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Charles J. Charles J. ColbournColbournComputer Science and EngineeringComputer Science and EngineeringArizona State University, Tempe, AZArizona State University, Tempe, AZ
Constructions ofCovering Arrays
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001202010120110210211210222221112000
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
It is well known that
CAN(2,k,v) ≤ CAN(2,k,v-1) – 1.
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020120220102120110210211210202222011120000
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
Proof 1:
Make the first rowconstant byrenaming symbols.
Then delete it.
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001*00*00101*00110*010*011*010**0*0*0*111*000
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
Proof 2:
Change all of largestsymbol in eachcolumn to * = “don’tcare”
Then fill in * withentries from first row.
Then delete first row.
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020120220102120110210211210202222011120000
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
First renamesymbols and deletefirst row.
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2*1222*1212*11*21*21121**222*111
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
Second replace allelements in thedeleted row by *
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211222112121111211211211*222*111
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
Now move top rowelements into *positions and deletetop row.
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211222112121111211211211*222
Challenge: Deleting a SymbolChallenge: Deleting a Symbol
This works in generaland shows that
CAN(2,k,v) ≤ CAN(2,k,v-1) – 2.
In fact it works formixed coveringarrays by removingone level from eachfactor.
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Challenge: Deleting a SymbolChallenge: Deleting a Symbol
Is it always the case for k,v ≥ 2 that
CAN(2,k,v) ≤ CAN(2,k,v-1) – 3?
For mixed CAs too?
True for OAs from the projective plane.
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A Testing ProblemA Testing Problem
• The user is presented with nparameters (“factors”), each havingsome finite number of values (“levels”).
• The j’th factor has sj levels; continuousfactors are modelled by a finite numberof intervals.
• Initially, we assume that levels forfactors can be selected independently.
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Covering ArraysCovering Arrays
• A covering array is an N x k array.• Symbols in column j are chosen from an
alphabet of size sj
• Choosing any N x t subarray, we find everypossible 1 x t row occurring at least once; t isthe strength of the array.
• Evidently, the number N of rows must be atleast the product of the t largest factor levelsizes
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Covering ArraysCovering Arrays
• In general this is not sufficient. For constant t> 1 and factor level sizes, the number of rowsgrows at least as quickly as log n.
• Indeed, even for t=2, every two columns ofthe covering array must be distinct
• and this alone suffices to obtain a log n lowerbound.
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Covering ArraysCovering Arrays
CAλ(N;t,k,v)– An N x k array where each N x t sub-array contains
all ordered t-sets at least λ times.
100101
010110001101000000111110
CA(6;2,5,2)
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Covering ArraysCovering Arrays
• The goal, given k, t, and the sj’s, is tominimize N. Or given N, t, and the sj’s,to maximize k.
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Covering ArraysCovering Arrays
• Research on the problem has fallen into fourmain categories:– lower bounds– combinatorial/algebraic constructions
• direct methods• recursive methods
– probabilistic asymptotic constructions– computational constructions
• exact methods• heuristic methods
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Basic Combinatorial MethodsBasic Combinatorial Methods
• Consider the problem of constructing acovering array of strength two, with g levelsper factor, and k factors.
• We could hope to have as few as g2 rows(tests), and if this were to happen then every2-tuple of values would occur exactly once (astronger condition than ‘at least once’).
• If we strengthen the condition to ‘exactlyonce’, the covering array is an orthogonalarray of index one.
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Orthogonal ArraysOrthogonal ArraysOAλ(N;t,k,v) -An N x k array where each N x t sub-
array contains all ordered t-sets exactly λ times.
11110011010110010110101011000000
OA(8;3,4,2)
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Orthogonal ArraysOrthogonal Arrays
• For strength two, an orthogonal array of indexone with g symbols and k columns exists– only when k ≤ g+1,– if k ≤ g+1 and g is a power of a prime.
• For primes, form rows of the array byincluding (i,j,i+j,i+2j,…,i+(g-1)j) for all choicesof i and j, doing arithmetic modulo g asneeded.
• For prime powers, the symbols used arethose of the finite field.
• For non-prime-powers, lots of openquestions!
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Direct MethodsDirect Methods
• OAs provide a direct construction of coveringarrays.
• Another direct technique chooses a group ong symbols, and forms a ‘base’ or ‘starter’array which covers every orbit of t-tuplesunder the action of the group.
