Download Peer-to-Peer Data Structures

Peer-to-Peer Data Structures
Christian Scheideler
Dept. of Computer Science
Johns Hopkins University
Peer-to-peer data structures
Internet
Data structures dispersed over dynamic set
of potentially large number of peers
Peer-to-Peer Data Structures
2
Outline of Talk
How to design peer-to-peer forms of
classical data structures like hash tables,
skip lists, and search trees?
How to formulate a general model for the
design of peer-to-peer data structures?
Peer-to-Peer Data Structures
3
Sequential data structures
Hash table
Skip list
2-3 tree
[Pugh ‘90]
[Hopcroft ’70]
h
How to design peer-to-peer forms of these
data structures?
Peer-to-Peer Data Structures
4
Parallel data structures
Based on PRAM model or derivatives:
Multiple processing units (1,2,…,n)
P
P ..
local computation (1 step)
.. P
read, write (1 step)
requests synchronized
Linear addressable memory
Peer-to-Peer Data Structures
5
Parallel Data Structures
Parallel hash tables:


n Insert/Delete operations: O(log * n) time
n Search operations: O(1) time
[Gil, Matias, Vishkin ’91], [Bast, Hagerup ’91]
Parallel skip lists:

n Insert/Delete/Search operations: O(log n)
[Gabarro, Martinez, Messeguer ’94]
Parallel 2-3 trees:

