Consider the following variant of the balanced allocation...

Consider the following variant of the balanced allocation paradigm: $n$ balls are placed sequentially in $n$ bins. Each ball comes with $d$ choices, chosen independently and uniformly at random from the $n$ bins. When a ball is placed, we are also allowed to move balls among these $d$ bins to equalize their load as much as possible. Show that the maximum load is still at least $\ln \ln n / \ln d-O(1)$ with probability $1-o(1 / n)$ in this case.

Key Concepts

Balanced Allocation Paradigm

This concept refers to distributing items (balls) into containers (bins) in such a way that the load, or number of items per container, is as evenly distributed as possible. In theoretical computer science and probability, this paradigm is used to study load balancing, resource distribution, and related performance guarantees in randomized settings.

Multiple-Choice Allocation

This principle provides each item with several options (d choices) for placement rather than assigning it to a single, random bin. This flexibility, central to the 'power of two choices' phenomenon, typically leads to a more balanced distribution and a reduction in the maximum load. The concept generalizes to any number of choices and is crucial in understanding how additional randomness and selection can improve distribution outcomes.

Maximum Load Analysis

This involves studying the behavior of the heaviest loaded bin (i.e., the maximum number of balls in any bin) after all allocations occur. In the context of these allocation problems, establishing lower bounds on the maximum load, often expressed in terms of logarithmic functions of the total number of bins, is key for analyzing the worst-case performance of load balancing strategies.

Sequential Allocation with Rebalancing

This concept pertains to the process where items are allocated one after another, and for each allocation, there is an opportunity to move items among a selected set of bins to even out the load. Although rebalancing can mitigate disparities, probabilistic analysis shows that there are inherent limitations; even with optimal moves, there exists a nontrivial lower bound on the maximum load.

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