We consider a coded distributed computing problem in a ring-based communication network, where $N$ computing nodes are arranged in a ring topology and each node can only communicate with its neighbors within a constant distance $d$. To mitigate the communication bottleneck in exchanging intermediate values, we propose new coded distributed computing schemes for the ring-based network that exploit both ring topology and redundant computation (i.e., each map function is computed by $r$ nodes). Two typical cases are considered: all-gather where each node requires all intermediate values mapped from all input files, and all-to-all where each node requires a distinct set of intermediate values from other nodes. For the all-gather case, we propose a new coded scheme based on successive reverse carpooling where nodes transmit every encoded packet containing two messages traveling in opposite directions along the same path. Theoretical converse proof shows that our scheme achieves the optimal tradeoff between communication load, computation load $r$, and broadcast distance $d$ when $N\gg d$. For the all-to-all case, instead of simply repeating our all-gather scheme, we delicately deliver intermediate values based on their proximity to intended nodes to reduce unnecessary transmissions. We derive an information-theoretic lower bound on the optimal communication load and show that our scheme is asymptotically optimal under the cyclic placement when $N\gg r$. The optimality results indicate that in ring-based networks, the redundant computation $r$ only leads to an additive gain in reducing communication load while the broadcast distance $d$ contributes to a multiplicative gain.

翻译：本文研究环形通信网络中的编码分布式计算问题，其中$N$个计算节点按环形拓扑排列，每个节点仅能与距离不超过常数$d$的邻居节点通信。为缓解交换中间值的通信瓶颈，我们针对环形网络提出新的编码分布式计算方案，该方案同时利用环形拓扑与冗余计算（即每个映射函数由$r$个节点计算）。我们考虑两种典型场景：全收集（每个节点需要所有输入文件映射的全部中间值）和全交换（每个节点需要来自其他节点的不同中间值集合）。针对全收集场景，我们提出基于连续反向拼车的新型编码方案，节点传输的每个编码数据包均包含沿相同路径反向传输的两条消息。理论逆证明表明，当$N\gg d$时，该方案实现了通信负载、计算负载$r$与广播距离$d$之间的最优权衡。针对全交换场景，我们并未简单重复全收集方案，而是根据中间值与目标节点的邻近性进行精细投递以减少冗余传输。我们推导了最优通信负载的信息论下界，并证明在循环放置策略下当$N\gg r$时该方案具有渐近最优性。最优性结果表明：在环形网络中，冗余计算$r$仅能带来通信负载的加性增益，而广播距离$d$可产生乘性增益。