Generating derivative superstructures for systems with high configurational freedom

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by  Wiley S. Morgan

Generating derivative superstructures for systems with high configurational freedom
Wiley S. Morgan
Department of Physics and Astronomy, Brigham Young University, Provo Utah 84602 USA
Gus L. W. Hart
Department of Physics and Astronomy Brigham Young University, Provo Utah 84602 USA
Rodney W. Forcade
Department of Mathematics Brigham Young University, Provo Utah 84602 USA
Modeling potential alloys requires the exploration of all possible configurations of atoms. Ad-
7 ditionally, modeling the thermal properties of materials requires knowledge of the possible ways
1 of displacing the atoms. One solution to finding all symmetrically unique configurations and dis-
0
placements is to generate the complete list of possible configurations and remove those that are
2
symmetrically equivalent. This approach, however, suffers from the combinatorial explosion that

n happens when the supercell size is large, when there are more than two atom types, or when
a there are multiple displaced atoms. This problem persists even when there are only a relatively
J small number of unique arrangements that survive the elimination process. Here, we extend an
9 existing algorithm1–3 to include the extra configurational degree of freedom from the inclusion of
displacement directions. The algorithm uses group theory to eliminate large classes of configura-

] tions, avoiding the combinatoric explosion. With this approach we can now enumerate previously
i
c inaccessible systems, including atomic displacements.
s
-
l
r I. INTRODUCTION of freedom, such as phonon models.13,14 There are nu-
t
m merous techniques available for modeling these systems
including cluster expansion (CE)15 and a recently devel-
. In computational material science, one frequently
t oped“smallsetoforderedstructures”(SSOS)method.16
a needs to list the “derivative superstructures”4 of a given
m However, the accuracy of these methods is still linked to
lattice. A derivative superstructure is a structure with
the number of unique configurations being modeled. In
- lattice vectors that are multiples of a “parent lattice”
d other words, if the model is trained on a small set of
and have atomic basis vectors constructed from the the
n configurations then it will not be able to make accurate
lattice points of the parent lattice. For example, many
o predictions. Increasingthenumberofconfigurationsused
c phases in metal alloys are merely “superstructures” of
to train the models can improve their predictive powers.
[ fcc,bcc,orhcplattices(L1 ,L1 ,B2,D0 ,etc.). When
0 2 19 Increasing the number of structures being used requires
modelingalloysitisnecessarytoexploreallpossiblecon-
1 a more efficient enumeration technique than those cur-
figurations and concentrations of atoms within these su-
v rently available.
2 perstructures. Whendeterminingifamaterialisthermo-
Leveraging the basic concepts of the algorithm pre-
8 dynamically stable, the energies of the unique arrange-
sented in Ref. 3, we altered the algorithm to have more
3 ments are compared to determine which has the lowest
favorable scaling in multinary cases. The basic idea is to
2 energy.
0 imaginetheenumerationasatreesearchandemploytwo
Derivative superstructures are found using combina-
. new ideas: (1) “partial colorings” and (2) stabilizer sub-
1 toric searches1–3,5–8, comparing every possible combina-
groups. Sec. III illustrates the algorithm with a concrete
0
tion of atoms to determine which are unique. However,
7 example.
these searches can be computationally expensive for sys-
1 The concept of partial colorings is to skip entire
tems with high configurational freedom and are some-
: branches of the tree that are symmetrically equivalent
v times impractical due to the large number of possible
i to previously visited branches. A partial coloring is an
X arrangements.
intermediate level in the tree (see Fig. 1) where config-
r Theinefficiencyofcombinatoricsearchesmakesfinding urations are not yet completely specified. It frequently
a
theuniquederivativesuperstructuresalimitingfactorin happensthat symmetric redundancy can be identified at
searches for high entropy alloys (HEA)9–11. The config- an early, “partially colored” stage, avoiding the need to
urational complexity of HEAs prevents them from phase descend further down the tree.
separating;thissamecomplexitymakeslistingeverypos- Stabilizer subgroups further increase the efficiency of
sible arrangement of atoms impractical with current al- the new algorithm. Any symmetrically-equivalent full
gorithms. colorings further down the current branch will have the
Otherproblemsimpairedbytheinefficiencyofcurrent same partial coloring. Thus, the only symmetries that
enumeration methods include modeling materials that are relevant are those that leave the current partial col-
have disorder in their structures, such as site-disordered oring unchanged. These symmetries form a (stabilizing)
solids12 orthatincludeatomicdisplacementsasadegree subgroupofthefullgroup. Thissignificantlyimpactsthe
2
efficiency because the stabilizer subgroup is often much III. TREE SEARCH
smaller than the full group.
