TY - JOUR

T1 - Testing computability by width-two OBDDs

AU - Ron, Dana

AU - Tsur, Gilad

N1 - Funding Information:
We would like to thank the anonymous reviewers of this paper for their helpful comments. We would also like to thank Joshua Brody, Kevin Matulef, and Chenggang Wu, for noting an error in a previous version of one of our lower bounds, and for bringing to our attention their paper [5], as well as [3]. The work of D. Ron was supported by the Israel Science Foundation (grant number 246/08).

PY - 2012/2/24

Y1 - 2012/2/24

N2 - Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is "far" (for a prespecified distance measure) from every object with that property. In this work, we consider the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, in two settings. In the first setting, the order of the variables is fixed and given to the algorithm, while in the second setting it is not fixed. That is, while in the first setting we should accept a function f if it is computable by a width-2 OBDD with a given order of the variables, in the second setting we should accept a function f if there exists an order of the variables according to which a width-2 OBDD can compute f. Width-2 OBDDs generalize two classes of functions that have been studied in the context of property testing: linear functions (over GF(2)) and monomials. In both these cases membership can be tested by performing a number of queries that is independent of the number of variables, n (and is linear in 1, where is the distance parameter). In contrast, we show that testing computability by width-2 OBDDs when the order of variables is fixed and known requires a number of queries that grows logarithmically with n (for a constant ), and we provide an algorithm that performs O(logn) queries. For the case where the order is not fixed, we show that there is no testing algorithm that performs a number of queries that is sublinear in n.

AB - Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is "far" (for a prespecified distance measure) from every object with that property. In this work, we consider the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, in two settings. In the first setting, the order of the variables is fixed and given to the algorithm, while in the second setting it is not fixed. That is, while in the first setting we should accept a function f if it is computable by a width-2 OBDD with a given order of the variables, in the second setting we should accept a function f if there exists an order of the variables according to which a width-2 OBDD can compute f. Width-2 OBDDs generalize two classes of functions that have been studied in the context of property testing: linear functions (over GF(2)) and monomials. In both these cases membership can be tested by performing a number of queries that is independent of the number of variables, n (and is linear in 1, where is the distance parameter). In contrast, we show that testing computability by width-2 OBDDs when the order of variables is fixed and known requires a number of queries that grows logarithmically with n (for a constant ), and we provide an algorithm that performs O(logn) queries. For the case where the order is not fixed, we show that there is no testing algorithm that performs a number of queries that is sublinear in n.

KW - OBDDs

KW - Property testing

UR - http://www.scopus.com/inward/record.url?scp=84856224506&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2011.11.007

DO - 10.1016/j.tcs.2011.11.007

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AN - SCOPUS:84856224506

SN - 0304-3975

VL - 420

SP - 64

EP - 79

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -