Testing Monotonicity Over Graph
Products
Shirley Halevy, Eyal Kushilevitz
Computer Science Department, Technion, Haifa 32000, Israel;
e-mails: shirleyh@cs.technion.ac.il; eyalk@cs.technion.ac.il
Received 8 January 2006; accepted 25 April 2007; received in final form 18 June 2007
Published online 2 May 2008 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/rsa.20211
ABSTRACT: We consider the problem of monotonicity testing over graph products. Monotonicity
testing is one of the central problems studied in the field of property testing. We present a testing
approach that enables us to use known monotonicity testers for given graphs G1, G2, to test monoto-
nicity over their product G1 × G2. Such an approach of reducing monotonicity testing over a graph
product to monotonicity testing over the original graphs, has been previously used in the special
case of monotonicity testing over [n]d for a limited type of testers; in this article, we show that this
approach can be applied to allow modular design of testers in many interesting cases: this approach
works whenever the functions are boolean, and also in certain cases for functions with a general
range. We demonstrate the usefulness of our results by showing how a careful use of this approach
improves the query complexity of known testers. Specifically, based on our results, we provide a new
analysis for the known tester for [n]d which significantly improves its query complexity analysis in
the low-dimensional case. For example, when d = O(1), we reduce the best known query complexity
from O(log2 n/
) to O(log n/
). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 33, 44–67, 2008
Keywords: property testing; monotonicity testing; graph products
1. INTRODUCTION
The classical notion of decision problems requires an algorithm to distinguish objects having
some property P from those objects which do not have the property. Property testing is a
relaxation of decision problems, where algorithms are only required to distinguish objects
Correspondence to: S. Halevy
An extended abstract of this paper appeared in ICALP 2004 [23].
© 2008 Wiley Periodicals, Inc.
44

TESTING MONOTONICITY OVER GRAPH PRODUCTS 45
having the propertyP from those which are at least “
-far” from every such object. The main
goal of property testing is to avoid “reading” the whole object (which requires complexity
at least linear in the size of its representation); i.e., to make the decision by reading a small
(possibly, randomly selected) fraction of the input (e.g., a fraction of size polynomial in 1/

and poly-logarithmic in the size of the representation) and still having a good (say, at least
2/3) probability of success. The notion of property testing was introduced by Rubinfeld
and Sudan [29] and since then attracted a considerable amount of attention. Property testing
algorithms (or testers) were introduced for problems in graph theory (e.g. [2, 7, 17–20,
22, 25, 26]), monotonicity testing (e.g. [4, 8–10, 13, 14, 16, 24]), and other properties (e.g.
[3, 5, 12, 21, 29]; the reader is referred to surveys by Ron [28], Goldreich [15], and Fischer
[11] for a presentation of some of this work, including some connections between property
testing and other areas).
In this article we focus on testing monotonicity of functions defined over graph products.
Monotonicity has been one of the central problems studied in the field of property testing,
e.g., [4, 8–10, 13, 14, 16]. We begin with a short survey of the previous work regarding
monotonicity testing. Let f : V → A be a function, where V is the set of vertices of some
directed graph G = (V , E) and let ≤A be a total order on A. We say that such a function f is
monotone with respect to G if for every u, v ∈ V whenever u ≤G v (i.e., there is a directed
path in G from u to v) then f (u) ≤A f (v). Monotonicity of general functions is a basic
property and, as such, attracted much attention: efficient testers were presented for certain
classes of graphs (e.g. [4, 8, 9, 13, 14, 16]), and hardness results show that monotonicity
cannot be tested for all graphs using poly-logarithmic number of queries (even for boolean
functions) [13]. One family of graphs for which efficient monotonicity testers were presented
is the d-dimensional hypercube, that is [n]d . A partial order is defined on the domain [n]
in the natural way (for y, z ∈ [n]d , we say that y ≤ z if each coordinate of y is bounded by
the corresponding coordinate of z)1. A function f over the domain [n]d is monotone with
respect to [n]d (in short, monotone) if z ≥ y implies f (z) ≥ f (y). Testing algorithms were
developed to deal with both the low-dimensional and the high-dimensional cases. In what
follows, we survey some known results relevant to our work.
1.1. Related Work
In the low dimensional case, d is considered to be small compared with n (and, in fact, it is
typically a constant); a successful algorithm for this case is typically one that is polynomial
in 1/
and in log n. The first paper to deal with this case is by Ergün et al. [9] who presented
an algorithm for the line (i.e., the case d = 1) whose query complexity is O( log n


), and
showed that this query complexity cannot be achieved without using membership queries
(that is, by using only queries of the function at randomly drawn points of the domain); in
other words, we have to allow the tester to make queries at points of its choice, computed
based on the information it learned during the previous steps of its execution. The algorithm
of [n] was generalized for any fixed d in [4], with O( (2 log n)d


) query complexity. For the
case d = 1, there is a lower bound showing that monotonicity testing (for some constant
)
indeed requires (log n) queries [9,10]. In the high dimensional case, d is considered as the
main parameter (and n might be as small as 2); a successful algorithm is typically one that is
polynomial in 1/
and d. This case was first considered by Goldreich et al. [16], who showed
1In the case d = 1 this yields a linear order.
Random Structures and Algorithms DOI 10.1002/rsa