Once a supercell has been selected, the remainder of
the enumeration algorithm resembles a tree search in
which each branch corresponds to a specific configura-
II. SUPERCELL SELECTION AND THE
tion of atoms within the supercell, many of which are
SYMMETRY GROUP
not fully populated and are called partial colorings (see
Fig. 2). The partial colorings are identified using a vec-
The first step in enumerating derivative superstruc-
tor that indicates their locations within the tree. Once
tures is the enumeration of unique supercells. This step
a partial coloring is constructed, the stabilizer subgroup
has already been solved by Hart and Forcade1, but due
forthatpartialcoloringisfound. Thestabilizersubgroup
to its importance to the algorithm we provide a brief
allowsforthecomparisonofbrancheswithinthetreeina
overview.
manner that minimizes the number of group operations
The supercells, of size n, are found by constructing all
used. These tools, (partial colorings and the stabilizer
Hermite Normal Form (HNF) matrices whose determi-
subgroup), are used to “prune” branches of the tree as
nant is n. An HNF matrix is an integer matrix with the
they are being constructed, eliminating large classes of
following form and relations:
arrangements at once.
We will use a 2D lattice of 9 atoms as an illustrative
 
a 0 0 example of the algorithm. The lattice will be populated
b c 0,0≤b<c,0≤d<f,e<f (1) with the following atomic species; 2 red atoms, 3 yel-
d e f low atoms, and 4 purple atoms. A subset of the possible
arrangementsofthissystemisshowninFig. 2. Thecon-
where acf = n. The HNFs determine all possible the cepts illustrated with this 2D example are equally appli-
supercellsforthesystem. Forexample,considera9-atom cable in 3D.
cell, then n = 9 and a, c, f are limited to permutations
of (1,3,3) and (1,1,9). Then following the rules for the
values of b, d, and e, every HNF for this system can A. Partial Colorings
be constructed. These HNFs represent all the possible
supercells of size n of the selected lattice. Some of these Whensearchingforalluniqueconfigurations,itisuse-
are equivalent by symmetry, so the symmetry group of ful to know, a priori, how many configurations are ex-
the parent lattice is used to eliminate any duplicates. pected. Arecentlydevelopednumericalalgorithmforthe
Next, we convert the symmetries of the lattice to a P´olyaenumerationtheorem17–19allowsonetoquicklyde-
list of permutations of atomic sites. There is a one-to- termine the memory requirements of storing the unique
one mapping between the symmetries of the lattice and arrangements. For the 9 atom system considered here,
atomicsitepermutations,i.e.,thegroupsareisomorphic. the P´olya algorithm predicts that there are 24 unique
The mapping from the symmetry operations to the per- arrangements to be found.
mutationgroupisaccomplishedusingthequotientgroup The algorithm places atomic species on the lattice ac-
G = L/L(cid:48), where L is the lattice, constructed from the cording to their concentrations. In this case, the red
unitcell,andL(cid:48) isthesuperlattice,constructedfromthe atoms have the lowest concentration and are placed in
supercell. The quotient group G is found directly from the first two sites of the cell creating the first 1-partial
the Smith Normal Form (SNF) matrices, which can be coloring (a partial coloring is a configuration with only a
constructed fromthe HNFs via astandard algorithm us- subsetoftheatomsdecoratingthelattice). Thisisshown
ing integer row and column operations. Thus S =UHV in the leftmost configuration, labeled (0,•,•), in the sec-
whereU andV areintegermatriceswithdeterminant±1 ond row of Fig. 1. The general procedure is to apply
and S is the diagonal SNF matrix, where each positive the symmetry group to each partial coloring in order to
integer diagonal entry divides the next one down. The make comparisons between partial colorings and deter-
(cid:76) (cid:76)
group, G, is then G = Zs1 Zs2 Zs3, where si is ith mine if they are symmetrically equivalent. For example,
diagonal of the SNF and Zsi represents the cyclic group in Fig. 1, the configuration labeled (1,•,•) is equivalent
of order n. to configuration (0,•,•) by a translation of the lattice.
