*Here is the last guest post by Andrew Winslow from ICALP 2014. Enjoy it! Thanks to Andrew for his conference reports.*

This [Friday, 11 July 2014] is the last day of ICALP!

### George Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos and Paul Spirakis. Determining Majority in Networks with Local Interactions and very Small Local Memory

This talk focused on a problem in distributed computing on a graph with stateful nodes. Given a graph with nodes colored red and blue, a*population protocol*is a set of rules that define how the colors of a pair of endpoints of an edge can be transformed. For instance, a population protocol might include a rule to turn the colors of a pair of red nodes sharing to both be blue. One is also allowed to add additional colors not found in inputs, and the total number of colors that appear in the input and population protocol is the number of

*states*of the protocol.

A population protocol for the

*majority problem*transforms the colors of all (or all but a (1-ε) fraction) of the nodes into the color appearing most frequently in the input graph. In may be possible at any point during the execution of the protocol that more than one rule can be applied to the endpoints of more than one edge, and it is assumed that either the rules are chosen randomly (a

*randomized scheduler*) or advesarially subject to some basic requirements, like that any rule that can be applied is applied after a finite amount of time (a

*fair scheduler*). One can also ask about the amount of time (i.e. the number of rule applications) needed for a majority to be reached.

It was previously known that a 3-state protocol for majority exists in the case that both the graph is a clique and the ratio of the occurrence counts of most common color over least common color is ω(sqrt(n)log(n))[1]. In this work, Mertzios, Nikoletseas, Christoforos, and Spirakis constructed a four-state protocol that correct computes majority with probability 1 on any graph and with any difference between the number of occurrences of the two colors, assuming a fair scheduler. They also prove that the 3-state protocol described in [1] is strictly worse, failing to converge to the correct majority with high probability for an infinite family of graphs (graphs consisting of a clique at the end of a chain), even with a probabilistic scheduler.

[1] D. Angluin, J. Aspnes, and D. Eisenstat. A simple population protocol for fast robust approximate majority. Distributed Computing, 21(2): 87-102, 2008.

### Karl Bringmann, Fabian Kuhn, Konstantinos Panagiotou, Ueli Peter and Henning Thomas. Internal DLA: Efficient Simulation of a Physical Growth Model.

Karl gave the talk. This work is about improving the speed of sampling the results of the following process, called*internal diffuse limited aggregation*or

*IDLA*:

- Place a particle at origin in the square lattice.
- Take a random walk with this particle until a lattice location not containing another particle is found.
- Place this particle at the empty location.

The IDLA process comes from statistical physics, and natural problems in this area related to understanding the distribution of shapes generated by this process. The naive approach to randomly sample the result of the IDLA process takes O(n^2) time, since the expected radius is Theta(n^0.5), a randomly walk from the origin takes Theta(n) expected time to reach an empty location (usually on the boundary), and n particles must be simulated.

Bringmann, Kuhn, Panagiotou, Peter, and Thomas improve this to O(nlog^2(n)) expected time and O(n^0.5*log(n)) expected space with high probability. The idea of the algorithm is to efficiently carry out the random walk by making large

*jumps*, carrying out a significant portion of the walk in O(1) expected time. An upper bound on the number of jumps can then be obtained by computing the time expected to leave the region containing previous points (with radius O(sqrt(n)*log(n))).