Mark
Dow
Geek
art
Minimal
arrays
containing all subarray combinations of symbols: De Bruijn sequences
and tori
For
n symbols in
m groups, what are the smallest
arrays that contain all combinations of adjacent elements?
This is a
symbolic
combinatorics question. It shares some characteristics with the
necklace
problem.
I know very little about this topic, and have not attempted to
formalize the problem in standard mathematical language. I don't even
understand the standard mathematical language used. The
observations and algorithms below were derived from tinkering and
intuition.
Since I wrote most
of this, I found that these
problems have been worked on, and many of the simpler cases fully
characterized. The onedimensional cases are called De Bruijn
sequences, and the twodimensional cases De Bruijn tori:
1D
2symbol, every pair
1D
2symbol, every triplet
1D
2symbol, every quadruplet
1D
2symbol, every quintuplet
1D 3symbol, every triplet
1D 3symbol, every sextet
1D 4symbol, every pair
1D 4symbol, every triplet
2D
2symbol, every 2x2
2D 3symbol, every 2x2
2D 4symbol, every 2x2
2D
2symbol, every 3x3
Corners
tilings
Programs and code
References
This problem came up
when designing motifs for aperiodic tesselations,
for example
ThueMorse
patterns. In a 1D 2symbol ThueMorse sequence
all pairs of symbols occur (
aa,
ab,
ba,
bb
) but no triplets (of symbols or sets of sequential symbols). Extended
to 2D and more symbols, aperiodic sequences can have subarrays of
symbols that occur infrequently. [To Do: 4symbol cyclic system
example.] It would be nice to be able to draw all possible
neighborhoods (sets of adjacent motifs) with no repetitions.
It is convenient to consider the symbols as
sequential integers starting at
0
because the possible pairs can be
enumerated as all possible integers with a base of the number of
symbols.
[To Do: Draw corresponding directed graphs for all 1D cases.]
1D 2symbol, every pair (k =
2, n = 2)
There are four unique pairs of two symbols.
Using {0,1}
as symbols they correspond to the binary (base 2) numbers between 0 and
3:
{ 00, 01, 10, 11 }
A bit of trial and error yields a string with all adjacent pairs
and no duplications:
[ 0 0 1 1 0 ]
The mirror and complements of this string also are solutions:
[ 0 1 1 0 0 ]
[ 1 0 0 1 1 ]
[ 1 1 0 0 1 ]
Note that the first symbol is the same as the last, so they can wrap
around as a length four ring. The solutions
are all cyclic permutation of a single ring:
[ ... 0 0 1 1 ... ] 

The graphical representation (right) uses black and white to represent
0 and 1 respectively, and is read clockwise from the top. It is the
only solution, as any mirror or complement is the same as a cyclic
permutation of this solution.
1D 2symbol, every
triplet (k = 2, n = 3)
There are eight unique triplets of two
symbols. Using {0,1}
as symbols they correspond to the binary (base 2) numbers between 0 and
7:
{ 000, 001, 010, 011, 100,
101, 110, 111 }
A solution can be found by starting with [ 0 0 0 ], and
iteratively add a symbol to the right with these conditions:
(a) Choose the symbol that will form a triple that is not yet in the
string
(b) If both symbols can be added to result in a new triplet, use the
one that will most closely balance the number of each symbol.
(c) If the number of symbols is
balanced, arbitrarily select the next symbol as 0.
Here's the sequence of adding to the string, with the appropriate
conditions:
[ 0 0 0
]
[ 0 0 0 1
] (a)
[ 0 0 0 1 1
] (b)
[ 0 0 0 1 1
1
] (b)
[ 0 0 0 1 1 1
0 ] (a), because 111 is
already in the string
[ 0 0 0 1 1 1 0
1 ] (b), because both
100 and
101
aren't
in the string, but there are more 0's than 1's
[ 0 0 0 1 1 1 0 1 0
] (a) Note that the last symbol is the
first symbol wrapped around.
[ 0 0 0 1 1 1 0 1 0
0 ] (a)
Note that
the last two symbols are the first symbols wrapped around.
Condition (c) is not used here, but will be for longer substring
solutions.
Does this algorithm result in a De Bruijn sequence
for all
2symbol strings,
of any length? Is there a generalization for more than 2 symbol
strings? Does it matter what triplet is used as the initial state?
Similar to the "every
pair" solution, the first two
symbols are the
same as the last, so all triplets occur in the eight term
ring:
[ ... 0 0 0 1 1 1 0
1 ... ] 

