1 Replication Strategies in Unstructured Peer-to-Peer Networks Edith Cohen, Scott Shenker ACM SIGCOMM Computer Communication Review, Proceedings of the 2002 conference on Applications, technologies, architectures, and protocols for computer communications, vol. 32 issue 4 Presentation by Tony Sung, MC Lab, IE CUHK 16th December 2004

2 Introduction What is an Unstructured P2P Network?  Centralized  Decentralized Structured Unstructured

3 Introduction Locating Objects in an Unstructured P2P Network  Probing  How to Reduce Probe Count? No Probing is better than Random Probing By Replication

4 Introduction Current Replication Strategies … Implicit Objective of the Paper: “ Designs an explicit replication strategy. ” “ What is the optimal way to replicate data? ”

5 Introduction Two Starting Points Uniform Replication Proportional Replication

6 Paper’s Outline  Introduction  Model and Problem Statement Defining an Allocation and the Expected Search Size Bounded Search Size and Insoluble Queries Heterogeneous Capacities and Bandwidth  Allocation Strategies Uniform and Proportional Characterizing Allocations Between Uniform and Proportional  The Square-root Allocation How much we can gain?  Square-root* and Proportional* Allocations Square-root* Allocation Proportional* Allocation  Distributed Replication Path Replication Replication with Sibling-number Memory Replication with Probe Memory Obtaining the Optimal Allocation Simulations  Conclusion

7 Today’s Outline  Introduction  Model and Problem Statement Defining an Allocation and the Expected Search Size Bounded Search Size and Insoluble Queries Heterogeneous Capacities and Bandwidth  Allocation Strategies Uniform and Proportional Characterizing Allocations Between Uniform and Proportional  The Square-root Allocation How much we can gain?  Square-root* and Proportional* Allocations Square-root* Allocation Proportional* Allocation  Distributed Replication Path Replication Replication with Sibling-number Memory Replication with Probe Memory Obtaining the Optimal Allocation Simulations  Conclusion

8 Model & Problem Statement n nodes capacity ρ total capacity R = nρ query rate q = q 1 ≥ q 2 ≥ … ≥ q m Σq i = 1 m distinct data replica r 1 r2r2 rmrm Σr i = R allocation p = (r 1 /R, r 2 /R, …, r m /R) allocation strategy: q → p

9 Model & Problem Statement n nodes capacity ρ total capacity R = nρ m distinct data query rate q = q 1 ≥ q 2 ≥ … ≥ q m replica r 1 r2r2 rmrm allocation p = (r 1 /R, r 2 /R, …, r m /R) bounds for m : R ≥ m ≥ρ bounds for p i : u ≥ p i ≥ l l = 1/R u = n/R = ρ -1 expected search size: optimization problem: Monotonicity:

10 Allocation Strategies, Uniform & Proportional Minimizes the required maximum search size Thus minimizes system resources spent on insoluble queries Minimizes maximum utilization rate. More relevant when the replication is of copies rather than of pointers

11 Allocation Strategies, Uniform & Proportional Expected Search Size A q (p) Uniform A q (p)= 1/ρΣ(q i /p i ) = 1/ρΣq i m = m/ρ Proportional A q (p)= 1/ρΣ(q i /p i ) = 1/ρΣ1 = m/ρ

12 Allocation Strategies, Characterizing Allocations Consider space allocations for two items p i, p j and q i, q j Range of allocation defined by x, 0 < x < 1, p i /(p i +p j ) = x p j /(p i +p j ) = (1-x) x = q i /(q i +q j ) [ Proportional] or 0.5 [Uniform] ESS proportional to q i /x + q j /(1-x) and is convex. ESS min occurs atwhich is independent of p.

13 Allocation Strategies, Characterizing Allocations Consider space allocations for two items p i, p j and q i, q j

14 Allocation Strategies, Between Uniform & Prop.

15 Allocation Strategies, Between Uniform & Prop.

16 Allocation Strategies, Short Conclusion  ESS of Uniform and Proportional Allocation is equal, and is equal to m/ρ  For one special case ( m=2 ), ESS is a convex function and is minimum for a square-root allocation  For any allocation p that lies between Uniform and Proportional, its ESS is at most m/ρ.  If p is different from Uniform or Proportional then its ESS is strictly less than m/ρ.

17 The Square-root Allocation

18 How much can we gain?  For uniform and proportional allocation, ESS= m/ρ  For Square-root allocation, ESS= (Σq i 1/2 ) 2 /ρ which depends on the query distribution  Define gain factor as ESS uniform /ESS SR It is shown that ESS uniform /ESS SR ≤ m(u + l - mlu) When l = 1/m or u = 1/m, the only legal allocation is p i = 1/m, and gain factor = 1 If l << 1/m, and gain factor is roughly mu.

19 How much can we gain?

20 How much can we gain?

21 Materials Left  Natural extension of Square-root and Proportional Allocation that are defined when l is fixed for a maximum search size. Similar Results  Distributed Replication Protocols for achieving Square-root Allocation Path replication, converges but unstable Replication with sibling-number memory, better Replication with probe memory, better Confirmed with Simulation

22 Conclusion  Modeled different replication strategies Uniform Proportional In-between, especially Square-root  Uniform and Proportional forms two extremes of all legal allocations  ESS is smaller in-between  Square-root is optimal