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Replication Strategies in Unstructured Peer-to-Peer Networks

Replication Strategies in Unstructured Peer-to-Peer Networks. Edith Cohen AT&T Labs-research. Scott Shenker ICIR. Peer-to-peer Networks. Peers are connected by an overlay network. Users cooperate to share files (e.g., music, videos, etc.). (Search in) Basic P2P Architectures.

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Replication Strategies in Unstructured Peer-to-Peer Networks

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  1. Replication Strategies in Unstructured Peer-to-Peer Networks Edith Cohen AT&T Labs-research Scott Shenker ICIR Sigcomm 2002

  2. Peer-to-peer Networks • Peers are connected by an overlay network. • Users cooperate to share files (e.g., music, videos, etc.) Sigcomm 2002

  3. (Search in) Basic P2P Architectures • Centralized: central directory server. (Napster) Supports versatile queries, scope, legal troubles • Decentralized: search is performed by probing peers • Structured (DHTs): (Freenet, Can, Chord,…)location is coupled with topology - search is routed by the query.Scope, Only exact-match queries, tightly controlled overlay. • Unstructured:(Gnutella, FastTrack); search is “blind” - probed peers are unrelated to query. Resilient to transient peers; versatile queries; Harsh scope/scalability tradeoff. Sigcomm 2002

  4. (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) Sigcomm 2002

  5. Question: how to use replication to improve search efficiency in unstructured networks with a proactive replication mechanism ? Sigcomm 2002

  6. Search and replication model 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: probe hosts, uniformly at random, until the query is satisfied (or the search max size is exceeded) • Replication: Each host can store up to r copies (or keys=metadata+pointer) of items. Goal: minimize average search size (number of probes till query is satisfied) Sigcomm 2002

  7. Search size • Query is soluble if there are sufficiently many copies of the item. • Query is insoluble if item is rare or non existent. What is the search size of a query ? • Insoluble queries: maximum search size • Soluble queries: number of probes until 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. Sigcomm 2002

  8. Search Example 2 probes 4 probes Sigcomm 2002

  9. Expected Search Size (ESS) • m items with relative query rates • q1> q2> q3> … > qm. Siqi = 1 • Allocation : p1, p2, p3,…, pmSipi = 1 ith item is allocated pi fraction of storage. (keys placed in pir fraction of hosts) • Search size for ith item is a Geometric r.v. with mean Ai = 1/(r pi ). • ESS is SiqiAi = (Siqi / pi)/r Sigcomm 2002

  10. Example: 3 items, q1=1/2, q2=1/3, q3=1/6 Uniform Proportional 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 Sigcomm 2002

  11. 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 ? Sigcomm 2002

  12. 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 … Sigcomm 2002

  13. ESS under Uniform and Proportional Allocations (soluble queries) • Lemma: The ESS under either Uniform or Proportional allocations is m/r • Independent of query rates (!!!) • Same ESS for Proportional and Uniform (!!!) • Proof… Proportional: ASS is(Siqi / pi)/r = (Siqi / qi)/r = m/r Uniform: ASS is(Siqi / pi)/r = (Sim qi)/r = (m/r) Si qi = m/r Sigcomm 2002

  14. Proportional and Uniform are the worst “reasonable” strategies (!!!) Space of Possible Allocations Definition: Allocation p1, p2, p3,…, pm 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, p i+1/p 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”) Sigcomm 2002

  15. Worse than prop/uni More popular item gets less. SR Better than prop/uni Worse than prop/uni More popular gets more than its proportional share Space of allocations on 2 items Uniform Proportional p2/p1 q2/q1 Sigcomm 2002

  16. So, what is the best strategy for soluble queries ? Sigcomm 2002

  17. Square-Root Allocation pi is proportional to square-root(qi) • Lies “In-between” Uniform and Proportional • Theorem: Square-Root allocation minimizes the ESS (on soluble queries) • Minimize Siqi / pi such thatSipi = 1 Sigcomm 2002

  18. How much can we gain by using SR ? Zipf-like query rates Sigcomm 2002

  19. OPT is a hybrid of Uniform and SR Tuned to balance cost of soluble and insoluble queries. • OK • SR is best for soluble queries • Uniform minimizes cost of insoluble queries What is the optimal strategy? Sigcomm 2002

  20. 10^4 items, Zipf-like w=1.5 All Soluble 85% Soluble All Insoluble SR Uniform Sigcomm 2002

  21. We now know what we need. How do we get there? Sigcomm 2002

  22. Replication Algorithms • 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 • 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. Desired properties of algorithm: Sigcomm 2002

  23. Property of the process: <Ci> average value of C used to replicate ith item. Claim: If <Ci>/<Cj> remains fixed over time, and <Ci>, <Cj> >e, then pi/pjg qi <Ci>/qj <Cj> 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) Sigcomm 2002

  24. Algorithm for square-root allocation needs to have <Ci> equal to or converge to a value inversely proportional to Creation/Deletion Process Corollary: If then Sigcomm 2002

  25. 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). Sigcomm 2002

  26. Alg1: 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 Sigcomm 2002

  27. Simulation Delay = 0.25 * copy lifetime; 10000 hosts Path replication Sibling number Hosts with copy time Sigcomm 2002

  28. 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. Sigcomm 2002

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