In computer networks, participants may cooperate in processing tasks, so that loads are balanced among them. We present local distributed algorithms that (repeatedly) use local imbalance criteria to transfer loads concurrently across the participants of the system, iterating until all loads are balanced. Our algorithms are based on a short local deal-agreement communication of proposal/deal, based on the neighborhood loads. They converge monotonically, always providing a better state as the execution progresses. Besides, our algorithms avoid making loads temporarily negative. Thus, they may be considered anytime ones, in the sense that they can be stopped at any time during the execution. We show that our synchronous load balancing algorithms achieve $epsilon$-Balanced state for the continuous setting and 1-Balanced state for the discrete setting in all graphs, within $O(n D log(n K/epsilon))$ and $O(n D log(n K/D) + n D^2)$ time, respectively, where $n$ is the number of nodes, $K$ is the initial discrepancy, $D$ is the graph diameter, and $epsilon$ is the final discrepancy. Our other monotonic synchronous and asynchronous algorithms for the discrete setting are generalizations of the first presented algorithms, where load balancing is performed concurrently with more than one neighbor. These algorithms arrive at a 1-Balanced state in time $O(n K^2)$ in general graphs, but have a potential to be faster as the loads are balanced among all neighbors, rather than with only one; we describe a scenario that demonstrates the potential for a fast ($O(1)$) convergence. Our asynchronous algorithm avoids the need to wait for the slowest participants activity prior to making the next load balancing steps as synchronous settings restrict. We also introduce a self-stabilizing version of our asynchronous algorithm.
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