Covering Arrays with Row Limit: Bounds and Constructions Nevena - PowerPoint PPT Presentation

Covering Arrays with Row Limit: Bounds and Constructions Nevena Franceti´ c Supervised by Prof. P. Danziger and Prof. E. Mendelsohn Discrete Maths Research Group Monash University September 22, 2014.

Covering arrays N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering arrays ... is a test suite ... N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering arrays ... is a test suite ... ... for verification of interactions between components. N. Franceti´ c (Monash) CARL s September 22, 2014.

Example N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Components: outline, shadow, blinking, hidden N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) What interactions? Pairwise interactions. N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) What interactions? Pairwise interactions. � 4 � 2 2 = 24 tests. Testing two at time: it would take 2 N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Testing can be done in only 5 iterations. N. Franceti´ c (Monash) CARL s September 22, 2014.

Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Parameters of a Covering Array A covering array is characterized by: k : the number of components (columns) v : the number of levels for each component (alphabet size) t : strength ⇒ testing interactions between t columns N. Franceti´ c (Monash) CARL s September 22, 2014.

Parameters of a Covering Array A covering array is characterized by: k : the number of components (columns) v : the number of levels for each component (alphabet size) t : strength ⇒ testing interactions between t columns Goal: find the smallest number of rows, called size N of a covering array. Size of an optimal CA is usually denoted by CAN ( t , k , v ). N. Franceti´ c (Monash) CARL s September 22, 2014.

Some facts about covering arrays N. Franceti´ c (Monash) CARL s September 22, 2014.

Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) N. Franceti´ c (Monash) CARL s September 22, 2014.

Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) The exact size of a covering array is ONLY known for one family of arrays (Kleitman and Spencer, 1973; Katona, 1973): � N − 1 � CA ( N ; 2 , k , 2) exists for all k ≤ � N � 2 − 1 N. Franceti´ c (Monash) CARL s September 22, 2014.

Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) The exact size of a covering array is ONLY known for one family of arrays (Kleitman and Spencer, 1973; Katona, 1973): � N − 1 � CA ( N ; 2 , k , 2) exists for all k ≤ � N � 2 − 1 Finding an optimal covering array is NP-complete when extra constraints are imposed even when v = 2 (Maltais and Moura, 2011). N. Franceti´ c (Monash) CARL s September 22, 2014.

More on covering arrays http://www.pairwise.org/tools.asp contains: 39 software tools for constructing CA s both commercial and open source N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 0 0 − 1 − 0 0 − 1 1 1 − 0 1 0 − − 1 0 − 0 0 0 − 1 0 1 0 − − 1 − 0 − 1 0 1 − 1 − 0 1 1 1 − 1 − 1 − 0 0 1 0 − − 1 1 − 1 0 − 0 − 0 1 1 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − 5 1 0 1 0 − − 6 1 − 0 − 1 0 7 1 − 1 − 0 1 8 1 1 − 1 − 1 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − v = the alphabet size; the 5 1 0 1 0 − − number of different values assigned to a column 6 1 − 0 − 1 0 7 1 − 1 − 0 1 8 1 1 − 1 − 1 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − v = the alphabet size; the 5 1 0 1 0 − − number of different values assigned to a column 6 1 − 0 − 1 0 7 1 − 1 − 0 1 w = row limit; the number of 8 1 1 − 1 − 1 non-empty cells in a row 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 k = the number of columns 1 0 0 − 1 − 0 (components) 2 0 − 1 1 1 − v = the alphabet size; the 3 0 1 0 − − 1 number of different values 4 0 − 0 0 0 − assigned to a column 5 1 0 1 0 − − w = row limit; the number of 6 1 − 0 − 1 0 7 1 − 1 − 0 1 non-empty cells in a row 8 1 1 − 1 − 1 t = strength 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 k = the number of columns 1 0 0 − 1 − 0 (components) 2 0 − 1 1 1 − v = the alphabet size; the 3 0 1 0 − − 1 number of different values 4 0 − 0 0 0 − assigned to a column 5 1 0 1 0 − − w = row limit; the number of 6 1 − 0 − 1 0 7 1 − 1 − 0 1 non-empty cells in a row 8 1 1 − 1 − 1 t = strength 9 − 0 0 1 0 − N = size; goal find minimum N 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) Theorem ( Gargano et al.(1993)) lim sup CAN (2 , k , v ) = Θ (log k ) . k →∞ N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) Theorem ( Gargano et al.(1993)) lim sup CAN (2 , k , v ) = Θ (log k ) . k →∞ Ω (log k ) = CAN (2 , k , v ) ≤ CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) N. Franceti´ c (Monash) CARL s September 22, 2014.