and in any cyclic permutation. This and its mirror/complement are
the
only solutions.
The mirror or
complement
(cyclic permutation of symbols) of this
ring is another unique solution  the mirror solution is the same as
the
complement. Considered as a 3D necklace (allowing a rotation about an
axis in the plane), there is only a single solution.
Ted demonstrated a nice method of finding solution
to some of these problems by
construction of a bilaterally symmetric directed graph. Solutions are Hamiltonian
cycles
on the graph; each vertex is visited once and the path returns to a
starting vertex. The general problem of finding Hamiltonian cycles on
graphs is well studied, with a complexity class
of NPcomplete in FNP.
While this particular graph is planar, the
graphs for longer substrings or more symbols are not. But any
graph for similar problems has symmetries that might reduce the
problem to polynomial
time. There is a reflection symmetry about the
vertical centerline, both in the graph layout and in the substring
triplets, and there is an antisymmetric reflection symmetry about the
horizontal centerline. These are De Bruijn graphs.
This directed graph can be mapped onto the corners
of a cube, and a solution will visits all
corners of the cube on a restricted set of directed edges:
Diagram
of all
triplets as corner coordinates (3tuples) of
a unit cube, and allowed neighbors as a graph on the cube (left). There
are
two
solutions, corresponding to two
Hamiltonian cycles
on this graph (center and right).
Treated as 3dimensional graphs, the two
solutions
are related by a
parity
transformation, equivalent to a mirroring and a rotation by pi
radians. Thus they are
achiral.
This is equivalent to the complement, or swapping of the
symbols 0 and 1.
The blue edge in the left figures
correspond
to the length 4 substrings (0 1
1 0) and (1 0 0 1) that can't appear in the solutions: if these
edges are traversed there is no way of reaching (1,1,1) and (0,0,0)
respectively without visiting these edge vertices twice.
This graph on a cube is related to Gray codes, in
that they also form a Hamiltonian cycle on the
corners of an ncube, where each bit is viewed as one dimension. But in
the case of Gray codes the graph is
undirected and the edges are all edges of the cube (no diagonals).
There is a unique Eularian cycle
on the left graphs,
including edges 000>000 and 111>111 (not shown), that here
gives the cycle
...0000101111010011... . These paths include all combinations
exactly twice.
1D 2symbol, every
quadruplet (k = 2, n = 4)
Using the algorithm described in "every
triplet", all sixteen possible quadruplets:
{ 0000, 0001, 0010, 0011,
0100, 0101, 0110, 0111, 1000,
1001, 1010, 1011, 1100, 1101, 1110, 1111 }
appear in this string:
[ 0 0 0 0 1 1 1 1 0
1 0 1 1 0 0 1 0 0 0 ]
The sixteen term ring is:
[ ... 0 0 0 0 1 1 1 1 0 1
0 1 1 0 0 1 ... ]
Is this is a unique minimal solution, up to cyclic
permutations and
mirroring? No! Ted Bell found another:
[ ... 0 0 0 0 1 0 0 1 1 1
1 0 1 0 1 1 ... ]
Ted found this as a Hamiltonian cycle
on a bilaterally symmetric directed graph. [To
Do: Draw directed graph.]
By brute force search,
I found the 16
solutions
0000100110101111
0000100111101011
0000101001101111
0000101001111011
0000101100111101
0000101101001111
0000101111001101
0000101111010011
0000110010111101
0000110100101111
0000110101111001
0000110111100101
0000111100101101
0000111101001011
0000111101011001
0000111101100101
with, 4? unique solutions up to mirroring and cyclic permutation of
the two symbols.
1D 2symbol, every
quintuplet (k = 2, n = 5)
Using the same procedures as for the "every
triplet" string:
[ 0 0 0 0 0 1 1 1 1