Oncethesupercellshavebeenfoundandtheirsymme- At this stage we only have one partial coloring so it is
trygroupshavebeenconvertedtotheisomorphicpermu- unique and no comparisons need to be made, however
tation group, the algorithm can begin finding the unique the symmetry group is still applied to find the stabilizer
arrangements of atoms within each supercell in a tree subgroup described in section IIIB.
searchframework. Thisisaccomplishedbytreatingeach Comparisons between configurations are made by us-
supercellwithitssymmetrygroupasaseparateenumer- ing a hash function. In computer science, any data set
ation problem. The results of the enumeration across all canbeplacedina hashtablewhichassociatesa hash, or
supercellsarethencombinedtoproducethefullenumer- label, with the data. In our case, the configurations are
ation. listedwithinthehashtableintheordertheyarecreated.
3
( , , )
(0, , ) (1, , ) (2, , ) (3, , ) (11, , ) (16, , ) (24, , ) (35, , )
(0, , ) (0, , ) (3, , ) (3, , ) (3, , ) (0, , )
A B
FIG.1. (Coloronline)Theemptylatticeand8ofthe36configurationswithonlyredatomsareshownfortheexamplediscussed
in section III. Above each partial coloring is a vector that indicates it’s location in the tree, i.e. (x ,x ,x ), where the x s
r y p i
are integers that indicate which arrangement of that color is on the lattice and a • means that no atoms of that color have
beenplacedyet. Beloweachconfigurationiseitherthelabelofasymmetricallyequivalentconfiguration,alongwiththegroup
operation that makes them equivalent, or the letters A and B. A and B are the branches that are built from the 1-partial
colorings that are unique and are displayed in Fig. 2
The hash function then maps the configuration to a vec- assigned yet then the x s are replaced by dots indicating
i
tor of integers with an entry for each species, color, in an empty vector site.
the system. The hash function used is similar to the one Thehashfunctionisaone-to-onemappingbetweenthe
described in Ref. 3. However, due to its importance in configurations to the location vectors. These numbers
this algorithm, we provide an overview of how the hash are constructed by considering each color separately and
function works. building a binary string of the color and the remaining
The hash function for the algorithm uses the princi- empty lattice sites, where the color is a 1 and the empty
ples of combinatorics to uniquely identify each partial site is a 0 within the string. From the binary string, we
coloring using an integer vector. Its construction starts can then use a series of binomial coefficients to find the
by determining the number of possible ways to arrange x ’s. Thebinomialcoefficientsarefoundbytakingeach0
i
the colors on the lattice. The number of possible config- inthestringthathas1’stotherightofitandcomputing
urations can be found using the multinomial coefficient, (cid:0) p (cid:1), where p is the number of digits to the right and
q−1
whichisequivalenttotheproductofbinomialcoefficients q is the number of 1’s to the right of the 0. Summing
for each individual color: thebinomialsforqualifyingzeroproducesanumberthat
tellsushowmanyconfigurationscamebeforethecurrent
(cid:18) (cid:19)
n
C = =C C ...C = one.
a ,a ,..,a 1 2 k
1 2 k Asanexampleofthehashfunctionthatconstructsthe
(2)
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
n n−a n−a −a −...−a location vector, consider configuration (3,19,0) of Fig.
1 ... 1 2 k ,
a a a 2B. The construction begins with the red atoms repre-
1 2 k
sented as the following binary string (1,0,0,0,1,0,0,0,0),
where n is the number of sites in the unit cell and where every atom that is not red has been represented
a ,a ,...,a are the number of atoms of species i such by a 0 and the red atoms by a 1. This string has 3 zeros
1 2 k
that (cid:80) a =n. The binomials determine the number of that have a single 1 to their right, the first zero has 7
i i
ways to place the atoms of each color within the lattice digitstoitsright,thesecondhas6atomstoitsrightand
once the previous colors have been placed. By assigning the third has 5 atoms to its right. The resultant sum of
each partial coloring an integer, xi, from 0 to Ci − 1, binomials is xr = (cid:0)70(cid:1)+(cid:0)60(cid:1)+(cid:0)50(cid:1) = 1+1+1 = 3. This
where i is the color, we can build a vector that identifies result is the first entry in our location vector.
