Replication Strategies in Unstructured Peer-to-Peer Networks

Edith Cohen Scott Shenker

This is a modified version of the original presentation by the authors

Search in Basic P2P Architectures

• Centralized: central directory server. (Napster)

• Decentralized:– Structured (DHTs): Only exact-match queries, tightly

controlled overlay.– Unstructured: (Gnutella, FastTrack); search is “blind” -

probed peers are unrelated to query.

Replication in P2P architectures

• No proactive replication (Gnutella)– Hosts store and serve only what they requested– A copy can be found only by probing a host with a

copy• Proactive replication of “keys” (= meta data +

pointer) for search efficiency (FastTrack, DHTs)• Proactive replication of “copies” – for search

and download efficiency, anonymity. (Freenet)

QUESTION How to use replication to improve search efficiency in unstructured networks with a proactive replication mechanism ?

Search and replication model

• Search: probe hosts, uniformly at random, until the query is satisfied (or the search max size is exceeded)

Goal: minimize average search size (number of probes till query is satisfied)

• Replication: Each host can store up to copies (or keys=metadata+pointer) of items.

Unstructured networks with replication of keys or copies. Peers probed (in the search and replication process) are unrelated to query/item - Probe success likelihood can not be better, on average, than random probes.

Search size

What is the search size of a query ?• Insoluble queries: maximum search size• Soluble queries: number of nodes a query need to visit

until the answer is found. We look at the Expected Search Size (ESS) of each item.

The ESS is inversely proportional to the fraction of peers with a copy of the item.

• Query is soluble if there are sufficiently many copies of the item.

• Query is insoluble if item is rare or non existent.

Search Example

2 probes 4 probes

Notations• m items with relative query rates • n nodes (peers), each has a uniform capacity • R = n is the total available space• ri = number of copies of item i. Thus pi = ri/R is the

fraction of the total space allocated to item i.

i pi = 1• qi = normalized query rate for item i. Thus

i qi = 1

Notations• Allocation p = (r1/R, r2/R, …, rm/R)

• A replication strategy is a mapping from q to p.

• Assumption R ≥ m ≥ . (If m < , then one can copy every item in all the nodes. If R < m then no allocation can store a copy of all m objects)

Expected Search Size (ESS)

• Allocation : p1, p2, p3,…, pm i pi = 1• ri/n = pi is the fraction of hosts storing a copy of i

• m items with relative query rates

q1 > q2 > q3 > … > qm. i qi = 1

• Search size for ith item is a geometric r.v. with mean Ai = 1/( pi).

• ESS is i qi Ai = (i qi / pi)/

Uniform and Proportional Replication

Two natural strategies:• Uniform Allocation: pi = 1/m

•Simple, resources are divided equally

• Proportional Allocation: pi = qi•“Fair”, resources per item

proportional to demand• Reflects current P2P practices

Uniform and Proportional Replication

Example: 3 items, q1=1/2, q2=1/3, q3=1/6

q1 > q2 > q3

Uniform Proportional

Basic Questions• How do Uniform and Proportional allocations

perform/compare ?• Which strategy minimizes the Expected Search

Size (ESS) ?• Is there a simple protocol that achieves

optimal replication in decentralized unstructured networks ?

Insoluble queries• Search always extends to the maximum allowed

search size.• If we fix the available storage for copies, the

query rate distribution, and the number if items that we wish to be “locatable”, then

• The maximum required search size depends on the smallest allocation of an item. Thus,

• Uniform allocation minimizes this maximum and thus the cost induced by insoluble queries.

What about the cost of soluble queries? Answer is more surprising …

Uniform and Proportional Allocations (ESS for soluble queries)

Lemma: The ESS under either Uniform

or Proportional allocations is m/

– Independent of query rates (!!!)

– Same ESS for Proportional and Uniform

(!!!)

