Decomposition oftwo-dimensional microlaser patterns

Hans Raj Nahata and Miles Murdocca

For an ordinary individually addressable microlaser array, a separate control line is used for eachmicrolaser, which requires a large number of control lines for even a small array. An organization thatreduces the width of the control stream and simplifies packaging is matrix addressing, in whichmicrolasers are arranged at the crossings of horizontal and vertical control lines. We consider theproblem of decomposing arbitrary two-dimensional microlaser patterns into matrix-addressablepatterns that are applied time sequentially to realize the target pattern. We present a mathematicalmodel for the decomposition process and present an algorithm for optimal decomposition. We alsoconsider bake factor, in which no more than N microlasers in a neighborhood of M 1where N , M2 areenabled, which avoids thermal overload by limiting the density of enabled microlasers. We concludewith a case study and show that, for completely arbitrary two-dimensional patterns, the averagenumber of time-sequential patterns is less than the number of rows in a square array. r 1996 OpticalSociety of America

1. Introduction

Two-dimensional 12D2 arrays ofmicrolasers aremanu-factured in two primary configurations: individu-ally addressable1 and matrix addressable,2 as illus-trated in Fig. 1. Eachmicrolaser in the individuallyaddressable array has a ground 1n2 terminal and apositive 1p2 terminal. All the microlasers share thesame ground, but a separate p contact is provided foreach microlaser. An 8 3 8 array thus requires 64 pcontacts, as indicated by the numbered bonding padsat the edges of the array. For small arrays, indi-vidual addressing works well, but the complexitybecomes unmanageable as the arrays scale to largesizes, and so an alternative configuration is neededthat scales more gracefully.For the matrix-addressable array, each row of

microlasers shares the same ground. For the eightrows shown in Fig. 11b2, there are eight independentn lines that are each connected to a distinct bondingpad. The p lines are connected to the columns in asimilar manner, and so there are eight independentp lines, which connect the p contacts of the eightmicrolasers in a column. To enable a microlaser at

The authors are with the Department of Computer Science, HillCenter, Rutgers University, New Brunswick, New Jersey 08903.Received 6 July 1995; revised manuscript received 14 Novem-

ber 1995.0003-6935@96@081195-10$06.00@0r 1996 Optical Society of America

location 1i, j2, in which i identifies a row and jidentifies a column, the corresponding i row and jcolumn bonding pads must be enabled. The nground is applied to the row pad, and the p potentialis applied to the column pad. If a potential isapplied to more than one pad, then the correspond-ing collection of microlasers is enabled. In Fig. 11b2,a potential is applied across rows 2, 3, and 5 andcolumns 3, 4, and 7, which enables the nine microla-sers at the corresponding crosspoints. Note thatonly six bonding pads are used, as opposed to thenine bonding pads for the same individually address-able configuration shown in Fig. 11a2.An advantage of thematrix-addressable configura-

tion is that for an N2 increase in the size of an array,the bonding pad complexity increases by only 2N,which allows for a simplified electronic interface.A disadvantage is that the user loses a degree offreedom in selecting combinations of microlasers toenable or disable. For example, in Fig. 11b2, there isno combination of bonding pads that can be poweredso that a checkerboard pattern is formed on thearray. Despite the limited number of possible on–off combinations for a matrix-addressable array, thecomplexity of the electronic addressing is simplified,which is an important practical consideration.Here we describe an algorithm for decomposing

arbitrary patterns for a 2D microlaser array into aset of matrix-addressable subpatterns, which, whenapplied in succession, achieve the desired target

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119

Fig. 1. 1a2 Individually addressable microlaser array, 1b2matrix-addressable microlaser array.

pattern. For discussion purposes, we assume thatthere is a flip-flop that maintains the state of eachmicrolaser 1allowing time-sequential pattern buildup2that can be set, reset, or toggled. The flip-flop is notneeded if the pattern buildup is recorded by anothermeans 1such as with film or a detector array2. Webegin by describing a mathematical model for theproblem. We then develop an algorithm for theoptimal decomposition of arbitrary patterns intomatrix-addressable subpatterns. We relate the de-composition method to the outer product of matrixmultiplication and to singular value decomposition1SVD2. We conclude with a case study of decomposi-tion applied to randomly generated patterns for an8 3 8 array and show that the expected number ofmatrix-addressable subpatterns that compose anarbitrary pattern is approximately 5.6 for an 8 3 8array.

