We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or $\varepsilon$-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than $\varepsilon$, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least $\varepsilon,$ then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter $\varepsilon$. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. `06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms `08) and Eden et al. (ICALP `17). We conclude with a discussion of our model and open questions raised by our work.

中文翻译：

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擦除弹性次线性时间图算法

我们研究亚线性时间算法，该算法采用邻接表表示的部分擦除图作为输入。我们的算法对输入图进行度和邻查询，并处理邻接项中指定比例的对抗性擦除。我们专注于两个计算任务：测试图是否已连接或距离连接远，并估计平均度。为了测试连通性，我们发现了一个阈值现象：当擦除的分数小于$ \ varepsilon $时，可以有效地测试此属性（时间独立于图形的大小）；当删除的分数至少为$ \ varepsilon，$时，则需要数量与图表示大小成线性关系的查询。我们的抗擦除算法（对于无擦除的特殊情况）是对标准属性测试模型中先前已知的连通性算法的改进，并且对邻近参数$ \ varepsilon $具有最佳依赖性。为了估算平均程度，我们的结果在模型中针对此计算任务的查询复杂度之间进行了“插值”，没有在两种不同设置下进行擦除：仅由Feige（SIAM J. Comput.`06）进行程度查询，以及度数查询和邻居查询，由Goldreich和Ron（Random Struct。Algorithms '08）和Eden等研究。（ICALP`17）。我们以对模型的讨论和工作中提出的开放性问题作为结尾。