Covering Arrays with Row Limit Ottawa-Carleton Discrete Mathematics - - PowerPoint PPT Presentation

Covering Arrays with Row Limit

Ottawa-Carleton Discrete Mathematics Days University of Ottawa Nevena Franceti´ c Supervised by Prof. E. Mendelsohn and Prof. P. Danziger

Department of Mathematics University of Toronto

May 13-14, 2011.

Overview

1

Introduction Definition Some equivalent objects Goal

2

w = o(k) Bounds Constructions

3

w ∼ ck, 0 < c ≤ 1 Bounds

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 2 / 19

Covering array with row limit (CARL)

CARL(N = 12; t = 2, k = 6, v = 2: w = 4) : components 1 2 3 4 5 6 1 ∅ 1 ∅ t 2 ∅ 1 1 1 ∅ e 3 1 ∅ ∅ 1 s 4 ∅ ∅ t 5 1 1 ∅ ∅ 6 1 ∅ ∅ 1 r 7 1 ∅ 1 ∅ 1 u 8 1 1 ∅ 1 ∅ 1 n 9 ∅ 1 ∅ s 10 ∅ 1 1 ∅ 1 11 ∅ ∅ 1 1 12 ∅ 1 ∅

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 3 / 19

Covering array with row limit (CARL)

CARL(N = 12; t = 2, k = 6, v = 2: w = 4) : components 1 2 3 4 5 6 1 ∅ 1 ∅ t 2 ∅ 1 1 1 ∅ e 3 1 ∅ ∅ 1 s 4 ∅ ∅ t 5 1 1 ∅ ∅ 6 1 ∅ ∅ 1 r 7 1 ∅ 1 ∅ 1 u 8 1 1 ∅ 1 ∅ 1 n 9 ∅ 1 ∅ s 10 ∅ 1 1 ∅ 1 11 ∅ ∅ 1 1 12 ∅ 1 ∅

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 4 / 19

Covering array with row limit (CARL)

CARL(N = 12; t = 2, k = 6, v = 2: w = 4) : components 1 2 3 4 5 6 1 ∅ 1 ∅ t 2 ∅ 1 1 1 ∅ e 3 1 ∅ ∅ 1 s 4 ∅ ∅ t 5 1 1 ∅ ∅ 6 1 ∅ ∅ 1 r 7 1 ∅ 1 ∅ 1 u 8 1 1 ∅ 1 ∅ 1 n 9 ∅ 1 ∅ s 10 ∅ 1 1 ∅ 1 11 ∅ ∅ 1 1 12 ∅ 1 ∅ In general, CARLλ(N; t, k, {v1, v2, . . . , vk}: w).

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 4 / 19

Covering array with row limit (CARL)

CARL(N = 12; t = 2, k = 6, v = 2: w = 4) : components 1 2 3 4 5 6 1 ∅ 1 ∅ t 2 ∅ 1 1 1 ∅ e 3 1 ∅ ∅ 1 s 4 ∅ ∅ t 5 1 1 ∅ ∅ 6 1 ∅ ∅ 1 r 7 1 ∅ 1 ∅ 1 u 8 1 1 ∅ 1 ∅ 1 n 9 ∅ 1 ∅ s 10 ∅ 1 1 ∅ 1 11 ∅ ∅ 1 1 12 ∅ 1 ∅ In general, CARLλ(N; t, k, {v1, v2, . . . , vk}: w). Applicable for testing in studies in: pharmacology, medicine, agriculture, chemistry, etc.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 4 / 19

Equivalent objects

w = k: CA(N; t, k, v). v = 1: t − (k, w, λ) covering (V , B):

C1 C2 C3 C4 Ck v

...

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 5 / 19

Equivalent objects

w = k: CA(N; t, k, v). v = 1: t − (k, w, λ) covering (V , B):

C1 C2 C3 C4 Ck v

...

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 5 / 19

Equivalent objects

w = k: CA(N; t, k, v). v = 1: t − (k, w, λ) covering (V , B):

C1 C2 C3 C4 Ck v

...

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 6 / 19

What is the optimal size of a CARL?

Given t ≤ w ≤ k and v, we need to determine the smallest possible size of a CARL(t, k, v : w), denoted by CARLN(t, k, v : w). Also, we need to construct these objects.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 7 / 19

A lower bound

Theorem (Sch¨

• nheim bound, (Franceti´

c et al.))

