Ergodic Effects in Token Circulation Additive combinatorics meets - - PowerPoint PPT Presentation

Ergodic Effects in Token Circulation

Additive combinatorics meets distributed load balancing

Adrian Kosowski1 Przemysław Uznański2

1Inria Paris, France 2ETH Z¨

urich, Switzerland

HALG 2018 Open in Acrobat Reader to properly see the animations.

Setting

Distributed token propagation:

graph (bidirected, unweighted), n vertices, m edges k identical tokens synchronous rounds local rules for token propagation

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Edge patrolling

schedule k tokens in a graph each edge is traversed every ≤ τ steps minimize idle time τ. centralized solution: τ = Θ(m/k).

Main result:

Extremely simple local rule of propagation with Θ(m/k) edge idle time ..for wide range of parameters ..after initial grace period.

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Cumulative local fairness rule

stronger than fairness rule of

Rabani Sinclair Wanka [FOCS’98] Sauerwald Sun [FOCS’12]

Goal:

• 



t≤T

Lt(e)   − T k 2m

• ≤ σ

∀τ > 2m k σ :

• t≤τ

Lt(e) > 0 time separating traversals of e: Idle time (e) ≤ 2m

k σ

Idle time is a good measure of token dispersion when k ≤ m.

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Results

cumulative fairness =

      

Round-robin; Rotor-router; Eulerian walker; Propp machine; Chip firing; Distributed ant; Sandpile model;

• Det. random walks;

[Propp]; [Priezzhev, Dhar, Dhar, Krishnamurthy ’96]; [Cooper, Doerr, Spencer, Tardos ’07]; [Yanovski, Wagner, Bruckstein ’03], . . .

Main result:

gcd(k, 2m) = 1 ⇓ Every Θ( m

k ) steps every

edge is visited at least once!

Other bounds:

• O(gcd(k, 2m) m

k )

O( m

k ) = O(1) for k ≥ ( 1 2 + ε)m

O(Diam · m

k )

• O(√n · m

k )

• O(

√ k · m

k )

O( m

k ) for trees

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Example of limit trajectory

Eulerian circulation.

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Example of limit trajectory

Many circulations.

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Proof ingredients

x is a self-intersection of the cycle: for some e, e and ϕx(e) share starting point.

Lemma:

The set X of self-intersections satisfies: ∀f ≥1 ∃x∈X (f · x) ∈ 2 3m, 4 3m

• Bohr(X, 1/6) = {0}

= ⇒ X + X + . . . + X

• O(log2 m)times

= Z2m Proof: similar to [Tao, Vu].

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