the location, (x ,x ,...,x ), of the configuration within The second entry in the location vector is constructed
1 2 k
the tree. For example, there are C = (cid:0)9(cid:1) = 36 ways to for the yellow atoms. The bit string representation of
r 2
place the red atoms on the empty lattice. After the red theyellowatomsis(0,1,0,1,1,0,0), thereareonly7digits
atoms are placed then there remain C =(cid:0)7(cid:1)=35 ways because the 2 red atoms have already been placed, so
to place the yellow atoms on the remayining3lattice sites. x = (cid:0)6(cid:1)+(cid:0)4(cid:1) = 15+4 = 19. The last entry in the
This leaves C =(cid:0)4(cid:1)=1 way to place the purple atoms loycation2vecto1r is built for the purple atoms which have
p 4
on the lattice. Within Fig. 1 and 2, the vector locations the bit string (1,1,1,1), so x = 0. The location vector
p
have the form (x ,x ,x ) and if the color has not been is complete once all atoms within a configuration have
r y p
4
(0, , )
A
(0,0, ) (0,1, ) (0,2, ) (0,3, ) (0,4, ) (0,5, ) (0,11, ) (0,15, ) (0,16, ) (0,17, ) (0,18, ) (0,19, ) (0,20, ) (0,21, ) (0,22, ) (0,34, )
(0,1, ) (0,16, ) (0,16, ) (0,15, )
(0,0,0) (0,1,0) (0,2,0) (0,3,0) (0,4,0) (0,11,0) (0,15,0)(0,16,0) (0,18,0) (0,19,0) (0,20,0) (0,21,0)
(3, , )
B
(3,0, ) (3,1, ) (3,2, ) (3,3, ) (3,4, ) (3,6, ) (3,8, ) (3,9, ) (3,11, ) (3,19, )(3,20, ) (3,21, ) (3,23, )(3,24, ) (3,26, ) (3,34, )
(3,0, ) (3,19, ) (3,2, ) (3,21, )
(3,0,0) (3,1,0) (3,2,0) (3,3,0)(3,4,0) (3,8,0) (3,9,0) (3,11,0) (3,19,0) (3,21,0) (3,23,0)(3,24,0)
FIG. 2. (Color online) Here the A and B branches of the tree from Fig. 1 are shown. Each branch starts with the initial 1-
partialcoloringthebranchisbuiltfrom((0,•,•)and(3,•,•)respectively). Thebranchesthenshowaselectionofthe2-partial
coloringsforthatbranch,andtheuniquefullcoloringsthatarefound. AsinFig. 1thevectorsthatindicatetheconfigurations
location in the tree are displayed above the configurations and the symmetrically equivalent labels appears beneath them. In
thisfiguretheactionsthatmaketheconfigurationshavebeenexcludedduetotheircomplexity. Forexample,Theconfiguration
labeled (0,5,•) is equivalent to the (0,1,•) configuration by a rotation about the vertical followed by a translation to the left.
In the B branch configuration (3,19,0) is outlined for reference because it is used as an example later in the text.
been included. then the corresponding configuration has already been
The location vectors allow us to determine if a config- visited.
uration is unique by checking if an element of the sym-
metry group maps the configuration to a configuration
withasmallerlocationvector. Thesymmetryoperations
B. The Stabilizer Subgroup
map a configuration’s location to a second, equivalent
location. Uniqueness is determined by comparing the
original and mapped locations for the configuration; if Thealgorithmisefficientbecausetheentiresymmetry
the mapped configuration has already been enumerated, group does not need to be applied to a partial coloring,
that is, if x > x , then the configuration is only the stabilizer subgroup of the partial coloring one
original mapped
not unique because it is equivalent to one we have al- level up the tree is needed. The stabilizer subgroup is
ready visited. For example, configuration (2,•,•) shown found when the symmetry group was applied to the par-
in Fig. 1 can be turned into configuration (0,•,•) by a tial coloring one level higher up the tree, so finding the
180 degree rotation about the diagonal. Since (2,•,•) stabilizersubgroupcostsnothingcomputationally. Asan
and (0,•,•) are equivalent we conclude that (2,•,•) is example of an element of the stabilizer subgroup, con-
notuniquebecause2>0. Insummary, ifanyelementof sider the cell (3,•,•), displayed in Fig. 1, and reflect it
thesymmetrygroupmakesthelocationvector“smaller”, about the diagonal; the red atoms are unaffected. This
5
also unique, and records its stabilizer subgroup. Next,
it places the purple atoms to get the configuration at
(0,0,0); this configuration is saved, then the algorithm
backs up to the 2-partial coloring level to consider the
configuration (0,1,•) and find its stabilizer subgroup.
