## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 15, 2016, 12:15 pm

Duration: 30 minutes

Location: CAB G11

Speaker: Przemek Uznański

We revisit the notion of load balancing schemes with local token propagation. We investigate the setting when the number of tokens is small when compared to the graph size, that is $k \ll m$. In such a scenario of low density of tokens, it is natural to expect that in addition to loads being well balanced across the network, the cumulated loads (loads summed over fixed, arbitrarily long, intervals of time) are also balanced.

In this setting we analyze a class of deterministic schemes. The starting point is round-robin token propagation (also known as the rotor-router model). This process, for a graph of $m$ edges with $k$ tokens, is known to reach in polynomial time a recurrent state, and in this work we exhibit new properties of this recurrent state.

For the considered class of processes, we obtain a series of additive bounds on the edge cumulated load discrepancy. Our results apply to specific graph classes and also show that, in general, the cumulated edge load discrepancy is $\widetilde O(1)$ whenever $gcd(k,m) = \widetilde O(1)$. In particular, this result holds always when the number of edges of the graph is a prime number.

Our results also yield as a corollary bounds on edge idle time of the process, i.e., the longest possible time between two consecutive appearances of a unit of load on an edge, taken over all edges. The task of circulating $k$ tokens to minimize edge idle time is a fundamental one in the study of the patrolling problem on networks. While the best centralized offline solution to the patrolling problem trivially achieves $\Theta(m/k)$ idle time, it is an open problem how to construct good distributed solutions. Our results on cumulative discrepancy automatically translate into a distributed solution to patrolling with $\widetilde O(1)$ competitive ratio on idle time, for all of the covered cases.

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