2. Background

The decomposition method is based on binary matri-ces and symmetric patterns, as defined below:

Binary Matrix: Bn3n 5 1bi, j2 is a matrix in whicheach element bi, j is either 1 or 0. A binary matrix isalso called a 0–1 matrix. A one-dimensional 11D2case of a binary matrix is a binary vector.Symmetric Patterns: Pn3n 5 1pi, j2 is a binary

matrix with the following constraint3 for all pairs of1i, j2 and 1k, l2:

;1i, j2, 1k, l2 i, j c c k, j

pi, j 5 1, pi,l 5 1,

⇔pk,l 5 1, pk, j 5 1, i, l c c k, l

where ; indicates ‘‘for all’’ and ⇔ implies ‘‘in bothdirections’’; that is, for all 1i, j2 and 1k, l2 pairs thatform diagonal corners of a rectangle, there is another

6 APPLIED OPTICS @ Vol. 35, No. 8 @ 10 March 1996

pair, 1i, l2 and 1k, j2, that is also part of the pattern.Note that a symmetric patternmay have interveningpoints between the corners that are not part of thepattern, that it may have zero extent in any dimen-sion, and that it may have any number of points aslong as the above relationships are satisfied.Examples of symmetric patterns are shown in Fig.

21a2. Any matrix with a single 1 cell, a single

Fig. 2. 1a2 Examples of symmetric patterns, 1b2 a binary matrixthat is not a symmetric pattern.

column of 1’s, or a single row of 1’s is a symmetricpattern. For comparison, Fig. 21b2 shows a binarymatrix that is not a symmetric pattern. Note thatall nonzero rows of a symmetric pattern are the sameas are all nonzero columns.

A. Symmetric Patterns Compose Arbitrary Patterns

The key relationship between symmetric patternsand the decomposition of 2D microlaser patterns isthat any pattern that can be applied to a 2D array ofmatrix-addressable microlasers in a single step is asymmetric pattern. Conversely, every symmetricpattern can be applied to a 2D array of matrix-addressable microlasers in a single step. Our objec-tive is to decompose an arbitrary pattern into theminimal number of symmetric patterns that, whenapplied in succession, enable the target pattern.As an example, consider the 5 3 5 matrix shown inFigure 3, in which 16 of the 25 microlasers areenabled, forming a nonsymmetric pattern. Theoriginal pattern can be decomposed into three sym-metric patterns as shown.An arbitrary pattern can be expressed as a sum of

symmetric patterns. A decomposition always ex-ists; observe that a pattern with a single nonzeropoint is a symmetric pattern, by definition. There-fore any pattern can be expressed as a sum ofsymmetric patterns, each having only one nonzero

Fig. 3. Example of decomposition of a nonsymmetric pattern intothree symmetric patterns.

point. This is illustrated in Fig. 4, for one example.Note also that a decomposition need not be unique.In Fig. 5, a binary matrix is decomposed in twodifferent ways.The principal algebra used for the decomposition

process in the subsections below is Boolean AND–OR;that is, for two Boolean variables a and b, therelationships shown in Fig. 6 must hold. The ANDoperation produces a 1 when a and b are both 1 andproduces a 0 otherwise. The OR operation 1alsoreferred to as the logical sum2 produces a 0 when aand b are both 0, and produces a 1 otherwise. Thisalgebra is applied over binary matrices and binaryvectors, in a componentwise sense.