CARLNλ(t, k, v : w) ≥ vk w v(k − 1) w − 1 · · · λv(k − t + 1) w − t + 1

• . . .
• .

CARL(12; 2, 6, 2: 4)

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 8 / 19

Sch¨

• nheim bound for fixed w and t = 2

Corollary

CARLN(2, k, v : w) ≥ vk w v(k − 1) w − 1

• .

Alon, Caro and Yuster (1998): bound is optimal for large k and w fixed. Heinrich and Yin (1999): w = 3. N.F., P. Danziger, E. Mendelsohn: w = 4 with regular excess graph.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 9 / 19

Sch¨

• nheim bound for fixed w and t = 2

Corollary

CARLN(2, k, v : w) ≥ vk w v(k − 1) w − 1

• .

Alon, Caro and Yuster (1998): bound is optimal for large k and w fixed. Heinrich and Yin (1999): w = 3. N.F., P. Danziger, E. Mendelsohn: w = 4 with regular excess graph. Any t ≥ 3: R¨

• dl(1985): w is fixed, v = 1, the bound is optimal.
• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 9 / 19

CARLs with w = o(k1/(t+1))

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 10 / 19

CARLs with w = o(k1/(t+1))

Theorem (Sch¨

• nheim bound)

CARLN(t, k, v : w) ≥ vk w v(k − 1) w − 1 · · · v(k − t + 1) w − t + 1

• . . .
• .
• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 10 / 19

CARLs with w = o(k1/(t+1))

Theorem (Sch¨

• nheim bound)

CARLN(t, k, v : w) ≥ vk w v(k − 1) w − 1 · · · v(k − t + 1) w − t + 1

• . . .
• .

Extending the proof of R¨

• dl(1985) using the interpretation of given by

Alon and Spencer(2000), we get the following:

Theorem (N.F.)

If limk→∞ wt+1

k

= 0, then CARLN(t, k, v : w) = k

t

• w

t

vt(1 + o(1)) = k(k − 1) · · · (k − t + 1) w(w − 1) · · · (w − t + 1)vt(1 + o(1)).

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 10 / 19

PARLs with w = o(k1/(t+1))

Also, we can extend this result to the Packing Arrays with Row Limit (PARL):

Theorem (N.F.)

If limk→∞ wt+1

k

= 0, then PARLN(t, k, v : w) = k

t

• vt

w

t

(1 + o(1)).

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 11 / 19

CARLs with w = o(k)

Theorem (N.F.)

If limk→∞ w

k = 0, then

CARLN(t, k, v : w) ≤ k

t

• w

t

vt

• 1 + ln

w t

• .
• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 12 / 19

CARLs with w = o(k)

Theorem (N.F.)

If limk→∞ w

k = 0, then

CARLN(t, k, v : w) ≤ k

t

• w

t

vt

• 1 + ln

w t

• .

Conjecture (N.F.)

Let w be a non-decreasing function of k. When limk→∞ w

k = 0, there

exists an optimal CARL(t, k, v : w) meeting the Sch¨

• nheim bound for

large enough k.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 12 / 19

Wilson construction

Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type 6

k−9 6 91, (V , G, B).

...... ...... ...... .........

...

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 13 / 19

Wilson construction

Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type 6

k−9 6 91, (V , G, B).

Let {b1, b2, b3, b4} ∈ B. Let v = 3.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 14 / 19

Wilson construction

Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type 6

k−9 6 91, (V , G, B).

Let {b1, b2, b3, b4} ∈ B. Let v = 3. · · · · · · b1 b2 b3 b4 · · · · · · ∅ ∅ 1 1 1 1 ∅ ∅ ∅ ∅ 1 2 2 2 ∅ ∅ ∅ ∅ 1 3 3 3 ∅ ∅ ∅ ∅ 2 1 3 2 ∅ ∅ ∅ ∅ 2 2 1 3 ∅ ∅ ∅ ∅ 2 3 2 2 ∅ ∅ ∅ ∅ 3 1 2 3 ∅ ∅ ∅ ∅ 3 2 3 1 ∅ ∅ ∅ ∅ 3 3 1 2 ∅ ∅

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 14 / 19

Wilson construction

Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type 6

k−9 6 91, (V , G, B).

