In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Associate Editor of the Journal of Computer and System Sciences

ion Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Pei-Hsin Ho

Automated Technology for Verification and Analysis

We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC1 has no more power than arithmetic circuits of polynomial size and degree nO(log log n) (improving the trivial bound of nO(log n)). Connections are drawn between TC1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation. We show that over the algebra ( ∑ ∗ , max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O(log2 n) depth (and even to O(log n) depth if unbounded-fanin gates are allowed). This establishes that OptLOGCFL is in AC1. This is the first depth-reduction result for arithmetic circuits over a non-commutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in the general non-commutative setting. We define new notions called “short-left-paths” and “short-right-paths” and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible. This class also can be characterized using the AuxPDA model. Finally, we characterize the languages generated by efficient circuits over the semiring ( 2 ∑ ∗ , union, concat) in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.

Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds

Parallel technology boosts data processing in recent years, and parallel direct data processing on hierarchically compressed documents exhibits great promise. The high-performance direct data processing technique brings large savings in both time and space by removing the need for decompressing data. However, its benefits have been limited to data traversal operations; for random accesses, direct data processing is several times slower than the state-of-the-art baselines. This article proposes a novel concept, orthogonal processing on compression (orthogonal POC), which means that text analytics can be efficiently supported directly on compressed data, regardless of the type of the data processing – that is, the type of data processing is orthogonal to its capability of conducting POC. Previous proposals, such as TADOC, are not orthogonal POC. This article presents a set of techniques that successfully eliminate the limitation, and for the first time, establishes the near orthogonal POC feasibility of effectively handling both data traversal operations and random data accesses on hierarchically-compressed data. The work focuses on text data and yields a unified high-performance library, called POCLib. In a ten-node distributed Spark cluster on Amazon EC2, POCLib achieves 3.1× speedup over the state-of-the-art on random data accesses to compressed data, while preserving the capability of supporting traversal operations efficiently and providing large (3.9×) space savings.

POCLib: A High-Performance Framework for Enabling Near Orthogonal Processing on Compression

We show that a context-free grammar of size m that produces a single string w of length n (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for w of size O(m), whose unique derivation tree has depth O(log n). This solves an open problem in the area of grammar-based compression, improves many results in this area and greatly simplifies many existing constructions. Similar results are stated for two formalisms for grammar-based tree compression: top dags and forest straight-line programs. These balancing results can be all deduced from a single meta theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to O(log n) with the cost of a constant multiplicative size increase. Here, n refers to the size of the unfolding (or unravelling) of the circuit. In particular, this results applies to standard arithmetic circuits over (non-commutative) semirings.A long version of the paper can be found in [1].

Balancing Straight-Line Programs

We study the space complexity of querying regular languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. A trichotomy is shown: For every regular language the optimal space requirement is either in Theta(n), Theta(log(n)), or constant, where $n$ is the size of the sliding window.

Querying Regular Languages over Sliding Windows

https://www.eti.uni-siegen.de/ti/veroeffentlichungen/15-tree-compr.pdf

Constructing small tree grammars and small circuits for formulas

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem)
for the wreath product G \wr Z is NP-complete.

https://drops.dagstuhl.de/opus/volltexte/2018/8520/pdf/LIPIcs-STACS-2018-32.pdf/

Knapsack Problems for Wreath Products

In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.

Moses Ganardi

Papers

Balancing Straight-Line Programs

Querying Regular Languages over Sliding Windows

Constructing small tree grammars and small circuits for formulas

Knapsack Problems for Wreath Products

Automata theory on sliding windows