B. Equivalence Between Symmetric Patterns and theOuter Product

The above definition of a symmetric pattern is exact,but it does not expose the inherent structure of apattern. We now define symmetric patterns in anequivalent manner that makes a more direct connec-tion to matrix addressing.The outer product is a matrix product of two

vectors 1or, equivalently, a 1D row matrix R and a1D column matrix C 2. LetR13n 5 1ri2 and C13n 5 1cj2be two binary 1row2 vectors. The outer product Qn3nof these two vectors is RT 3 C , where RT is thetranspose of a row matrix into a column matrix.The notation for the outer product is Q 5 3R , C 415 RT 3 C 2.As an example, consider the outer product of R

and C shown below:

R 5 31 1 0 04,

C 5 30 1 0 14,

Q 5 3R , C 4 5 RT 3 C 5 31

1

0

04 3 30 1 0 14

5 30 1 0 1

0 1 0 1

0 0 0 0

0 0 0 04

Note that all the nonzero rows 1and nonzero columns2of Q are the same. Furthermore, the number ofnonzero rows 1or columns2 is directly related to thenonzero entries of R 1or C 2.

Fig. 4. Existence of decomposition shown as a sum of symmetric patterns that have only a single nonzero point.

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1198 APPLIED

Fig. 5. Decomposition need not be unique.

Every outer product Q of two binary vectors is asymmetric pattern P, and every nonzero symmetricpattern P can be expressed uniquely as an outerproduct of two binary vectors. In essence, the rowmatrix R and the column matrix C identify thebonding pads that need to be activated to enable asymmetric pattern onto a matrix-addressable array.As an example of this relationship, consider thefollowing:

30 1 0 1 1 1

0 1 0 1 1 1

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 1 1 1

0 0 0 0 0 0

4 ⇔R 5 31 1 0 0 1 04

C 5 30 1 0 1 1 14,

30 0 0

0 0 1

0 0 04 ⇔

R 5 30 1 04

C 5 30 0 14.

C. Properties of Symmetric Patterns

The outer-product formulation exposes properties ofsymmetric patterns that are important for the decom-position process, as described below.

Fig. 6. AND–OR Boolean algebra.

OPTICS @ Vol. 35, No. 8 @ 10 March 1996

1. Equivalence of Two Symmetric PatternsLet P1 5 3R1, C14 and P2 5 3R2, C24 be two nonzerosymmetric patterns. Then

P1 5 P2 ⇔ R1 5 R2, C1 5 C2.

Therefore it follows that an ordered pair of twobinary vectors 1R, C2 is equivalent to a patternP1P 5 Q 5 3R, C 4 5 RT 3 C 2 and vice versa.

2. Number of Enabled Cells in aSymmetric PatternThe number of 1’s in the pattern P is the same as theproduct of the number of 1’s inR and the number of1’s in C , where P 5 3R , C 4.

3. Permutations on Symmetric PatternsBe permuting the rows and the columns of P, we canalways convert it to the form,

31’s 0

0 04 ,that is, for any given pattern P, there exist twopermutation matrices P and F such that

P 3 P 3 F 5 31’s 0

0 04 .4. Number of Symmetric PatternsThe total number of distinct symmetric patterns is2n12n 2 22 1 2, where n is the number of row 1orcolumn2 elements in a square matrix. This is givenby

12n 2 12 3 12n2 12 1 1.

3 > 4

number of non- number of non- the zerozero R vectors zero Cvectors symmetric pattern

Contrast this with the total number of distinctbinary matrices, which is 2n2!The total number of distinct symmetric patterns,

discounting the transposed patterns, is given by:

12n 2 1212n 2 22 1 1.

Define the size of a symmetric pattern as the numberof 1’s it contains. The number of symmetric pat-terns of a given sizem is given by

o3;1p,q200,p,q#n

pq5m 41np21

n

q2 .

Again, n is the extent of a row or a column in asquare matrix, and p and q are the number of 1’s inthe row and the column vectors, respectively.

