We consider the problem of distributed lossless computation of a function of two sources by one common user. To do so, we first build a bipartite graph, where two disjoint parts denote the individual source outcomes. We then project the bipartite graph onto each source to obtain an edge-weighted characteristic graph (EWCG), where edge weights capture the function's structure, by how much the source outcomes are to be distinguished, generalizing the classical notion of characteristic graphs. Via exploiting the notions of characteristic graphs, the fractional coloring of such graphs, and edge weights, the sources separately build multi-fold graphs that capture vector-valued source sequences, determine vertex colorings for such graphs, encode these colorings, and send them to the user that performs minimum-entropy decoding on its received information to recover the desired function in an asymptotically lossless manner. For the proposed EWCG compression setup, we characterize the fundamental limits of distributed compression, verify the communication complexity through an example, contrast it with traditional coloring schemes, and demonstrate that we can attain compression gains higher than $\% 30$ over traditional coloring.

翻译：我们研究了一个由两个信源向单一公共用户进行分布式无损函数计算的问题。为此，首先构建一个二分图，其中两不相交部分分别表示各信源的输出结果。随后将该二分图投影至每个信源上，从而获得边加权特征图（EWCG），其中边权值表征函数结构中信源输出需要区分的程度，这推广了经典特征图的概念。通过利用特征图概念、此类图的分数着色方法以及边权值，各信源分别构建反映向量值信源序列的多重图，确定此类图的顶点着色方案，对这些着色进行编码，并将其发送至用户。用户对接收信息执行最小熵解码，以渐近无损方式恢复所需函数。针对所提出的EWCG压缩框架，我们刻画了分布式压缩的极限性能，通过实例验证通信复杂度，与传统着色方案进行对比，并证明相较于传统着色可获得超过30%的压缩增益。