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 layered coding scheme, where the coding rate is reduced by introducing auxiliary random variables and local decoding is achieved by exploiting the expander graph code. Then we show the rate region is optimal under mild regularity conditions on source distributions. The proof relies on the reverse hypercontractivity and a rounding technique to construct auxiliary random variables. The rate region is strictly smaller than that for the classical problem without the constant locality constraint in most cases, which indicates that more rate is required in order to achieve lower coding complexity. Moreover, a coding for computing problem with side information is analogously studied. We also develop graph characterizations, which simplifies the computation of the achievable rate region.

翻译：考虑一个具有恒定解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，仅通过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。我们通过设计一种高效的分层编码方案建立了可达速率区域，其中通过引入辅助随机变量降低编码速率，并利用扩张图码实现局部解码。随后我们证明，在源分布的温和正则性条件下，该速率区域是最优的。证明过程依赖于逆超收缩性以及用于构造辅助随机变量的舍入技术。在多数情况下，该速率区域严格小于无恒定局部性约束的经典问题对应的速率区域，这表明实现更低编码复杂度需要更高的编码速率。此外，我们对带边信息的编码计算问题进行了类似研究，并提出了图论刻画方法，简化了可达速率区域的计算。