1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 ]
The 32term ring is:
[ ... 0 0 0 0 0 1 1 1 1 1
0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 ... ]
Is
this is a unique minimal solution, up to cyclic permutations and
mirroring? No, there are 2048 solutions, 512 solutions up to symbol
permutation (swapping 1 and 0) and mirroring. The can be found by a
brute force search of all strings represented by binary integers from 0
to 2^{27}. There are 2^{32} binary integers with 32
digits, but we can find only those with leading digits 000001... and
all cyclic permutations, symbol swaps, and mirrors will also be
solutions. This number of checks (2^{27}) in a brute force
search is nearing the limit of what is practicle on a serial processor.
1D 3symbol, every
triplet (k = 3, n = 3)
There are 216 solutions, but eliminating
permutations and mirrors leaves 12?
Text files with sequence lists, generated by brute
force search (see programs
and code):
1D 3symbol, every sextet
(k = 3, n = 6)
A universal cycle, by Anna Virágvölgyi
"A picture of an unwrapped cylinder.
The universal cycle "a b c a b a b a b c b c b c a b c a b c b c a c b
a c a b a c b c a c a c a b a b c a c a b c b" includes all possible
words of six length from the alphabet (a, b, c) in which no letters of
alphabet are paired. The picture is created by substituting stripes for
letters of the universal cycle. Due to of the nature of universal
cycles all possible diagonal striped square tiles (with six stripes 
each differ from its neighbour) can be found on this picture."
1D 4symbol, every pair (k =
4, n = 2)
There are 20736 solutions, but eliminating the 24
symbol permutations (all combinations of 4 symbols) and mirrors
leaves 20736/48 = 432 unique sequences.
Text files with sequence lists, generated by brute
force search (see programs and code):
k4_n2_no_permutes.txt
(List of 20736/4 = 5184 sequences that have
001
as a subsequence. Note that all
20736 sequences can be generated by with the permutations
01> 02, 03, 23, 32
.)
k4_n2_unique.txt
(List of 432 unique sequences up to symbol permutation and
mirroring.)
1D 4symbol, every
triplet (k = 4, n = 3)
Ted Bell sent me this solution, which he found by visually searching
it's graph for a Hamiltonian cycle:
... 0001110100202221211220030333232233131133013203210310231201230213 ...
This is way too long for my brute force (with principled lowest
estimate) programming approach.
The mother of one of my programming kids
taught me an interesting
trick: once you've found one solution there are ways of finding more
solutions, and perhaps all solutions. The basic transform for this one
goes like:
 all pairs of symbols happen several times. For example
the 00
pairs are:
0001110100202221211220030333232233131133013203210310231201230213
 the sequences between these pairs (for example 11101 and 20222121122)
can be swapped giving
0002022212112200111010030333232233131133013203210310231201230213
This works with any 1D solution, and I think it could be used
to count the
number of unique solutions. But it is still a difficult combinatorics
problem that I don't know how to solve.
2D 3symbol, every 2x2
Here is a solution for this case from "On the
de Bruijn Torus Problem", where they have used a constructive proof
(Theorem 1.3, Cock's construction). This is possible for any odd
number of symbols; all 2x2's, a dB_{k}( k^{2}, k^{2};
2, 2 ) member where k is odd. They step through the actual
construction, but it's hard for me to follow.
2 0 0 2 1 1 2 0 0
1 0 2 1 2 2 1 2 0
0 1 2 0 1 1 0 2 1
2 1 0 2 0 0 2 0 1
2 2 0 2 2 2 2 0 2
0 1 1 0 2 2 0 1 1
0 0 1 0 0 0 0 1 0
1 2 2 1 0 0 1 2 2
1 2 1 1 1 1 1 1 2