D. Merging of Symmetric Patterns

Symmetric patterns P1 and P2 are said to be merge-able if and only if their logical sum P 1equal toP1 1 P22 is also a symmetric pattern. Two symmet-ric patterns P1 1equal toR1

T 3 C12 and P2 1equal toR2

T 3 C22 are mergeable 1i.e., P1 1 P2 is also asymmetric pattern2 if and only ifR1 5 R2 or C1 5C2. Furthermore, the structure of the resultantsymmetric pattern P 1equal to P1 1 P2 5 RT 3 C 2is3

R1 5 R2 ⇒ 5R 5 R1

C 5 C1 1 C2,

C1 5 C2 ⇒ 5C 5 C1

R 5 R1 1 R2.

3. Decomposition Algorithm for Uniform Cost Model

In the first of the two decomposition methods, weassume that all symmetric patterns are equallydesirable, that is, the cost or penalty associated witheach symmetric pattern is the same. This differsfrom the second approach 1described in Section 42 inwhich localized thermal loads are reduced.

A. Algorithm

The problem of finding themost desirable decomposi-tion is essentially an optimization problem. Givena binary matrix B 1that is, a pattern2, we want todecomposeB into a sum of the minimum number ofsymmetric patterns 1Pi’s2. The objective function is

minPi’s

w.

The constraint is

oi51

w

Pi 5 B.

We define thematrix inclusion relation 1#2 as follows:A Boolean matrix A is said to be included in B if

and only ifA 1 B 5 B, that is,

A # B ⇔ B 5 A 1 B .

The decomposition algorithm has two phases.Given the binary matrix B ,

Phase 1. Find all symmetric patterns 1Pi’s2 in-cluded inB . Let this set be I.Phase 2. Compute C # I, such that C coversB

1that is, oP[C P 5 B 2. Note that I can be a verylarge set. Therefore we define I so that none of itsmembers can be contained in other members of I:

c P [ I ⇒ P # B .c For every nontrivial 1i, j, k2 triplet, there is no

symmetric pattern Pk that also contains two othersymmetric patterns Pi and Pj, that is, each symmet-ric pattern subsumes all symmetric patterns that itcontains:

Pi, Pj, Pk [ I ⇒ Pi 1 Pj fi Pk.

c P # B ⇒ 'Q [ I such that P # Q.

Let each element of I be called a prime implicant1symmetric pattern2 and let I be the set of primeimplicants. Note that

c I is not unique.c I covers B , that is, oP[I P 5 B .c Aminimal cover C for B need not be unique.c Phase 1 of the algorithm 1finding the set of

prime implicants2 is closely related to the Quine–McCluskey method of finding prime-implicant termsfor Boolean expressions.4 This is covered in Subsec-tion 3.B.

c Phase 2 of the algorithm is a case of theset-covering problem, which is covered in Subsection3.C.

B. Generating the Prime Implicants

To generate the set of prime implicants for B wefirst need to generate the set of base patterns H.Let H be the set of symmetric patterns for B , withthe following properties:

112 h [ H ⇒ h # B. 1A member of symmetricpattern setH is contained in binary matrix B .2122 P # B ⇒ 'K # H such that oh[k h 5 P. 1If

pattern P, which may or may not be symmetric, iscontained in binary matrix B , then there exists asubset of symmetric patterns K that is contained inH that composes P.2132 h1, h2 [ H ⇒ h1 1 h2 ” H. 1For any two

symmetric patterns in H, there is no pattern in Hthat contains both.2

Note that

112 The base set for any binary matrix is unique.122 By definition,H covers B 1that is,oh[H h 5 B2.

10 March 1996 @ Vol. 35, No. 8 @ APPLIED OPTICS 1199

The base set H for B is the set of all single 11symmetric2 patterns included inB . An example is:

31 0 1

0 0 0

1 0 14 5 3

1 0 0

0 0 0

0 0 04 1 3

0 0 1

0 0 0

0 0 04 1 3

0 0 0

0 0 0

1 0 04

1 30 0 0

0 0 0

0 0 14 .

The following procedure generates an optimaldecomposition. It closely resembles the classicalQuine–McCluskey method.4 The central idea is touse the merging rule on symmetric patterns in a waythat produces the smallest number of symmetricpatterns that reconstructs the original binary ma-trix.

