanced Binary Trees for ID

Management and Load Balance in Distributed Hash Tables, PODC 2004

• a general algorithm, needs locally centralized decisions






Assumptions

• We know:– n (the number of nodes) def. := 1/n– or at least log n– or at least an upper bound on log n

• No interaction during rebalancing• Any initial state of the network• The network organises itself and implements:

– join(where)– leave– insert(hash)






Lazy nodes leave

First phase:if (length < 1/10¢) leave

Minimizing balance = minimizing smoothness

Balance = the longest intervalSmoothness = ratio between the longest and the

shortest interval






Continous scheme

All the next phases:if (length < 4¢) either propose(random) or notif (length > 10¢) halve using the first proposal else forward (log n) times

Analysis (sketch)• There are (n) proposals • Interval longer than (¢log n) receives enough

proposals in one step (w.h.p.)• Interval shorter than (¢log n) receives enough

proposals from its predecessors






To Propose or Not to Propose

Many consecutive short intervals:

L R L R L R L R L R L R L R

L L R L L R R R L L R L L L






Continous scheme

All the next phases:if (length < 4¢) either propose(random) or notif (length > 10¢) accept(length / 10)if (length < 10¢) forward(log n)

Analysis (sketch)• There are (n) proposals • Interval longer than (¢log n) receives enough

proposals in one step (w.h.p.)• Interval shorter than (¢log n) receives enough

proposals from its predecessors






Managing the volunteers – 1st phase






Managing the volunteers – 2nd phase






Estimating n and

• Choose global k > (log n)• Each node inserts k markers into the network (n¢log n) markers are distributed uniformly• An interval of length contains (k) items w.h.p.• Instead of lengths we consider the number of

markers (weights)

• If a node should insert a normal item it replaces a marker with it ) load increased only if it was low






Results

• A scheme that– copes with insertions and deletions– works in constant number of rounds– each round takes logarithmic time– has optimal number of migrations(up to a constant

factor)– each node migrates at most once– has bounded communication per half-life






Dynamic Load Balancing in Distributed Hash Tables

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dynamic load balancing in distributed hash tables

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