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Replication Strategies in Unstructured Peer-to-Peer Networks Edith Cohen Scott Shenker This is a modified version of the original presentation by the authors

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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.

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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)

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

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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.

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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.

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Search Example 2 probes 4 probes

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Notations m items with relative query rates n nodes (peers), each has a uniform capacity R = n is the total available space r i = number of copies of item i. Thus p i = r i /R is the fraction of the total space allocated to item i. i p i = 1 q i = normalized query rate for item i. Thus i q i = 1

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Notations Allocation p = (r 1 /R, r 2 /R, …, r m /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)

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Expected Search Size (ESS) Allocation : p 1, p 2, p 3,…, p m i p i = 1 r i /n = pi is the fraction of hosts storing a copy of i m items with relative query rates q 1 > q 2 > q 3 > … > q m. i q i = 1 Search size for i th item is a geometric r.v. with mean Ai = 1/ ( p i ). ESS is i q i A i = ( i q i / p i )/

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Uniform and Proportional Replication Two natural strategies: Uniform Allocation: p i = 1/m Simple, resources are divided equally Proportional Allocation: p i = q i “Fair”, resources per item proportional to demand Reflects current P2P practices

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Uniform and Proportional Replication Example: 3 items, q 1 =1/2, q 2 =1/3, q 3 =1/6 q1 > q2 > q3 Uniform Proportional

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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 ?

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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 …

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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 (!!!)

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Proof outline Proportional: Average Search Size is ( i q i / p i )/ ( i q i / q i )/ m/ Uniform: Average Search Size is ( i q i / p i )/ ( i m q i )/ m/ i q i m/

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Space of Possible Allocations Definition: Allocation p 1, p 2, p 3,…, p m is “in-between” Uniform and Proportional if for 1 i < m, q i+1 /q i < p i+1 /p i < 1 Theorem1: 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 (!!!)

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q2/q1 p2/p1 Space of allocations on 2 items Worse than prop/uni More popular item gets less. Worse than prop/uni More popular gets more than its proportional share Better than prop/uni Uniform Proportional SR

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So, what is the best strategy for soluble queries ?

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Square-Root Allocation (p i ) is proportional to square-root of (q i ) Lies “In-between” Uniform and Proportional Theorem: Square-Root allocation minimizes the ESS (on soluble queries) Minimize i q i / p i such that i p i = 1

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How much can we gain by using SR ? Zipf-like query rates

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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?

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UniformSR 10^4 items, Zipf-like w=1.5 All Soluble 85% Soluble All Insoluble

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We now know what we need. How do we get there?

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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:

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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) average value of C used to replicate i th item. Claim: If / remains fixed over time, and, , then p i /p j q i /q j Property of the process:

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Creation/Deletion Process If then Corollary: Algorithm for square-root allocation needs to have equal to or converge to a value inversely proportional to

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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).

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Algorithm 1: Path Replication Number of new copies produced per query,, is proportional to search size 1/p i Creation rate is proportional to q i Steady state: creation rate proportional to allocation p i, thus

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Simulation Path replication Sibling number Hosts with copy time Delay = 0.25 * copy lifetime; hosts

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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.

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