Once this process has been repeated for all 34 partial
FIG. 3. (Color online) The configuration (3,0,•), shown on colorings in the vector (0,xy,•) (0≤xy ≤34=Cy), the
the left, is acted on by a reflection about the diagonal re- algorithm retreats to the 1-partial coloring level shown
sultinginconfiguration(3,6,•),shownontheright. Because in the second row of Fig. 1 and finds that (1,•,•) and
the symmetry group operation is a stabilizer for the configu- (2,•,•)areequivalentto(0,•,•). Itthenbeginstobuild
ration (3,•,•) the red atoms were not affected. A stabilizer the (3,•,•) branch, of Fig. 2 B, in the same manner as
is a group element that leaves the set invariant. The yellow
the (0,•,•) branch.
atoms, however, were mapped to a different configuration.
Since there are only two unique 1-partial colorings for
This means we can use just the stabilizer subgroup for the
thissystemthealgorithmiscompleteoncebothbranches
(3,•,•)configurationtocompareallthe2-partialcoloringsof
that originate from these 1-partial colorings have been
the form (3,x ,•), where (0 ≤ x ≤ C −1), because any
y y y
other group operation would map us to a different branch of explored. In the end, 24 unique configurations are found
the tree. (shown in Fig. 2A and 2B), in agreement with the pre-
diction from the P`olya enumeration algorithm.
means that a reflection about the diagonal is a mem-
ber of the stabilizer subgroup for the 1-partial coloring C. Extension to Include Additional Degrees of
(3,•,•). In general, only a small subset of the symmetry Freedom
group will be in the stabilizer subgroup for any partial
coloring.
Having established the algorithm, we will now ad-
The stabilizer subgroup leaves the desired n-partial dress its extension to include displacement directions.
coloring unchanged, where n is the depth in the tree. These enumerations are more difficult because includ-
Once another color is added (making an (n+1)-partial ing displacement directions changes the action of the
coloring) the stabilizer subgroup for the n-partial color- group. Displacement directions simply indicate the di-
ing becomes the only group operations that can be ap- rection that an atom could be displaced off the lattice.
plied without affecting the n-partial coloring. In other Theenumerationofstructuresthatincludedisplacement
words, if we were to use any other group elements we directions can be used to build databases20 of possible
wouldbecomparingconfigurationsthatwealreadyknow structures with displacements included.
are equivalent on the n-partial coloring level. Our algorithm changes only slightly if displacement
Onceauniquen-partialcoloringanditsstabilizersub- directions are included in the enumeration. First, the
grouphavebeenfound,thealgorithmproceedsdownthe atoms that will be displaced are treated as a different
branch to the (n+1)-partial colorings, see Fig. 2. To atomicspeciessothateachdisplacedatom’suniqueloca-
check the uniqueness of the (n+1)-partial colorings, the tionscanbedetermined(seeFig. 4foranexamplewhere
stabilizer subgroup from the n-partial coloring are used yellow displaced atoms are replaced with the red atoms
and the stabilizer subgroup for the (n+1)-partial color- from the example system used above). Once the arrows
ingsarestored. Whenauniqueconfigurationisfoundon havebeenreplacedbyatomicspecies,thealgorithmpro-
the(n+1)levelanothercolorisadded,makingthe(n+2)- ceeds as normal until a full configuration is found. The
partial colorings, and the process starts over again. algorithmthenrestoresthearrowsandusesthestabilizer
The algorithm proceeds down a branch of the tree un- subgroupofthefullconfigurationtocheckforequivalent
til a unique full configuration is found, such as (0,0,0) of arrow configurations.
Fig. 2. When the full configuration is found, the algo- In order to determine if the combined arrow and color
rithm backs up one level and considers the next partial configuration is unique, each group element has to be
coloring. When no partial colorings are available on a paired with a second set of permutations that determine
level, the algorithm backs up until it finds a level with how the symmetry operation affects the arrows. The ef-
untestedpartialcolorings. Inthismanner,theentiretree fect on the arrows is represented as a permutation of
is explored but only sections with unique configurations the numbers 0 to d−1, where each number represents
are explored in detail. a different displacement direction up to the d directions
For an example of the complete algorithm, consider beingconsidered. Forexample,ifweconsiderthesystem
Figs. 1 and 2. The algorithm starts at (•,•,•) then in Fig. 4, we have two atoms being displaced along one
builds the 1-partial coloring at (0,•,•), which is unique of the 6 cardinal directions, then any arrow could have
by virtue of being the first partial coloring considered values of between 0 and 5 where each integer has an as-
on this level, and records its stabilizer subgroup. The sociated direction; up=0, right=1, down=2, left=3, into
yellowatomsarethenaddedtotheconfigurationtobuild the page=4, and out of the page=5. The initial arrow
the 2-partial coloring at (0,0,•), of Fig. 2 A, which is vector, shown in the figure, is (up,up) and is represented
6
as (0,5) has been considered. At this point all possible
arrangements that have the first arrow pointing up have
been considered, so the second arrow is set to point up
andfirstarrowisrotatedtomakethearrangement(1,0).
