We consider the problem of deterministic load balancing of tokens in the discrete model. A
set of n processors is connected into ad-regular undirected network. In every timestep, each
processor exchanges some of its tokens with each of its neighbors in the network. The goal
is to minimize the discrepancy between the number of tokens on the most-loaded and the
least-loaded processor as quickly as possible. In this work, we identify some natural
conditions on deterministic load-balancing algorithms to improve upon the long-standing …

In [17], Rabani et al. suggest a framework to analyze a wide class of discrete neighborhood
load balancing algorithms in regular graphs (it can be adapted to non-regular graphs). The
scheme compares the discrete balancing algorithm with its continuous version, and the
difference is used to bound the so-called error that occurs due to the rounding. Their results
hold for round-fair algorithms where any node u having load xt (u) at time t sends either⌊ xt
(u)/(d+ 1)⌋ or⌈ xt (u)/(d+ 1)⌉ tokens over its edges. They show that the discrepancy is …