抗擦除性能测试
Erasure-Resilient Property Testing
摘要：
属性测试器形成了一类重要的次线性算法。在标准属性测试模型中，算法通过返回所有查询域点处的函数值的oracle访问输入函数。在许多现实情况下，由于隐私问题，或者当某些值被错误或对手删除时，oracle可能无法在某些域点上显示函数值。我们发起了一项研究的属性测试仪，是有弹性的存在adversarially擦除的功能值。属性P的α擦除弹性ε测试器被给定参数α，ε in（0，1），以及对函数f的oracle访问，使得至多函数值的α分数已经被擦除。测试仪不知道某个点是否被擦除，除非它查询该点。如果有一种方法可以将值分配给擦除点，使得所得函数满足P，则测试人员必须以高概率接受。如果对于擦除点的所有值分配，所得到的函数必须在非擦除畴点的至少ε分数中改变以满足P。弹性性能测试仪用于测试大类性能。对于某些属性，可以通过将标准测试程序用作黑盒来获得可恢复擦除的测试程序。但是，还有更多具有挑战性的属性，所有已知的测试人员都依赖于查询特定的点。如果这个点被擦除，所有这些测试器都坏了。我们给出了有效的擦除弹性测试几类这样的属性，包括单调性，Lipschitz属性，凸性。最后，我们描述了一个属性，可以在标准模型中使用O（1/2）查询进行ε测试，而在擦除弹性模型中测试它需要输入大小的查询多项式数。
英文摘要：
Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function via an oracle that returns function values at all queried domain points. In many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. We initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester for a property P is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of the function values have been erased. The tester does not know whether a point is erased unless it queries that point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies P. It has to reject with high probability if, for all assignments of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the nonerased domain points to satisfy P. We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by using standard testers as a black box. But there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several classes of such properties including monotonicity, the Lipschitz property, and convexity. Finally, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.
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