Fast Computation of Population Protocols with a Leader

1. Fast Computation of Population Protocols with a Leader Dana Angluin(Yale), James Aspnes(Yale), David Eisenstat(Princeton)

2. The trend • Centralized systems • Distributed Systems • WSN and mobile devices and RFID • Smart molecules?

3. Miniature sensors moving around

4. When sensors “meet” Before C(u)=a, C(v)=b After C(u)= a’, C(v)=b’ u v a, b ➝ a’, b’ initiator responder Asymmetric interaction configuration configuration

5. When sensors “meet” 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) u v initiator responder (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

6. 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

8. 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?

9. 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 anyk >nε, c1.n lnk ≤ T(k) ≤ c2.n lnk with probability at least (1 − n−c)

10. Epidemic Proof hint. Uses instances of the “Coupon Collectors problem” and then uses Chernoff bounds …

11. 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.logn) local interactions, in order that it outlasts the completion of an epidemic with high probability.

12. Phase clock protocol Clock phase ∈ {0,1,2,…,m-1} Let b = any agent, x, y are clock phases (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.logn) apart with high probability

13. Phase clock protocol Lemma 2. Let phase istart at interaction t. Then there is a constant a such that for sufficiently large n, phase (i + 1) starts before interaction t + a.nlnnwith probability at most n−1/2.

14. Phase clock protocol m-1 infects m-1 0 m-1 infects m-1 m-1 m-1

15. Phase clock protocol Theorem 1. For any fixed c, d > 0, there exists a constantm 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.nlnnwith probability at least 1 − n−c.

16. 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).

17. Duplication When the initial number of active agents a is close to n/2 , duplication may take as much asθ(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.

18. 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 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 andbinactive agents has not converged after (2c+1).n lnn interactions is at most n−c.

19. 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)

20. 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

21. 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 the spatial dispersion of agents.