The top row and left column are repeated at the bottom and
right. 


It's about as symmetric as a 3 symbol set can be:
bilaterally symmetric
about the vertical axes between the fourth and fifth column, and about
the center of the ninth column. The 0's and 2's are exactly the same
pattern, with 2's shifted two units up relative to the 1's. I don't
know how many unique solutions exist.
2D 2symbol, every 2x2
There are sixteen possible 2x2 subarrays with
two symbols. Using {0, 1} as
symbols they correspond to the binary (base 2) numbers between 0 and 15:
{ [ 0 0 [
0 0 [
0 0 [
0 0 [
0 1 [
0 1 [
0 1 [
0 1
0 0 ],
0 1 ],
1 0 ],
1 1 ],
0 0 ],
0 1 ],
1 0 ],
1 1 ],
[ 1 0 [
1 0 [
1 0 [
1 0 [
1 1 [
1 1 [
1 1 [
1 1
0 0 ],
0 1 ],
1 0 ],
1 1 ],
0 0 ],
0 1 ],
1 0 ],
1 1 ], }
I gave up on using a tesselation on a torus (wraparound in both
directions), because it appeared to be ugly, requiring many duplicates.
7/1/08 But Ted Bell didn't, and showed that there aren't duplicates:
Brigid's cross (octomino) is a torroidal solution. I was fussing with the big hierarchical set of solutions last night, I got too frustrated and decided to count the number of 2x2 squares in a torroidal matrix of size 4.
It's 16, so I thought I'd try to fit the 16 possible 2x2 2 symbol squares into it. Only a symmetrical solution would do and Brigid's cross is one of them, I think the others are all row and column swaps.
0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0

Putting the left column and top row on top of the originals (to show all subarrays when tesselated on a torus):
0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 0 ___________ 0 1 0 0  0

This array's tiling is symmetric and periodic with respect the
symbols 0/1, or black/white.

The tesseract's
adjacency diagram (which faces are mutually
adjacent) can be mapped onto a 4x4 toroidal grid. So the tesseract
faces can be colored using this pattern such that
each of the 16 vertices (which are all at intersections of four faces)
has
a uniquely colored neighborhood (counting each of four orthogonal
orientations as different).
[To Do: A stereo projection of a tesseract (similar to this) with
labelled vertices and a corresponding 4x4 adjacency grid (see bottom of
this). Try
illustrating the coloring of faces.]
[To Do: Construct a single projected "corner" of a tesseract, composed
of four interpenetrating parallelpipeds. This corresponds to a cube's
single corner projected onto a plane, composed of three diamonds.]
To construct a "nonwrapping" array that contains
all subarrays, a 5x5
array seems natural because it has exactly 16 2x2 subarrays. It
has 25 elements so must include an unequal number of the two symbols 
it didn't seem possible since the set of 2x2's is equivalent with a
swap of symbols so there must be an equal number of symbols. But there
is another solution:
[ 0
0 0 0 1
0
0 1 0 1
1 1 1 1
0
1
1 0 1 0
0 0 0 0 1 ]
There are 140's and 111's, but the
central elements (with more 1's) appear
in more subarrays that the side and corner elements (with more 0's). Are
there other 5x5 solutions?
All columns, but no rows, are 1D 2symbol solutions. The top
and bottoms rows are the same, so the 5x4 array (without the top or
bottom row) tesselates a
cylinder. The left and right columns are complements, so it and its
complement forms an 8x4 tesselation on a torus that has a uniform
distribution of all 2x2 elements:
0
0 0 0 1 1 1 1
0
0 1 0 1 1 0 1
1
1 1 1 0 0 0 0
1
1 0 1 0 0 1 0
The
8x4 tile.

Tesselation of the tile. 


Any 5x5 subset of these tesselations will be a
minimal solution (contain all 2x2 subarrays), and any 4x8 subset of
the latter will tesselate in the same way.
The periodicity of the tesselations is curious. Both
the rows and the columns of the first are periodic on the string 0 1 1
1 or its complement 1 0 0 0 (and cyclic permutations). The columns are
all periodic on 0 0 1 1 and its complement, but the rows
are periodic on either 0 0 0 0 1 1 1 1 or 0 0 1 0 1 1 0 1 or their
complements (and cyclic permutations).
It turns out that there are offset tesselations of
these 4x8 arrays that also have an equal distribution of 2x2
subarrays. For example:
1
0 0 1
0
0 1 1
0
0 1 1
1
1 0 0
0 1 1 0
1
1 0 0
1
1 0 0
0
0 1 1
A subset of the tesselation (above) rotated by pi/2 radians, and
its pi rotation.

