Dynamic Necklace Splitting
Rishi Advani, Abolfazl Asudeh, Mohsen Dehghankar, Stavros Sintos
TL;DR
This work generalizes necklace splitting to dynamic settings, enabling real-time updates (relocation, insertion, deletion) while preserving fair color quotas among $k$ agents. The authors develop a suite of linear-time algorithms for the two-color case (including adjacent swaps and arbitrary relocations) that achieve the optimal $2(k-1)$ cuts, and extend linear-time relocation results to multi-color scenarios under certain conditions; they also prove MinNodeMaxFlow is NP-complete and propose batch-relocation strategies that leverage a neighborhood-tree flow to maintain near-optimal cuts. To support dynamic updates, they introduce robust data structures (doubly linked necklaces, neighborhood graphs, and agent-color mappings) and derive several practical approaches, including a dense-swap/jump framework for many colors and a dynamic approximate algorithm with polylogarithmic per-bead time for $n=2$. The proposed methods have direct implications for fair hash maps, load balancing, and bucketization in dynamic data systems, providing both exact and approximate solutions with proven guarantees and scalability considerations.
Abstract
The necklace splitting problem is a classic problem in fair division with many applications, including data-informed fair hash maps. We extend necklace splitting to a dynamic setting, allowing for relocation, insertion, and deletion of beads. We present linear-time, optimal algorithms for the two-color case that support all dynamic updates. For more than two colors, we give linear-time, optimal algorithms for relocation subject to a restriction on the number of agents. Finally, we propose a randomized algorithm for the two-color case that handles all dynamic updates, guarantees approximate fairness with high probability, and runs in polylogarithmic time when the number of agents is small.