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Fast Computation of Population Protocols with a Leader. Dana Angluin ( Y ale), James Aspnes (Yale), David Eisenstat (Princeton). The trend. Centralized systems Distributed Systems WSN and mobile devices and RFID Smart molecules?. Miniature sensors moving around. When sensors “meet”. - PowerPoint PPT Presentation
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Fast Computation of Population Protocols with a Leader
DANA ANGLUIN (Yale), JAMES ASPNES (Yale), DAVID EISENSTAT (Princeton)
The trend
• Centralized systems• Distributed Systems• WSN and mobile devices and RFID• Smart molecules?
Miniature sensors moving around
When sensors “meet”
vu
Before C(u)=a, C(v)=b
After C(u)= a’, C(v)=b’
Asymmetric interaction
initiator
a, b a’, b’➝
configuration configuration
responder
When sensors “meet”
vu
initiator
Sensors are (usually) anonymous
Initial state X = initial state of all the sensors
Final state Y = final state of all the sensors
Output Z = f(Y)
(A computation) given X, computes Y (or Z)
-- Interactions are random-- Execution is a sequence of configurations (Non-deterministic)-- No sensor has global knowledge-- Protocols are (generally) non-terminating
responder
More on population protocol
(Fair execution)
C C’ and C occurs infinitely often C’ occurs infinitely often➝ ⇒
(Stable computation of a predicate P(X))
Every fair execution converges to a configuration that reflects the
correct value of P
Epidemic
(One way epidemic)
State Space {0,1}: 1 = infected, 0 = susceptible
THE PROTOCOL: (x,y) (x, max(x,y))➝
(Question) Starting with a single infected agent, how many
interactions are needed to infect every agent?
Epidemic
Lemma 1. Let T(k) be number of interactions before a one-way
epidemic starting with a single infected agent infects k agents.
For any fixed ε > 0 and c > 0, there exist positive constants c1 and
c2, such that for sufficiently large n and any k > nε,
c1.n ln k ≤ T(k) ≤ c2.n ln k
with probability at least (1 − n−c)
Epidemic
Proof hint. Uses instances of the “Coupon Collectors problem”
and then uses Chernoff bounds …
Phase clock(Motivation) How can a leader figure out when the epidemic is
likely to have finished? A leader can only count the number of its
local Interactions before moving to the next phase.
(Overview of the result) The phase clock protocol suggests a way
for the leader to count off θ(n.log n) local interactions, in order
that it outlasts the completion of an epidemic with high
probability.
Phase clock protocolClock phase {0,1,2,…,m-1}∈
(x, b),(y, follower) (x, b),(max➝ * (x, y), follower)
(x, b), (x, leader) (x, b), (x + 1 mod m, leader)➝(x, b), (y, leader) (x, b), (y, leader) [y ≠ x]➝
*A follower in phase x copies the state from any initiator with phase in the
range [x+1, x+m/2] A round consists of m phases 0..m-1. Successive rounds
should be θ(n.log n) apart with high probability
Let b = any agent, x, y are clock phases
Phase clock protocol
Lemma 2. Let phase i start at interaction t. Then there is a
constant a such that for sufficiently large n, phase (i + 1)
starts before interaction t + a.n ln n with probability at
most n−1/2.
Phase clock protocol
infects
infects
m-1
0
m-1
m-1m-1
m-1
m-1
Phase clock protocol
Theorem 1. For any fixed c, d > 0, there exists a constant m such
that, for all sufficiently large n, the finite-state phase clock with
parameter m, starting from an initial state consisting of one leader
in phase 0 and n−1 followers in phase m−1, completes nc rounds of
m phases each, where the minimum number of interactions in any
of the nc rounds is at least d.n ln n with probability at least 1 − n−c.
Duplication
A duplication protocol has state space {(1, 1), (0, 1), (0, 0)} and
transition rules:
(1, 1), (0, 0) (0, 1), (0, 1)➝(0, 0), (1, 1) (0, 1), (0, 1)➝
A duplication protocol starting with a “active” agents in state (1, 1) and the rest in the null state (0, 0) converges to 2a “inactive” agents in state (0, 1), provided 2a < n (otherwise it converges to a population of mixed active and inactive agents with no agents left in the null state).
Duplication
When the initial number of active agents a is close to n/2 ,
duplication may take as much as θ(n2) interactions to converge, as
the last few active agents wait longer to encounter the last few null
agents. But for smaller values of a the protocol converges more
quickly.
DuplicationWhen the initial number of active agents a is close to n/2,
duplication may take as much as θ(n2) interactions to converge, as
the last few active agents wait to encounter the last few null agents.
But for smaller values of a the protocol converges more quickly.
Lemma 3. Let (2a+b) ≤ n/2. The probability that a duplication
protocol starting with a active agents and b inactive agents has not
converged after (2c+1).n ln n interactions is at most n−c.
Cancellation
A cancellation protocol has states {(0, 0), (1, 0), (0, 1)} and
transition rules:
(1, 0), (0, 1) (0, 0), (0, 0)➝(0, 1), (1, 0) (0, 0), (0, 0)➝
Cancellation
The cancellation maintains the invariant that the number of 1 tokens in the left-hand position minus the number of 1 tokens in the right-hand position is fixed. It converges when only (1, 0) and (0, 0) or only (0, 1) and (0, 0) agents remain.
As with duplication, the number of interactions to converge when (1, 0) and (0, 1) are nearly equally balanced can be as many as θ(n2), since we must wait in the end for the last few survivors to find each other
Food for thought
-- New computational model that may be naturally supported in certain settings. The authors propose “computation by epidemic”that can mimic many operations of a conventional register machine.
-- Are there ways to speed up some of these operations?
-- Solutions for new problems on this model
-- Can we eliminate the leader without drastically raising the cost?
-- New models of agent interaction to reflect the physical effects of thespatial dispersion of agents.