Decomposition Algorithm for Uniform Cost Model112 Input: The binary matrixB .122 Preprocessing step: Compute the base set H

for B .132 Let W 5 H. Mark each element of W uncov-

ered.142 While there exists two mergeable patterns P1,

P2 inW, such that P1 ‹ P2 and P2 ‹ P1,x Let P 5 P1 1 P2.x Mark P1 and P2 covered.x LetW 5 W < P.x Mark P uncovered.End.Let I be the set of all uncovered patterns of

W. This is the set of prime implicants forB .Having computed the set of prime implicants I, we

compute the minimal cardinality set C # I thatcoversB , or equivalently that covers the base setH,that is, the smallest number of symmetric patternsthat composeB :

minC [2l

1 0C 0 0oP[C

P 5 oh[H

h 5 B 2 .C. Set-Covering Problem

Reference 5 describes the nonweighted case of theset-covering problem in the hypergraph setting andpresents greedy heuristics. We cast the set-cover-ing problem in the form of a 2D matrix 1differentfrom the microlaser matrix2 in which the m rows inthe matrix correspond to elements in the base set Hand the n columns correspond to elements in theprime implicant set I. The problem is to find the setA that contains the fewest columns that interseteach row at least once, taking the weightswi for eachmember of the base set into account. The weightsare all equal to 1 for the uniform cost model.In the standard form of any linear-programming

1LP2 problem, the variables are assumed to be non-negative real numbers. In integer linear program-ming 1ILP2, the variables are assumed to be nonnega-

1200 APPLIED OPTICS @ Vol. 35, No. 8 @ 10 March 1996

tive integers. The standard ILP formulation forweighted set covering6 is:

Given the weight 1cost2 vector W [ Rm, A [50, 16m3n,

minX[50,16m

1WTX 0AX $ e2,

where e 5 516m 1a vector of all 1’s2. This is simply aformal way of stating that a solution contains thefewest prime implicants and minimizes the overallcost. For the uniform cost model, this reduces tothe fewest prime implicants that cover the base set.In the above setting, xi 5 1 means that the ith

prime implicant is in the covering set C, and ai, j 5 1means that the jth prime implicant covers the ithelement of the base set.We describe a method for computing the minimal

cover even when the costs are not uniform 1in whichcase it is the minimal cost set-cover problem2. Abrief sketch of the strategy is as follows7:

112 First relax the ILP to LP. 1Relax the integral-ity constaint.2122 Add slack variables to A. 1Change the in-

equalities to equalities.2132 Find a primal cover X*. This is the subset K

of the set of prime implicants I such thatc K coversB ,c No proper subset of K coversB ;

1The greedy strategy can be adapted here to gener-ate primal covers2.142 Construct the corresponding basis b to LP.

1Every primal cover corresponds to a basis.2152 Invert b to get b8. 1This step is highly opti-

mized. Because the special structure of A and theproperty of the primal cover, b8 could be obtainedmerely by the permutation of the rows and thecolumns of A.7,82

162 Compute the dual solution.172 Check the dual feasibility:

c If the solution is dual feasible, than STOP, asthe present solution is an optimal solution.c Otherwise,1a2 Compute the set of indicesQ that violate the

dual-feasibility conditions.1b2 To the present LP add a constraint of the

form oi[Q Xi $ 1.1c2 Go to the step of computing the primal cover.

Note the following:

c All intermediate solutions are integral.c The actual computation is in proving the

optimality, so heuristics may be appropriate.c The inversion of the basis matrix b is almost

a trivial task.8

4. Decomposition Algorithm for NonuniformCost Model

In a system, not all symmetric patterns may beequally desirable because of thermal loads or otherpractical considerations. A symmetric pattern thathas dense nonzero clusters gives rise to local hotspots that may need to be reduced to meet systemconstraints. Here we present a model that incorpo-rates thermal bake-factor penalties associated withsymmetric patterns.Let P 35 1pi, j24 be a symmetric pattern, where w1P2

is the bake-factor penalty associated with P. Thereis more than one way to measure the bake factor.We define two classes of measures:

