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.