Sch¨ onheim lower bound Theorem CARLN λ ( t , k , v : w ) ≥ SB ( t , k , v : w ) � vk � v ( k − 1) � v ( k − t + 1) � �� SB ( t , k , v : w ) = · · · . . . . w w − 1 w − t + 1 CARL (12; 2 , 6 , 2: 4) N. Franceti´ c (Monash) CARL s September 22, 2014.

Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t N. Franceti´ c (Monash) CARL s September 22, 2014.

Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t CARLN ( t , k , v : w ) = Θ ( k t ) N. Franceti´ c (Monash) CARL s September 22, 2014.

Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t CARLN ( t , k , v : w ) = Θ ( k t ) k t lim k →∞ w ( k ) t log k = 0 SB ( t , k , v : w ( k )) lim = 0 log k k →∞ N. Franceti´ c (Monash) CARL s September 22, 2014.

Covering Arrays with Row Limit: Bounds and Constructions Nevena - PowerPoint PPT Presentation

Covering Arrays with Row Limit: Bounds and Constructions Nevena Franceti c Supervised by Prof. P. Danziger and Prof. E. Mendelsohn Discrete Maths Research Group Monash University September 22, 2014. Covering arrays N. Franceti c

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Covering Arrays with Row Limit: Bounds and Constructions Nevena - PowerPoint PPT Presentation

Covering Arrays with Row Limit: Bounds and Constructions Nevena Franceti c Supervised by Prof. P. Danziger and Prof. E. Mendelsohn Discrete Maths Research Group Monash University September 22, 2014. Covering arrays N. Franceti c

Sequence Covering Arrays Lower Bounds Upper Bounds Existence Results Charles J. Colbourn 1

A new construction of strength-3 covering arrays using linear feedback shift register (LFSR)

Covering Arrays with Row Limit Ottawa-Carleton Discrete Mathematics Days University of Ottawa

Chapter 7 Arrays Chapter Scope Array declaration and use Bounds checking Arrays as

Construction of covering arrays from Outline m-sequences Covering arrays Definition Research

Combinatorial Testing and Covering Arrays Lucia Moura School of Electrical Engineering and

Salvaging Weak Security Bounds for Blockcipher-based Constructions Thomas Shrimpton (University

Variable Strength Covering Arrays Lucia Moura School of Electrical Engineering and Computer

A Post-Processing Algorithm Applied to Reduce the Size of the Covering Arrays of the NIST

Covering Arrays on Graphs Karen Meagher Department of Mathematics and Statistics University of

Lower Bounds on Classical Ramsey Numbers constructions, connectivity, Hamilton cycles Xiaodong Xu

An Algorithm for Generating t-wise Covering Arrays from Large Feature Models Martin Fagereng

Oracle-free Testing with Two-layer Covering Arrays Rick Kuhn National Institute of

Pointer arithmetic arrays only arrays only Pointer arithmetic Can add or subtract an

Large Deviation Bounds A typical probability theory statement: Theorem (The Central Limit

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Objectives: Discuss arrays Syntax Multi-dimensional arrays Arrays

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Arrays Arrays and Methods Searching Sorting Arrays Reading: =&gt; Continue with

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