Distributed Object Location in a Dynamic Network

1
Distributed Object Location in a Dynamic Network

• Kirsten Hildrum, John D. Kubiatowicz, Satish Rao
  and Ben Y. Zhao

2
DOLR(Decentralized Object Location and Routing)
3
Requirements

• Deterministic location
• Routing locality routes should have low stretch
• Minimal storage and computational load
• Dynamic membership

4
Main Results
System Stretch Storage Node Insertion Metric
PRR, 1997 O(1) O(log n) -- Special
PRR this O(1) O(log n) O(log2 n) Special
Tapestry, 2000 Low in practice O(log n) -- General
Tapestry this Low in practice O(log n) O(log2 n) General
5
Additional Results

• Handle deletes
• Simultaneous insertions
• Algorithm with polylog stretch for general
  metrics.
• Similar to Bourgains use of subsets to embed a
  general metric into L2

6
Outline

• Background and previous work
• PRR/Tapestry
• Insertion
• Maintaining deterministic location
• Low stretch A nearest neighbor algorithm
• Related Work
• Conclusion

7
Neighbor Table
1
NodeID 2245
8
Basic Tapestry Mesh(from PRR97)
9
Use of Mesh for Object Location
10
Surrogate Routing

• Neighbor table will have holes
• Must be able to find unique root for every object
• Our solution try next highest.
• For 2176, try 2177
• For 213? Try 214?
• Need all Ø really are Ø -gt root unique.

11
Outline Again
12
Insertion

• Build own neighbor table
• Want Closest for each entry (note cannot copy
  from nearby node)
• Has to add self to other tables
• Fill in holes
• May replace further node

1
13
Finding the Closest Node

• Building table means finding nearest neighbor for
  each entry.
• Our algorithm find nearest level-i node for
  every i in process of finding overall nearest
• Easily extend to find other entries

14
Network Assumption

• Nearest neighbor is hard in general metric
• Assume the following
• Ball of radius 2r contains only a factor of c
  more nodes than ball of radius r.
• Also, b gt c2
• Both assumed by PRR
• Start knowing one node allow distance queries

15
Algorithm Idea

• Call a node a level i node if it matches the new
  node in i digits.
• The whole network is contained in forest of trees
  rooted at highest possible imax.
• Let listimax contain the root of all trees.
  Then, starting at imax, while i gt 1
• listi-1 getChildren(listi)
• Certainly, listi contains level i neighbors.

16
We Reach The Whole Network
NodeID 0xEF34
17
The Real Algorithm

• Simplified version ALL nodes in the network.
• But far away nodes are not likely to have close
  descendents
• ? Trim the list at each step.
• New version while i gt 1
• Listi-1 getChildren(listi)
• Trim(listi-1)

18
How to Trim

• Consider circle of radius r with at least one
  level i node.
• Level-(i-1) node in little circle must must point
  to a level-i node in the big circle
• Want listi had radius three times listi-1
  and listi 1 contains one level i

19
Animation
20
True in Expectation

• Want listi had radius three times listi-1
  and listi 1 contains one level i
• Suppose listi-1 has k elements and radius r
• Expect ball of radius 4r to contain kc2/b
• Ball of radius 3r contains less than k nodes, so
  keeping k all along is enough.
• To work with high probability, k O(log n)

21
Steps of Insertion

• Find node with closest matching ID (surrogate)
  and get preliminary neighbor table
• If surrogates table is hole-free, so is this
  one.
• Find all nodes that need to put new node in
  routing table via multicast
• Optimize neighbor table
• w.h.p. contacted nodes in building table only
  ones that need to update their own tables
• Need
• No fillable holes.
• Keep objects reachable

22
Need-to-know nodes

• Need-to-know a node with a hole in neighbor
  table filled by new node
• If 1234 is new node, and no 123s existed, must
  notify 12?? Nodes
• Acknowledged multicast to all matching nodes

23
Acknowledged Multicast Algorithm
Locates Contacts all nodes with a given prefix

• Create a tree based on IDs as we go
• Nodes send acks when all children reached
• Starting node knows when all nodes reached

The node then sends to any 5430?, any 5431?, any
5434?, etc. if possible
543??
5431?
5434?
54340
54345
24
Outline Again
25
Related Work

• Many peer-to-peer systems
• Not locality-aware
• Chord Stoica, Morris, Karger, Kaashoek,
  Balakrishnan
• CAN Ratnasamy, Francis, Handley, Karp, Shenker
• Pastry Rowstron and Druschel
• Polylog stretch, general metric
• Awerbuch and Peleg 1991 (not dynamic)
• Rajaraman, Richa, Vöcking, Vuppuluri 2001
• Constant stretch, special metric
• Plaxton, Rajaraman, Richa 1997 (not dynamic)
• Nearest Neighbor
• Karger and Ruhls 2002
• Thorup and Zwick 2001
• Clarkson 1997

26
Future Work

• Getting stretch is low in practical systems
• Transit stub networks
• Network structure inside stub
• Periodic optimization
• Better handling of inserts in presence of
  deletes

27
Extra/Backup Slides
28
Related Work
System Node Insertion Stretch Hops Metric
CAN, 2001 O(r) -- O(rn1/r) General
Chord, 2001 O(log2 n) -- O(log n) General
Pastry, 2001 O(log2 n) -- O(log n) General
Awerbuch Peleg, 1991 -- polylog polylog General
PRR, 1997 -- O(1) O(log n) Special
RRVV, 2001 polylog polylog polylog General
PRR this O(log2 n) O(1) O(log n) Special
Tapestry O(log2 n) Low in practice O(log n) General
29
Maintaining Object Availability

• Object requests continue
• New node resends requests for object it does not
  have via surrogate
• Established node always checks that that it is
  final destination.
• Sometimes-checking would require state.
• May get cycle for non-existent objects, but can
  be detected.

30
Body

• Finding the nearest neighbor
• Can query for distance between self and any given
  node.
• Assume a low expantion property (low dimentional)
• Karger and Ruhl, May 2002.
• Our algorithm not as good for general problem,
  but solves our particular problem better
• No additional storage needed
• Finds all levels of table

31
Picture
32
Introduction

• Object Location/DOLR slide
• Related Work table
• Not much done with stretch
• Our results
• Add insertion to PRR/Tapestry
• Deletes
• Parallel inserts
• Solve nearest neighbor problem inprocess
• Similar to Karger and Ruhl
• Result for general metric spaces