Wethengobacktoincreasingthelastentryinthevector
to create new arrangements in order to determine if any
FIG. 4. (Color online) To include displacement directions of them are unique until (1,5) is reached. The process
to the algorithm we represent the atoms to be displaced by is repeated until all possible the arrangements, i.e., all
a unique color and then convert them back once a unique 2-tuplesof0...(d−1),havebeenconsidered. Onceallthe
configuration is found. In this figure two displaced yellow vectorshavebeenconsidered,thealgorithmgoesbackup
atomsarerepresentedbyredatomsuntilthepreviousportion the tree to find the next unique configuration of colors.
ofthealgorithmiscomplete,thentheyarereplacedbyarrows
In this manner, discrete displacement directions can
again for the arrow enumeration.
be added to the configurations. In this example, adding
arrows to the system increases the number of possible
arrangements to 45360 (the number of possible arrange-
as (0,0). mentsforjusttheatomsis1260). However,theresultant
The comparison of the rotated and unrotated arrows number of unique arrangements is only 663.
is achieved using a hash function. This function gives
eacharrowconfigurationauniquelabelthatcorresponds
to the order it is constructed within the algorithm. IV. CONCLUSION
This hash function takes a vector of arrow directions
(a0,a1,a2,....,ak), where ai is an integer from 0 to d−1 Our previous algorithms1–3 explored configuration
indicating the direction of the ith arrow and k+1 is the space by comparing all possible configurations of the
number of arrows, and finds: atoms to eliminate those that were symmerically equiva-
lent. These algorithms were only effective for systems
k with relatively small amounts configurational freedom
(cid:88)
xa = aidi (3) duetothecombinatoricexplosionthatoccurforsystems
i=0 with high configurational freedom and were incapable of
enumerating systems that included displacement direc-
Thisgiveseacharrowarrangementauniqueintegerlabel tions.
thatwecanthencomparebetweensymmetryoperations. With this new algorithm, it is now possible to find
As was the case for the configurations, if the effect of a the unique arrangements of systems with high configu-
symmetry operation results in a relationship of x > rational freedom. The systems now accessible include k-
old
x , then the the arrow configuration is not unique and nary alloys and structures with displacement directions.
new
can be ignored. This is accomplished by using an approach which closely
The stabilizer subgroup for the unique color configu- resembles a tree search, in which large classes of config-
ration are used to map the arrows to new directions and urations are eliminated at a time. In this manner, we
the hash function is used to compare the original and are able to avoid the combinatoric explosion which im-
mapped arrows. After an arrow arrangement is checked, pedes the performance of the previous algorithms. This
thealgorithmthenincreasesthemagnitudeofthelasta algorithm’sabilitytoefficientlydeterminetheuniquear-
k
in the vector by 1 and checks it for uniqueness with the rangements of these systems enables more effective com-
stabilizer subgroup. If increasing the magnitude of a putational studies.
k
would cause it to be greater than the value of d−1 then This research was funded by ONR grant MURI
a becomes0againanda isincreasedby1. Thispro- N00014-13-1-0635. Thisalgorithmhasbeenimplemented
k k−1
cess is repeated until all the entries in the arrow vector in the enumlib package and is available for public use21.
are equal to d−1.
For example, the initial arrow vector for the system
V. ACKNOWLEDGEMENTS
shown in Fig. 4 is (up,up) and is represented as (0,0). It
isfoundtobeuniquesinceitisthefirstarrangement. For
thenextarrangementthearrowontherightisrotatedto Thisworkwassupportedunder: ONR(MURIN00014-
point to the right creating the arrangement represented 13-1-0635).
as (0,1). This arrangement is also checked to see if it is
unique. The right most arrow continues to be rotated
every time a new arrangement is constructed until it is VI. REFERENCES
pointingoutofthepageandthearrangementrepresented
7
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