• Then applying the action of the group to thestarter array and retaining all distinct rowsyields a covering array (typically exhibitingmuch symmetry as a consequence of thegroup action).
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Direct MethodsDirect Methods
• An example(-,0,1,3,0,2,1,4)
• Form eight cyclic shifts• Add a column of 0 entries• Develop modulo 5• Add the 6 constant rows (with – in last
column) to getCA(46;2,9,6)
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Direct MethodsDirect Methods
• Develop modulo 5• Add 6 constant rows (with – in last column)
0-412031000-412031010-412030310-412000310-412020310-410120310-404120310-
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Direct MethodsDirect Methods
• Stevens/Ling/Mendelsohn: FromPG(2,q) delete a point to obtain a frameresolvable q-GDD of type (q-1)(q+1).Extend a frame pc and fill in “don’t care’’positions to get a CA(2,q+2,q-1) with q2-1 rows.
• (C, 2005) Can be extended to get aCA(2,q+1+x,q-x) for all nonnegative x.Relies only on having a row with notwice-covered pair.
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Direct MethodsDirect Methods
• Sherwood: Rather than use the field asa group of symmetries, use partial testsuites build from the field and acompact means of determining when tsuch partial suites cover all possibilities.
• Sherwood, Martirosyan, C (2006): manynew constructions for t=3,4,5
• Walker, C (preprint): and for t=5,6,7.
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Recursive MethodsRecursive Methods
B
• A simple example (the Roux (1987) method).
B
AAA is a strength 3 covering array, 2 levelsper factor.
B is a strength 2 covering array, 2 levelsper factor.
The bottom contains complementaryarrays.
The result is a strength 3 covering array.
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Generalizing RouxGeneralizing Roux
• Extensions by– Chateauneuf/Kreher (2001) to t=3, all g– Cohen/C/Ling (2004) to t=3, adjoining more than
two copies, all g– Hartman/Raskin (2004) to t=4– Martirosyan/Tran Van Trung (2004) to all t under
certain assumptions– Martirosyan/C (2005) to all t, all g.– C/Martirosyan/Trung/Walker(2006) for t=3, t=4.
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Roux for twoRoux for two
• Prior to the Roux construction for t ≥ 3, Poljakand Tuza had studied a direct productconstruction when t=2.
• This forms the basis of methods of Williams,Stevens, and Cohen & Fredman.
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Roux for twoRoux for two
• Let A be a CA(N;2,k,v) and B a CA(M;2,f,v)
is a CA(N+M;2,kf,v).
AA A………
b1b1b1b1 b2b2b2b2bfbfbfbf
………
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Roux for twoRoux for two
• Stevens showed that when each array has vconstant rows, the resulting array has vduplicated rows and hence v rows can beremoved.
• A recent extension (CMMSSY, 2006) showsthat even when the arrays have “nearlyconstant” rows, again v rows can beeliminated.
• And an extension to mixed CAs.
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Roux for twoRoux for two
• Let O be the all zero matrix• Let C be a matrix with v rows, all of which are
constant and distinct• An SCA(N;2,k,v) A looks like
A1
OC
A2
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Roux for twoRoux for two
Let A be a SCA(N;2,k,v), B a SCA(M;2,f,v)minus v rows forming C,O
A1 A2A1 A2 A1………
b1b1b1b1 b2b2b2b2bfbfbfbf………
OO C O C O
has M+N-v rows
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PHF and PHF and TuranTuran Families Families
• Of particular note, but not enough time todiscuss in detail:– Bierbrauer/Schellwat (1999): use a “perfect hash
family” of strength t whose number of symbolsequals the number of columns of the CA.Substitute columns for symbols. Asymptotically thebest thing since sliced bread.
– Hartman (2002): Turan families used much likeabove but more accurate for arrays with fewsymbols.
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Four Values Per FactorFour Values Per Factor
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Six Values Per FactorSix Values Per Factor
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Ten Values Per FactorTen Values Per Factor
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13 Values Per Factor13 Values Per Factor
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TablesTables
• For more tables than you can shake astick at (and updates of the ones here),see– Colbourn (Disc Math, to appear) for t=2– C/M/T/W (DCC, to appear) for t=3, 4– Walker/C (preprint) for t=5
• We need better *general* directconstructions for small t, better recursionsfor large t.
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ThanksThanks