We consider a distributed coding for computing problem with constant decoding locality, i.e. with a vanishing error probability, any single sample of the function can be approximately recovered by probing only constant number of compressed bits. We establish an achievable rate region by designing an efficient coding scheme. The scheme reduces the required rate by introducing auxiliary random variables and supports local decoding at the same time. Then we show the rate region is optimal under mild regularity conditions on source distributions. A coding for computing problem with side information is analogously studied. These results indicate that more rate has to be taken in order to achieve lower coding complexity in distributed computing settings. Moreover, useful graph characterizations are developed to simplify the computation of the achievable rate region.

翻译：我们研究具有恒定译码局部性的分布式编码计算问题，即：在误差概率可忽略的条件下，通过仅探测恒定数量的压缩比特即可近似恢复函数的任意单个样本。通过设计一种高效编码方案，我们建立了可达速率区域。该方案通过引入辅助随机变量来降低所需速率，同时支持局部译码。随后我们证明，在源分布满足温和正则性条件时，该速率区域是最优的。类似地，我们研究了具有边信息的编码计算问题。这些结果表明，在分布式计算场景中，为降低编码复杂度需要消耗更多速率。此外，我们开发了有用的图论刻画方法以简化可达速率区域的计算。