We initiate the study of Local Computation Algorithms on average case inputs. In the Local Computation Algorithm (LCA) model, we are given probe access to a huge graph, and asked to answer membership queries about some combinatorial structure on the graph, answering each query with sublinear work. For instance, an LCA for the $k$-spanner problem gives access to a sparse subgraph $H\subseteq G$ that preserves distances up to a factor of $k$. We build simple LCAs for this problem assuming the input graph is drawn from the well-studied Erdos-Reyni and Preferential Attachment graph models. In both cases, our spanners achieve size and stretch tradeoffs that are impossible to achieve for general graphs, while having dramatically lower query complexity than worst-case LCAs. Our second result investigates the intersection of LCAs with Local Access Generators (LAGs). Local Access Generators provide efficient query access to a random object, for instance an Erdos Reyni random graph. We explore the natural problem of generating a random graph together with a combinatorial structure on it. We show that this combination can be easier to solve than focusing on each problem by itself, by building a fast, simple algorithm that provides access to an Erdos Reyni random graph together with a maximal independent set.

翻译：我们启动了平均情况输入下局部计算算法的研究。在局部计算算法（LCA）模型中，我们被允许对巨大图进行探针访问，并被要求回答关于图上某种组合结构的成员查询，每个查询都以亚线性工作量完成。例如，针对$k$-路问题的LCA提供对稀疏子图$H\subseteq G$的访问，该子图保持距离至多$k$倍。我们为此问题构建了简单的LCA，假设输入图来自已广泛研究的Erdos-Reyni和优先连接图模型。在这两种情况下，我们的子图在大小和拉伸之间达到了对一般图不可能实现的权衡，同时查询复杂度显著低于最坏情况下的LCA。我们的第二个结果研究了LCA与局部访问生成器（LAG）的交集。局部访问生成器提供对随机对象（例如Erdos Reyni随机图）的高效查询访问。我们探讨了生成随机图及其上组合结构的自然问题。我们展示了这种组合可以比单独处理每个问题更容易解决，通过构建一个快速、简单的算法，提供对Erdos Reyni随机图及其上最大独立集的访问。