Distributed Selfish Replication

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Distributed Selfish Replication

• Nikolaos Laoutaris
• Orestis Telelis
• Vassilios Zissimopoulos
• Ioannis Stavrakakis
• laoutaris,telelis,vassilis,ioannis_at_di.uoa.gr

Department of Informatics and Telecommunications,
University of Athens, Greece
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A Distributed replication group (Leff et al.,
IEEE TPDS 93)
origin server
access cost tl lttrlt ts

• Applications
• Content distribution
• Shared memory
• Network file systems

ts

• n nodes
• ? objects

tr
vj
tl
group
Cj vjs storage capacity rij vjs request rate
for obj. oi
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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 thats holding the
  object

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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) anda 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 vj replicates the Cj most popular objects
  according to the local demand rj
• requires only local information (the local
  request vector)

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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 toFind
  appropriate object placement strategies for
  distributed replication groups of selfish nodes

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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

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Mistreatment under SO
Lets get out of here!
an over-active node
1000 reqs/sec
10 reqs/sec
I can do better by following GL (replicate objs
1,2,3,4)
group
these nodes end up replicating potentially
irrelevant objects. They are mistreated by SO
mistreated nodes pursue GL and the group
disintegrates
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The problem with nodes following GL

• Poor performance under common scenarios

• Lets assume that the nodes
• have similar demand patterns
• are adjacent (tr?tl)
• 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

Uncooperativeness is harmful to both the social
and the local utility
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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?
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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)

no reason for a node to abandon the group then

• requires the exchange of a small amount of
  information

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The Distributed Selfish Replication (DSR) game

• nodes ? players
• n players
• local placements ? strategies
• player vj can choose among (N choose Cj) possible
  strategies
• global placement ?outcome of the game
• global placementsum 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?
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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 (v1,v2,,vn)
• 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

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Our first algorithm TSLS

• Two Step Local Search
• Step 0 (initialization)
• each node computes its GL placement

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TSLS (continued)
so a node might exchange some multiple objects
from its GL placement with unrepresented ones

• each node solves a 0/1 Knapsack problem
• unit-weight objects, value gij, integral knapsack
  capacity
• greedy solution ? optimal

• at the end of Step 1 of TSLS -gt Nash eq. plcmnt
• no node can benefit unilaterally

• proof
• vjs 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 vj
• only multiple objects are evicted during Step 1
• only unrepresented objects are inserted during
  Step 1

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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 -gt 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
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Eliminating the impact of ordering

• Suppose that the nodes are identical
• same capacity, demand pattern, request rate

• TSLSmerit-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 nodes turn does not have a large impact on the
  amount of gain that it gets

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TSLS(k) improving the TSLS fairness

• Same as TSLS but
• at Step 1 -gt 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 nodes has a diminishing effect
  on the amount of gain it receives
• for large k ? TSLS(k) reduces to TSLS

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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(Cmax/k)
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Main benefit ? reduced information

• centralized algorithm
• has to send up to nN (obj. id, obj. rate) pairs
  to a central node

• our protocol
• transmits up to SCj obj. ids

• large reduction on the amount of info sent
• typically SCj ltlt N
• obj ids encoded easily (can use Bloom filters)
• (obj. id, obj. rate) pairs harder to represent

• known placements ? perfect routing

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Example

• n2, N100, C1 C2 40, Zipf-like(0.8) demand,
  tl0, tr1, ts2, ?11

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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)

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The end

• Q ?