...... ...... ...... .........

...

CARL(2,6,v:4) CARL(2,6,v:4) CARL(2,6,v:4) CARL(2,9,v:4)

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 15 / 19

CARLs with w = o(k) and t = 2

We can apply the Wilson construction. From affine planes we get the following:

Theorem

There exists an optimal CARL(q3(q + 1); 2, q2, q : q) for all prime powers q.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 16 / 19

CARLs with w = o(k) and t = 2

We can apply the Wilson construction. From affine planes we get the following:

Theorem

There exists an optimal CARL(q3(q + 1); 2, q2, q : q) for all prime powers q.

Lemma (N.F)

Let q be a prime power. If there exists an optimal CARL((q2 − q)(q2 − 1); 2, q2 − q, q : q), then there exists an optimal CARL((q2 − q)(q + 1)(q3 + q2 − 1); 2, q3 − q, q : q).

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 16 / 19

CARLs with w = o(k) and t = 3

Theorem

Let q be a prime power and k ≡ 2, 4 (mod 6). There exists an optimal CARL(N; 3, k, q : 4) with N = k(k−1)(k−2)

24

q3.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 17 / 19

CARLs with w = o(k) and t = 3

Theorem

Let q be a prime power and k ≡ 2, 4 (mod 6). There exists an optimal CARL(N; 3, k, q : 4) with N = k(k−1)(k−2)

24

q3.

Theorem

Let q be a prime power. There exists an optimal CARLq(N; 3, q2 + 1, q; q + 1) of size N = q5(q2 + 1).

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 17 / 19

CARLs with w ∼ ck, 0 < c ≤ 1

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 18 / 19

CARLs with w ∼ ck, 0 < c ≤ 1

Corollary (Sch¨

• nheim bound)

CARLN(t, k, v : w = ck) ≥ vt ct .

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 18 / 19

CARLs with w ∼ ck, 0 < c ≤ 1

Corollary (Sch¨

• nheim bound)

CARLN(t, k, v : w = ck) ≥ vt ct . Generalizing the bound by Godbole et al.(1996), we get the following:

Theorem

CARLN(t, k, v : w) ≤ k

t

• w

t

vt

• 1 + ln
• tvt−1

k − 1 t − 1

• .
• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 18 / 19

CARLs with w ∼ ck, 0 < c ≤ 1

Corollary (Sch¨

• nheim bound)

CARLN(t, k, v : w = ck) ≥ vt ct . Generalizing the bound by Godbole et al.(1996), we get the following:

Theorem

CARLN(t, k, v : w) ≤ k

t

• w

t

vt

• 1 + ln
• tvt−1

k − 1 t − 1

• .

Also, we can use the same counting method as Cohen et al.(1997), to get an algorithmic bound.

Theorem

CARLN(t, k, v : w) ≤ k

t

• w

t

vt ln

• vt

k t

• .
• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 18 / 19

References

Alon, Noga, Yair Caro, and Raphael Yuster. 1998. Packing and covering dense graphs, Journal of Combinatorial Designs 6, no. 6, 451–472. Alon, Noga. and Joel H. Spencer. 2000. The probabilistic method, 2nd ed., John Wiley, New York. Cohen, David M., Siddhartha R. Dalal, and Gardner C. Patton. 1997. The AETG system: An approach to testing based on combinatorial design, IEEE Transaction on Software Engineering 23, no. 7, 437–444. Franceti´ c, Nevena, Peter Danziger, and Eric Mendelsohn. Covering arrays with row limit w = 4, Journal of Combinatorial Designs. Submitted in February 2011. Godbole, Anant P, Daphne E Skipper, and Rachel A Sunley. 1996. t-covering arrays: upper bounds and poisson approximations, Combinatorics, Probability and Computing 5, no. 2, 105–117. Heinrich, Katherine and Jianxing Yin. 1999. On group divisible covering designs, Discrete Mathematics 202, no. 1-3, 101–112. R¨

• dl, Vojtˇ
• ech. 1985. On a packing and covering problem, European Journal of

Combinatorics 6, no. 1, 69–78.

• N. Franceti´

c (University of Toronto) CARLs May 13-14, 2011. 19 / 19