In this paper, we study the problem of efficiently reducing geometric shapes into other such shapes in a distributed setting through size-changing operations. We develop distributed algorithms using the reconfigurable circuit model to enable fast node-to-node communication. Our study considers two graph update models: the connectivity model and the adjacency model. Let $n$ denote the number of nodes and $k$ the number of turning points in the initial shape. In the connectivity model, we show that the system of nodes can reduce itself from any tree to a single node using only shrinking operations in $O(k \log n)$ rounds w.h.p. and any tree to its minimal (incompressible) form in $O(\log n)$ rounds with additional knowledge or $O(k \log n)$ without, w.h.p. We also give an algorithm to transform any tree to any topologically equivalent tree in $O(k \log n+\log^2 n)$ rounds w.h.p. if both shrinking and growth operations are available to the nodes. On the negative side, we show that one cannot hope for $o(\log^2 n)$-round transformations for all shapes of $O(\log n)$ turning points: for all reasonable values of $k$, there exists a pair of geometrically equivalent paths of $k$ turning points each, such that $\Omega(k\log n)$ rounds are required to reduce one to the other. In the adjacency model, we show that the system can reduce itself from any connected shape to a single node using only shrinking in $O(\log n)$ rounds w.h.p.

翻译：本文研究在分布式环境下通过尺寸变换操作将几何形状高效约简为其他形状的问题。我们利用可重构电路模型开发分布式算法，以实现快速的节点间通信。本研究考虑两种图更新模型：连通性模型与邻接模型。令$n$表示节点数量，$k$表示初始形状的转折点数量。在连通性模型中，我们证明节点系统仅通过收缩操作即可在$O(k \log n)$轮内以高概率将任意树结构约简为单节点；若具备额外知识，可在$O(\log n)$轮内将任意树约简至其最小（不可压缩）形式，若无额外知识则需$O(k \log n)$轮（均以高概率达成）。当节点同时具备收缩与扩展操作时，我们给出在$O(k \log n+\log^2 n)$轮内以高概率将任意树转换为任意拓扑等价树的算法。在负面结果方面，我们证明对于所有具有$O(\log n)$个转折点的形状，无法期望实现$o(\log^2 n)$轮的转换：对于所有合理的$k$值，存在一对各含$k$个转折点的几何等价路径，使得将其中一条约简为另一条需要$\Omega(k\log n)$轮。在邻接模型中，我们证明系统仅通过收缩操作即可在$O(\log n)$轮内以高概率将任意连通形状约简为单节点。