Proof outline

Proportional: Average Search Size is

(i qi / pi)/(i qi / qi)/m/

Uniform: Average Search Size is

(i qi / pi)/(i m qi)/m/i qi m/

Space of Possible AllocationsDefinition: Allocation p1, p2, p3,…, pm is “in-between” Uniform

and Proportional if for 1i <m, q i+1/q i < p i+1/p i < 1Theorem1: All (strictly) in-between strategies are (strictly)

better than Uniform and Proportional

Theorem2: p is worse than Uniform/Proportional if for all i, q i+1/q i > 1 (more popular gets less) OR for all i, q i+1/q i > p i+1/p i (less popular gets less than “fair share”)

(These are unreasonable strategies)

Proportional and Uniform are the worst “reasonable” strategies (!!!)

q2/q1

p2/p1

Space of allocations on 2 items

Worse than prop/uniMore popular item gets less.

Worse than prop/uni

More popular gets more thanits proportional share

Better than prop/uni

Uniform

Proportional

SR

So, what is the best strategy for soluble queries ?

Square-Root Allocation(pi) is proportional to square-root of (qi)

m

jj

ii

q

qp

1

• Lies “In-between” Uniform and Proportional

• Theorem: Square-Root allocation minimizes the ESS (on soluble queries)

Minimize i qi / pi such that i pi = 1

How much can we gain by using SR ?w

i iq Zipf-like query rates

OK• SR is best for soluble queries• Uniform minimizes cost of insoluble queries

OPT is a hybrid of Uniform and SR

Tuned to balance cost of soluble and insoluble queries.

What is the optimal strategy?

UniformSR

10^4 items, Zipf-like w=1.5

All Soluble

85% Soluble

All Insoluble

We now know what we need.

How do we get there?

Replication Algorithms

• Fully distributed where peers communicate through random probes; minimal bookkeeping; and no more communication than what is needed for search.

• Converge to/obtain SR allocation when query rates remain steady.

• Uniform and Proportional are “easy” :– Uniform: When item is created, replicate its key in a fixed

number of hosts.– Proportional: for each query, replicate the key in a fixed

number of hosts

Desired properties of algorithm:

Model for Copy Creation/Deletion

• Creation: after a successful search, C(s) new copies are created at random hosts.

• Deletion: is independent of the identity of the item; copy survival chances are non-decreasing with creation time. (i.e., FIFO at each node)

<Ci> average value of C used to replicate ith item.Claim: If <Ci>/<Cj> remains fixed over time, and <Ci>, <Cj> , then pi/pj qi <Ci>/qj <Cj>

Property of the process:

Creation/Deletion Process

If i

i qC 1 jiji qqpp then

Corollary:

Algorithm for square-root allocation needs to have <Ci> equal to or converge to a value inversely proportional to iq

SR Replication Algorithms

• Path replication: number of new copies C(s) is proportional to the size of the search (Freenet)– Converges to SR allocation (+reasonable conditions)– Convergence unstable with delayed creations

• Sibling memory: each copy remembers the number of sibling copies,– Quickly “on target”– For “good estimates” need to find several copies.

• Probe memory: each peer records number and combined search size of probes it sees for each item. C(S) is determined by collecting this info from number of peers proportional to search size. – Immediately “on target”– Extra communication (proportional to that needed for search).

Algorithm 1: Path Replication• Number of new copies produced per query, <Ci>, is

proportional to search size 1/pi

• Creation rate is proportional to qi <Ci>• Steady state: creation rate proportional to allocation pi,

thus

iiiii ppqCq

ii qp

Simulation

Path replicationSibling number

Hosts with copy

time

Delay = 0.25 * copy lifetime; 10000 hosts

Summary• Random Search/replication Model: probes to “random”

hosts• Proportional allocation – current practice• Uniform allocation – best for insoluble queries• Soluble queries:

• Proportional and Uniform allocations are two extremes with same average performance

• Square-Root allocation minimizes Average Search Size• OPT (all queries) lies between SR and Uniform• SR/OPT allocation can be realized by simple algorithms.