Dynamic Load Balancing in Distributed Hash Tables

1. Dynamic Load Balancing in Distributed Hash Tables Marcin Bieńkowski Mirosław Korzeniowski Friedhelm Meyer auf der Heide

2. Ring / Hypercube Topologies .1 • 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) .12 .2 .23 .25 .3 .32 .5 .4 .45 Server Key

3. 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

4. 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 

5. 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

6. 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)

7. Lazy nodes leave First phase: if (length < 1/10¢) leave Minimizing balance = minimizing smoothness Balance = the longest interval Smoothness = ratio between the longest and the shortest interval

8. Continous scheme All the next phases: if (length < 4¢) eitherpropose(random) or not if (length > 10¢) halve using the first proposal elseforward (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

9. 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

10. Continous scheme All the next phases: if (length < 4¢) eitherpropose(random) or not if (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

11. Managing the volunteers – 1st phase

12. Managing the volunteers – 2nd phase

13. 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

14. 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

15. Dynamic Load Balancing in Distributed Hash Tables Thank you for your attention