ECCC-Report TR17-029https://eccc.weizmann.ac.il/report/2017/029Comments and Revisions published for TR17-029en-usSun, 19 Feb 2017 01:35:19 +0200
Revision 1
| An Adaptivity Hierarchy Theorem for Property Testing |
Tom Gur,
Clement Canonne
https://eccc.weizmann.ac.il/report/2017/029#revision1Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries.
In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is $k$-(round) adaptive if it makes queries in $k+1$ rounds, where the queries in the $i$'th round may depend on the answers obtained in the previous $i-1$ rounds. Then, we ask the following question:
Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity?
We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every $n\in \mathbb{N}$ and $0 \le k \le n^{0.99}$ there exists a property $\mathcal{P}_{n,k}$ of functions for which (1) there exists a $k$-adaptive tester for $\mathcal{P}_{n,k}$ with query complexity $\tilde{O}(k)$, yet (2) any $(k-1)$-adaptive tester for $\mathcal{P}_{n,k}$ must make $\Omega(n)$ queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
Sun, 19 Feb 2017 01:35:19 +0200https://eccc.weizmann.ac.il/report/2017/029#revision1
Paper TR17-029
| An Adaptivity Hierarchy Theorem for Property Testing |
Tom Gur,
Clement Canonne
https://eccc.weizmann.ac.il/report/2017/029Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all queries are independent of answers obtained from previous queries.
In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of ``rounds of adaptivity'' it uses. More accurately, we say that a tester is $k$-(round) adaptive if it makes queries in $k+1$ rounds, where the queries in the $i$'th round may depend on the answers obtained in the previous $i-1$ rounds. Then, we ask the following question:
Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity?
We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every $n\in \N$ and $0 \le k \le n^{0.99}$ there exists a property $\mathcal{P}_{n,k}$ of functions for which (1) there exists a $k$-adaptive tester for $\mathcal{P}_{n,k}$ with query complexity $\tildeO{k}$, yet (2) any $(k-1)$-adaptive tester for $\mathcal{P}_{n,k}$ must make $\Omega(n)$ queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
Sat, 18 Feb 2017 17:19:14 +0200https://eccc.weizmann.ac.il/report/2017/029