Measure 1: t-moment weight distribution oversquare templates 1of size k2:

wkt1P2 5 o

x51

n2k11

oy51

n2k11

1oi5x

x1k

oj5y

y1k

pi, j2t ,that is, within a neighborhood of size k 3 k on an n 3

n microlaser array, add up all the 1 elements withinthe neighborhood, raise that sum to the t power, andadd it into the total sum wk

t1P2. Variable t deter-mines the degree of variance from the desired value.For example,

w2331 0 1 0

1 0 1 0

0 0 0 0

1 0 1 04 5 30, w2

331 1 0 0

1 1 0 0

1 1 0 0

0 0 0 04 5 153.

Measure 2: threshold 1T2 on square templates 1ofsize k2:

wk,T 1P2 5 ox51

n2k11

oy51

n2k11

min31, max10, oi5x

x1k

oj5y

y1k

pi, j 2 T24 ,that is, wk,T 1P2 is 0 if the threshold T is not exceededwithin any neighborhood and is greater than 0 by thesum of the amounts that the neighborhoods exceedT. For example,

w2,331 0 1 0

1 0 1 0

0 0 0 0

1 0 1 04 5 0, w2,33

1 1 0 0

1 1 0 0

1 1 0 0

0 0 0 04 5 2.

Because the set-covering method described in Sec-tion 3 is general enough to take the weights intoaccount, only the first part of the decompositionalgorithm needs to be modified for the noniniform1weighted2 cost model:

112 Consider all symmetric patterns included inB.122 Choose some measure of penalty, such as

wkt1P2 or wk,T 1P2 and compute the weight of each

pattern.132 Continue the decomposition process as above

by setting up a 1larger2 set-covering problem andsolve it by the method sketched in the earlier sec-tions.

5. Exclusive- OR Algebra: Toggling the Patterns

In the sections above, we assumed that an AND–ORalgebra is used, that is, a symmetric pattern isenabled as a collection 1a logical OR2 of microlasersthat are enabled if and only if their correspondingrow and column bonding pads are activated.Here we look into a different exclusive-OR 1XOR2

algebra in which the state of a microlaser is toggled1changed from enabled to disabled or vice versa2 if itscorresponding row and column bonding pads areactivated. Physically, a flip-flop or some other de-vice should be attached to each microlaser so that itsstate can be retained and the XOR can be determinedlocally. Except for this assumption, the underlyingphysical model is the same as for the sections above.A summary of the XOR algebra is given in Fig. 7.

Note that

c The two-input XOR function is true 1produces a12 if and only if its inputs are unequal.

c In conventional digital electronics, the AND–ORalgebra is related to D flip-flops 1data latches2,whereas XOR is related to T flip-flops 1toggle latches2.

c XOR defines an Abelian group and, along withlogical AND, defines a finite field 1Galois modulo-2field2. As an example,

31 1 1 0

0 0 1 1

1 1 0 1

1 1 0 14 5 3

1 1 0 1

0 0 0 0

1 1 0 1

1 1 0 14 % 3

0 0 1 1

0 0 1 1

0 0 0 0

0 0 0 04 .

In order to develop an optimal decompositionmethodfor the XOR algebra, we need to borrow an importantconcept from matrix theory, which is covered inSection 6.

6. Singular Value Decomposition

SVD is a method of matrix decomposition that isappropriate for creating XOR symmetric patterns.

Fig. 7. XOR Boolean algebra.

