1 Distributed Selfish Replication Nikolaos Laoutaris Orestis Telelis Vassilios Zissimopoulos Ioannis Stavrakakis Department of Informatics and Telecommunications, University of Athens, Greece

2 A Distributed replication group (Leff et al., IEEE TPDS ‘93) vjvj trtr tsts tltl origin server group C j : v j ’s storage capacity r ij : v j ’s request rate for obj. o i access cost: t l <t r < t s n nodes Ν objects Applications Content distribution Shared memory Network file systems

3 Two main issues to address Object placement which objects to replicate in each node? …will be the focus of this talk Request routing how to find a node that replicates the requested object? … our object placement solution facilitates perfect routing routing to the closest node that’s holding the object

4 Two popular obj. placement strategies Socially Optimal (SO) placement strategy minimizes the average access cost in the entire group requires complete information (all request vectors) and a centralized algorithm Leff et al.: SO by casting the object placement problem as a capacitated transportation problem (polynomial complexity) SO appropriate under a single authority (e.g., CDN operator) Greedy Local (GL) placement strategy each node acting in isolation (completely uncooperative) node v j replicates the C j most popular objects according to the local demand r j requires only local information (the local request vector)

5 What happens when nodes are selfish? a selfish node: seeks to minimize its local access cost is a better model for applications with: multiple/independent authorities e.g., P2P, distributed web-caching our main research goal will be to: “Find appropriate object placement strategies for distributed replication groups of selfish nodes”

6 Why not use SO or GL? the SO strategy: can mistreat some nodes (example coming next) requires transmitting too much information the GL strategy: being uncooperative leads to poor performance

7 Mistreatment under SO group an over- active node 10 reqs/sec 1000 reqs/sec SO replicates the most popular objects locally (smaller id-> greater popularity) uses the storage capacity of all other nodes to replicate the next most popular ones these nodes end up replicating potentially irrelevant objects. They are mistreated by SO “I can do better by following GL” (replicate objs 1,2,3,4) “Lets get out of here!” … mistreated nodes pursue GL and the group disintegrates

8 The problem with nodes following GL Poor performance under common scenarios Uncooperativeness is harmful to both the social and the local utility Lets assume that the nodes: have similar demand patterns are adjacent (t r  t l ) then fetching an object locally or remotely costs the same If all nodes follow GL: they will be replicating the same few objects multiple times this is inefficient. Clearly they can do much better by: replicating different objects, and fetching the missing ones from their (adjacent) neighbors

9 The bottom line… Seems that a selfish node faces a deadlock (1) it cannot blindly trust the SO strategy because SO might mistreat him (2) it is not satisfied with the potentially poor performance of the (uncooperative) GL Research question: How can we claim the (freely) available “cooperation gain” without risking a mistreatment and do that without complete information?

10 The Equilibrium (EQ) placement strtgy is our approach for breaking the deadlock fills the gap between SO and GL in both: performance (access cost) required amount of information is based on the concept of pure Nash equilibrium from game theory forbids the mistreatment of any one node all nodes do at least as good as GL and typically much better (cooperation driven by selfish motives) requires the exchange of a small amount of information no reason for a node to abandon the group then

11 The Distributed Selfish Replication (DSR) game nodes  players n players local placements  strategies player v j can choose among (N choose C j ) possible strategies global placement  outcome of the game global placement=sum of the individual local placements reduction of access cost  payoff function DSR is a non-cooperative, non-zero-sum, n-player game pure Nash equilibria?

12 Our approach for finding EQ strategies for the DSR game starting with the DSR game in normal form we assume that nodes act sequentially following some pre-defined order (v 1,v 2,…,v n ) this resembles an extensive game formulation we use the ordering as a device for finding pure Nash equilibrium strategies for the original DSR game … in a distributed manner without requiring complete information

13 Our first algorithm: TSLS Two Step Local Search Step 0 (initialization): each node computes its GL placement g ij = r ij (t s -t l ),if o i not replicated in another node r ij (t r -t l ), if o i replicated in another node distance reduction with respect to the previous closer copy incomplete information only the strategies are revealed but not the payoff functions Step 1 (improvement): nodes line up; node v j : “observes” the placements of the other nodes proceeds to improve its GL placement according to the following definition of “excess gain”

14 TSLS (continued) each node solves a 0/1 Knapsack problem unit-weight objects, value g ij, integral knapsack capacity greedy solution  optimal at the end of Step 1 of TSLS -> Nash eq. plcmnt no node can benefit unilaterally proof: v j ’s OPT placement at the time of its turn to improve: remains OPT until the end of TSLS despite the changes performed from nodes that follow v j only multiple objects are evicted during Step 1 only unrepresented objects are inserted during Step 1 so a node might exchange some multiple objects from its GL placement with unrepresented ones

15 Comments on the use of ordering TSLS without ordering may never converge to an EQ placement nodes inserting/evicting the same objects indefinitely impact of ordering on individual gains: sometimes a certain turn (higher or lower) gives an advantage to a node identifying the OPT turn for a node requires knowing the remote payoff functions (not possible) when demand patterns (thus the payoffs also) are alike -> then higher turns (towards the end of Step 1) are better simple “merit based” protocol for deciding turns more important nodes getting a better turn

16 Eliminating the impact of ordering Suppose that the nodes are identical same capacity, demand pattern, request rate TSLS+”merit-based” protocol give some nodes an advantage (better turn) hard to justify since: nodes are identical thus lack any kind of difference in merit We would like to have an algorithm where: a node’s turn does not have a large impact on the amount of gain that it gets

17 TSLS(k): improving the TSLS fairness Same as TSLS but: at Step 1 -> up to k changes allowed k (multiple) objects belonging to the GL placement substituted by k (unrepresented) ones if more changes are desirable a node has to wait for the next round TSLS(k) requires multiple rounds to converge to EQ we show that convergence is guaranteed for small k  a node’s has a diminishing effect on the amount of gain it receives for large k  TSLS(k) reduces to TSLS

18 Distributed protocol Decide turn according to “merit” e.g., jth largest node getting the jth better turn Phase 0: compute GL placements all nodes in parallel each node to multicast its own Phase 1: improve the GL placements nodes lining up each one improving its GL plcmnt and multicasting the differences 1 round for TSLS, M rounds for TSLS(k) M  ceil(C max /k)

19 Main benefit  reduced information centralized algorithm has to send up to n*N (obj. id, obj. rate) pairs to a central node our protocol transmits up to Σ C j obj. ids large reduction on the amount of info sent typically Σ C j << N obj ids encoded easily (can use Bloom filters) (obj. id, obj. rate) pairs harder to represent to represent all the rate vectors aggregate storage capacity known placements  perfect routing

20 Example n=2, N=100, C 1 = C 2 =40, Zipf-like(0.8) demand, t l =0, t r =1, t s =2, ρ 1 =1

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22 Wrap up many content distribution applications involve selfish nodes previous socially optimal object placement solutions not suitable new EQ strategies: avoid mistreatment problems harness the freely available cooperation gain require limited information to be implemented only the local demand pattern remote placements (but not the remote demands)

23 The end Q ?