n Insert/Delete/Search operations: O(log n)
[Paul, Vishkin, Wagener ’83]
Peer-to-Peer Data Structures
6
Are we done??
Limitation of parallel designs: only
parallelize the way the data structure is
accessed, not the data structure itself!
Is it realistic to assume that a reliable
shared memory can be provided in a peerto-peer system?
If at all, the overhead is high!
( W(log n) time and work for read/write)
Peer-to-Peer Data Structures
7
Peer-to-peer scenario
Dynamic set of peers, no consecutive numbers
P
P ..
.. P
Reliable shared memory
Peer-to-Peer Data Structures
8
Peer-to-peer data structures
Basic requirements:
Remains well-connected even under massive failures.
Desirable: high expansion
U
N(U)
min |N(U)|
U
|U|
Low maintenance overhead.
Necessary: low degree
Allows efficient concurrent execution of operations.
Peer-to-Peer Data Structures
9
Sequential data structures
Hash table
Skip list
2-3 tree
[Pugh ‘90]
[Hopcroft ’70]
h
How to design peer-to-peer forms of these
data structures?
Peer-to-Peer Data Structures
10
Hash graphs
[Stoica, Morris, Karger, Kaashoek, and Balakrishnan ’01]
Idea: organize objects in cycle according to
hash function h:Names
[0,1).
0.28
0.43
Peer-to-Peer Data Structures
0.58
11
Hypercubic Shortcuts
1
0
+1/2
+1/4
fingers
+1/8
+1/16
x
Peer-to-Peer Data Structures
12
Hypercubic Pointer Structure
Peer-to-Peer Data Structures
13
Sequential data structures
Hash table
Skip list
2-3 tree
[Pugh ‘90]
[Hopcroft ’70]
h
How to design peer-to-peer forms of these
data structures?
Peer-to-Peer Data Structures
14
Skip Graphs
[Aspnes and Shah ’03]
Basic approach: organize objects in cycle
according to real names.
Apple
Banana
Peer-to-Peer Data Structures
Cherry
15
Skip Graphs
Shortcuts: pad each object with random bits
1
1
1
0
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
Peer-to-Peer Data Structures
0
16
Sequential data structures
Hash table
Skip list
2-3 tree
[Pugh ‘90]
[Hopcroft ’70]
h
How to design peer-to-peer forms of these
data structures?
Peer-to-Peer Data Structures
17
2-3 Graphs
[Awerbuch & Scheideler ’04]
Same basic approach: organize objects in
cycle according to real names.
Apple
Banana
Peer-to-Peer Data Structures
Cherry
18
Shortcuts: Intertwined Rings
bridge
Peer-to-Peer Data Structures
19
Insert and Delete
Inserting a node:
Peer-to-Peer Data Structures
20
Insert and Delete
Deleting a node:
Peer-to-Peer Data Structures
21
k-separated 2-3 Graph
In every level, bridges are k nodes apart.
How large does k have to be to guarantee
1/polylog expansion a?
Theorem: a = (1/2)
W(log n / k 2 )
min |N(U)|
U
N(U)
So k has to be non-constant ( W( log
).
U n )|U|
Do areas with old insertions/deletions have to be
revisited??
Peer-to-Peer Data Structures
22
k-separated 2-3 Graph
Rule: Choose k=6(d+3)
d: current degree of node initiating op.
Theorem:
degree: 2(log n +/- 2)
expansion: W(1/log n)
work for Insert/Delete: O(log3 n)
e
Similar DS but only with 1/n expansion in worst
case was proposed by [Harvey & Munro ’03]
Peer-to-Peer Data Structures
23
Outline of Talk
How to design peer-to-peer forms of
classical data structures like hash tables,
skip lists, and search trees?
How to formulate a general model for the
design of peer-to-peer data structures?
Peer-to-Peer Data Structures
24
Hierarchical vs. Flat
Internet
Peer-to-Peer Data Structures
25
How do objects find each other?
Internet
Peer-to-Peer Data Structures
26
Model (JXTA)
Application
searching, multicast
Middleware
shared space, DNS
Core
rendezvous service
Network
Peer-to-Peer Data Structures
27
Middleware Model
Actor
local computation (1 step)
leave P/M
spawn
P/M
notify (1 step for
reliable actors)
P/M
P/M
P/M
join(AID)
join (to
(un)register(AID)
random
bootstrap actor)
notify
Rendezvous
Simplified model,
sufficientservice
for most purposes
Peer-to-Peer Data Structures
28
Application Model
Common approach: shared space
may be unreliable!
Operations:
space
operations
Insert(addr, o)
Delete(addr)
Lookup(addr)
Join(peer)
Leave()
peer
operations
Available to application-level data structures
Peer-to-Peer Data Structures
29
Shared Space
Consistent hashing:
[Karger et al. ’97]
h:Addr
[0,1), g:Peers
[0,1)
Object o stored at peer p with min g(p) s.t.
h(o)<g(p)
0.34
0.28
0.43
Peer-to-Peer Data Structures
0.58
30
Shared Space
Solutions:
Chord, CAN, Pastry, Tapestry (~2001/02)
OpenHash, KBR, Bamboo (more open)
Problems:
How to deal with heterogeneous peers?
How to deal with adversarial peers?
Peer-to-Peer Data Structures
31
Heterogeneous Peers
Peers of non-uniform bandwidth:
Previous approach: cut into uniform peers, multi-tier
architecture (KaZaA)
Our approach [SPAA 2004] : PAGODA
Thm: Any concurrent multicast problem can be routed in
PAGODA with at most O(D+log n) more congestion than
with best possible network of degree D.
Peers of non-uniform memory size:
Previous approach: as above
Our approach [SPAA 2004] :
Thm: As long as peers are at most (1-e)-utilized, an O(e)
rate of insert/delete requests to peers can be sustained.
Peer-to-Peer Data Structures
32
Adversarial Peers
Assumption:
e-fraction of peers adversarial
Challenges:
Organization of peers in regions
Enforcement of random IDs
stochastic process / secure peer-to-peer random number generator
based on verifiable secret sharing
Enforcement of limited lifetime
Results: (only Join/Leave)
Trust-but-Verify [IPTPS 2004]
Group Spreading [ICALP 2004]
Enforced Spreading [submitted]
Peer-to-Peer Data Structures
33
Conclusion
In this talk:
Peer-to-peer forms of classical data structures
General model for peer-to-peer data structures
Challenges:
Data structures for unreliable shared space
Stability studies (reliable/adversarial peers)
Locally self-stabilizing data structures
Other applications: mobile ad-hoc, configware
Peer-to-Peer Data Structures
34
Questions ??