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We make use of results from classical matrix analy-sis:

c Amatrix Q [ Rm3m is said to be orthogonal ifQ TQ 5 I, where Q T is the transpose of Q and I isthe identity matrix 1all 1’s along the diagonal and 0’selsewhere2.

c If A is a real m 3 n matrix, then there existorthogonal matricesU and V :

U 5 3u1, u2, . . . , um4 [ Rm3m

V 5 3v1, v2, . . . , vn4 [ Rn3n,

such that

UTAV 5 o 5 diag1s1, s2, . . . , sp2 [ Rm3n,

p 5 min5m, n6,

where

s1 $ s2 $ · · · $sp $ 0;

si’s are the singular values of A and the vectors uiand vi are the ith left singular vector and the ithright singular vector, respectively. si1A 2 in the ithlargest singular value of A .

c If the SVD is given by the above definition andwe define r by

s1 $ s2 $ · · · $ sr . sr11 5 · · · sp 5 0,

then we have r as the rank of A , and A could beexpressed as a weighted linear combination of outerproducts:

A 5 oi51

r

siuiviT.

7. Singular Value Decomposition Based Algorithm

In order to translate these results to our problemdomain, we make a few observations:

112 XOR with AND defines a field 1Galois, modulo-2field2.122 Therefore any result that holds true for a real

field applies to a modulo-2 field also.132 In our case A, U, and V are all binary

matrices.142 If the rank of A is r, then

s1 5 s2 5 · · · sr 5 1,

sr11 5 · · · sp 5 0.

152 In particular we are dealing with squarematrices; therefore p 5 n andA,U, V [ B n3n.162 In the absence of a specialized algorithm for

finite fields 1in particular for amodulo-2 field2, we canuse the algorithm available for real fields, with aslight modification. The crucial and only changerequired regards the notion of 1 1addition, in real

1202 APPLIED OPTICS @ Vol. 35, No. 8 @ 10 March 1996

numbers2. For a modulo-2 field we use % 1addition,or XOR, as defined in congruent modulo-22, in placeof 1.172 Observe that the minimum number of outer

products required for expressing the given matrix 1inXOR algebra2 is the same as the rank of the matrix.Therefore we can break the overall process into twosteps. Given a pattern A ,

1a2 Compute the rank of A.1b2 Depending on the complexity of the SVD

algorithm, the rank of A and the dimension of Adecide if the outer-product expansion is economical1as opposed to a simple row-by-row XOR decomposi-tion, which is no greater than the number of rows2.

182 References 9 and 10 describe an algorithm forcomputing the SVD for real fields.

8. Connection Between AND–OR and XOR

Decompositions

In spite of the simplicity of the XOR decomposition,neither the AND–OR nor the XOR decomposition isgenerally better than the other in terms of theminimal number of symmetric patterns needed toreconstruct the initial pattern. As an example inwhich the AND decomposition is worse than the XORdecomposition, consider the following initial pattern:

31 1 1 0

0 0 1 1

1 1 0 1

1 1 0 14 .

Aminimal AND decomposition requires three symmet-ric patterns:

30 0 0 0

0 0 0 0

1 1 0 1

1 1 0 14 1 3

1 1 1 0

0 0 0 0

0 0 0 0

0 0 0 04 1 3

0 0 0 0

0 0 1 1

0 0 0 0

0 0 0 04 .

In contrast, a minimal XOR decomposition requiresonly two symmetric patterns:

31 1 0 1

0 0 0 0

1 1 0 1

1 1 0 14 % 3

0 0 1 1

0 0 1 1

0 0 0 0

0 0 0 04 .

Conversely, as an example in which the XOR decompo-sition is worse than the AND decomposition, considerthe following initial pattern:

31 1 1 0

1 1 1 1

1 1 1 1

0 1 1 14 .

Aminimal AND decomposition requires two symmet-ric patterns:

31 1 1 0

1 1 1 0

1 1 1 0

0 0 0 04 1 3

0 0 0 0

0 1 1 1

0 1 1 1

0 1 1 14 .

In contrast, a minimal XOR decomposition requiresthree symmetric patterns:

31 1 1 0

1 1 1 0

1 1 1 0

0 0 0 04 % 3

0 0 0 0

0 1 1 1

0 1 1 1

0 1 1 14 % 3

0 0 0 0

0 1 1 0

0 1 1 0

0 0 0 04 .

