Ergodic Effects in Token Circulation Additive combinatorics meets distributed load balancing Adrian Kosowski 1 Przemysław Uznański 2 1 Inria Paris, France 2 ETH 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 P. Uznański Additive combinatorics meets distributed load balancing 2 / 8
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. P. Uznański Additive combinatorics meets distributed load balancing 3 / 8
Cumulative local fairness rule stronger than fairness rule of Rabani Sinclair Wanka [FOCS’98] Sauerwald Sun [FOCS’12] Goal: � � � ∀ τ > 2 m � � k σ : L t ( e ) > 0 � � � − T k � � L t ( e ) ≤ σ t ≤ τ � � 2 m � � t ≤ T time separating traversals of e : Idle time ( e ) ≤ 2 m k σ Idle time is a good measure of token dispersion when k ≤ m . P. Uznański Additive combinatorics meets distributed load balancing 4 / 8
Results Round-robin ; Rotor-router ; Eulerian walker ; Propp machine ; cumulative fairness = Chip firing ; Distributed ant ; Sandpile model ; Det. random walks ; [Propp]; [Priezzhev, Dhar, Dhar, Krishnamurthy ’96]; [Cooper, Doerr, Spencer, Tardos ’07]; [Yanovski, Wagner, Bruckstein ’03], . . . Other bounds: Main result: � O ( gcd ( k , 2 m ) m k ) gcd ( k , 2 m ) = 1 O ( m k ) = O ( 1 ) for k ≥ ( 1 2 + ε ) m O ( Diam · m k ) ⇓ O ( √ n · m � k ) Every � Θ( m k ) steps every √ � k · m O ( k ) edge is visited at least once! O ( m k ) for trees P. Uznański Additive combinatorics meets distributed load balancing 5 / 8
Example of limit trajectory Eulerian circulation. P. Uznański Additive combinatorics meets distributed load balancing 6 / 8
Example of limit trajectory Many circulations. P. Uznański Additive combinatorics meets distributed load balancing 7 / 8
Proof ingredients x is a self-intersection of the cycle: for some e , e and ϕ x ( e ) share starting point. Lemma: The set X of self-intersections satisfies: � 2 � 3 m , 4 ∀ f ≥ 1 ∃ x ∈X ( f · x ) ∈ 3 m � = ⇒ X + X + . . . + X = Z 2 m Proof: � �� � similar to [Tao, Vu] . O ( log 2 m ) times Bohr ( X , 1 / 6 ) = { 0 } P. Uznański Additive combinatorics meets distributed load balancing 8 / 8
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