1. IPTPS 2005 Mirek Korzeniowski: Page 1 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Dynamic Load Balancing in Distributed Hash Tables Marcin Bieńkowski Mirosław Korzeniowski Friedhelm Meyer auf der Heide

2. IPTPS 2005 Mirek Korzeniowski: Page 2 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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).12.2.23.25.3.32.4.45.5 ServerKey.1

3. IPTPS 2005 Mirek Korzeniowski: Page 3 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 4 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 5 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 6 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 7 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 8 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Continous scheme All the next phases: if (length < 4¢  ) either propose(random) or not if (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

9. IPTPS 2005 Mirek Korzeniowski: Page 9 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 10 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Continous scheme All the next phases: if (length < 4¢  ) either propose(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. IPTPS 2005 Mirek Korzeniowski: Page 11 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Managing the volunteers – 1st phase

12. IPTPS 2005 Mirek Korzeniowski: Page 12 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Managing the volunteers – 2nd phase

13. IPTPS 2005 Mirek Korzeniowski: Page 13 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 14 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs 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. IPTPS 2005 Mirek Korzeniowski: Page 15 International Graduate School of Dynamic Intelligent Systems HEINZ NIXDORF INSTITUT University of Paderborn Algorithms and Complexity Dynamic Load Balancing in DHTs Dynamic Load Balancing in Distributed Hash Tables Thank you for your attention