9. Discussion

Although the matrix-addressable approach scalesmore gracefully in terms of pin count, the driverequirements per pin are greater, as an entire row orcolumn may need to be enabled, unless this con-straint is addressed by taking bake factor intoaccount. The effect of taking bake factor into ac-count may increase the number of symmetric pat-terns in the decomposition, however. Further, thetotal energy expended with these methods of decom-position may be greater than for individual address-ing. For example, the initial binary matrix shownin Fig. 3 has only 16 enabled microlasers, but thedecomposition into symmetric patterns requires 19enabled microlasers, although not all at once.In comparing the matrix-addressable approach

with the individually addressable approach, thequestion arises as to how much more time it takes toset up a matrix-addressable array than it takes foran individually addressable array 1which takes onlyone step, regardless of the initial pattern2. Figure 8shows a plot for an 8 3 8 array in which the numberof enabled microlasers varies from 1 to 63, and 1000random patterns are generated for each samplepoint. For an individually addressable array, the

Fig. 8. Decomposition results for randomly generated patternson an 8 3 8 matrix. 1000 sample points are taken for eachnumber of enabled microlasers 11–632.

number of activation patterns is only one for eachcase, as an arbitrary pattern can be applied directly.For amatrix-addressable approach, the best case is asingle activation pattern if the target pattern issymmetric, and the worse case is eight activationpatterns 1if eight microlasers are on a diagonal withdisabled microlasers elsewhere, for instance2. Asshown in the plot, the average for any number ofenabled microlasers is always better than the worsecase of eight. If we take the average of the aver-ages, assuming that any number of enabled microla-sers is equally likely, then the expected number ofactivation patterns is approximately 5.6.The expected value 15.62 is better than the worse

case 1eight activation patterns2, but still requires arelatively large number of activation patterns.Depending on the application, however, the targetpatterns may not be randomly distributed as theyare for the plot shown in Fig. 8. In fact, trulyrandom distributions are a rare behavior in commu-nication and computation in general, and we expectthat there will be a great deal of locality in applica-tions formicrolasers, especially for displays, in whichcase simple updates to the existing patterns mayfurther reduce the number of activation patterns.Because locality changes over time, the gains madepossible by locality will not be greatly offset by theeffects of laser aging, which is sometimes handled bythe removal of locality from the data stream 1which isnot necessary for this application2.The method as presented relies on set coverage,

which is a computationally hard problem 1of the classNP2 and is therefore not scalable to large arrays.However, the decomposition problem is tractable forsmall arrays 1such as 8 3 82, and simple changes tothe wiring pattern can ensure that a large arraybehaves as an array of tractable sized arrays. Forexample, the row–column addressing scheme can bechanged to a two-row–column addressing schemewith a nearly equivalent pin count but with a greatlysimplified decomposition. This can be done withoutincreasing the drive requirements for the pins byapplying bake-factor constraints to the decomposi-tion process.

10. Conclusion

We presented an optimal method for decomposingarbitrary 2D patterns into the minimal set of pat-terns that can be applied time sequentially to amatrix-addressable microlaser array. Two methodsare explored: AND–OR, in which the target patternis built up from a cleared array in an additivemanner; and XOR, in which the target pattern is builtup from a cleared array in an exclusive-OR 1toggled2manner. Neither approach is clearly better thanthe other when starting from a cleared array. Astudy of time-sequential buildup for an 8 3 8 arrayshows that the expected number of activation pat-terns for completely randomly occurring patterns is5.6, which is better than the worst case of eight.An open issue is how the expected value improves for

10 March 1996 @ Vol. 35, No. 8 @ APPLIED OPTICS 1203

real patterns when updates to the existing patternsare used.

RavindraAthale and Kannan Raj of GeorgeMasonUniversity are gratefully acknowledged for connect-ing the pattern decomposition methods to singularvalue decomposition. Konrad Zurl of the Univer-sity of Erlangen is gratefully acknowledged for draw-ing our attention to the greater drive requirementsof matrix-addressable arrays over individually ad-dressable arrays. This work was supported by theNational Science Foundation on grants MIP 92-24707 and ECS 93-12625.

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