An offset tesselation of the two tiles, with horizontal rows of the
same tile. By matching the 2x2 subarrays that occur on horizontal
boundaries, the equal frequency of subarrays is preserved.
[To Do: Make a rectangular region that periodically tiles the plane.]

A tile constructed by symbol replacement with a double diagonal motif:
(Rotated extension not shown.)

A frieze that based on this tiling and symbol replacement, followed by
coloring contiguous regions. 
A fractal elaboration of the frieze. [Broken link: synthetic ripples 
ripple distortion description and code.] 
[To Do: L_system_tiling( 'test', 1, 2, 1, 0,'zig_zag_eel_motif_5.png',
'2x2_2s_a.png', 0, 1 ); ]
Of course it works with a swap of the symbols, or with
rotations/mirrors of any 5x5 subset of the tesselations. Are there
other solutions? There are only so many 5x5 arrays and I tried just
about
every one that contained [ 0 0; 0 0 ] and
[ 1 1; 1 1 ].
2D 4symbol, every 2x2
There are 256 (4^{4}) possible 2x2's, and a
minimal toroidal solution
would fit on a 16x16 grid. Ted and I couldn't do this one. Can a brute
force
search find a solution?
We can imagine a simple brute force search algorithm
that requires somewhat less than 2^{31} checks to find (or not
find) a solution. Is this too large to be computationally tractable?
How much less to find at least one solution? What is a good algorithm,
and can it be guaranteed to
find all solutions?
There are several good reasons for having a positive
solution. The most basic is to make figures like this Truchetlike
tiling that show all combinations of 2x2 neighbors with no repeats.
I found a paper that constructs a dB_{4}(
4^{2}, 4^{2}; 2, 2 ), which is the formal notation for
this pattern. In "A Meshing
Technique for de Bruijn Tori". (One of the main topics of this
paper is to show
another result implying
(among other things) that there is also exists a 8 x 32 matrix with all
2x2's of 4 symbols.) The Section 2 "Meshing method"
describes the construction (due to Ivanyi and Toth, 1988).
The meshing method
uses a
1D 4symbol every pair De
Bruijn sequence to construct a 2D 4symbol every 2x2 De Bruin
torus. The 1D
sequences can be found by brute force searching of strings. Only
a fraction of these sequences, those that are "even", can be used with
the meshing
method. Here's what "even" and "odd" mean in this context: if
ab
and
ba occurs in a sequence, then if the distance
between the occurances is odd, then the sequence is odd. So
aba
is odd because
ab and
ba are one (index) apart. But
abcba
is also odd, as they are three apart. The subsequence
abba can
occur in an even
sequence.
Only 104 of 432 unique 1D 4symbol every pair De
Bruijn sequences are even.
Are all possible 2D 4symbol every pair De
Bruijn tori generated by this method? I don't know, and I don't know if
this is known.
0 0 1 1 0 2 1 3 3 1 2 0 3 2 2 3
0 0 0 1 0 0 0 1 0 3 0 2 0 3 0 2 0
0 0 0 0 1 0 2 0 3 0 1 0 0 0 2 0 3
1 0 1 1 1 0 1 1 1 3 1 2 1 3 1 2 1
1 1 0 1 1 1 2 1 3 1 1 1 0 1 2 1 3
0 0 0 1 0 0 0 1 0 3 0 2 0 3 0 2 0
2 2 0 2 1 2 2 2 3 2 1 2 0 2 2 2 3
1 0 1 1 1 0 1 1 1 3 1 2 1 3 1 2 1
3 3 0 3 1 3 2 3 3 3 1 3 0 3 2 3 3
3 0 3 1 3 0 3 1 3 3 3 2 3 3 3 2 3
1 1 0 1 1 1 2 1 3 1 1 1 0 1 2 1 3
2 0 2 1 2 0 2 1 2 3 2 2 2 3 2 2 2
0 0 0 0 1 0 2 0 3 0 1 0 0 0 2 0 3
3 0 3 1 3 0 3 1 3 3 3 2 3 3 3 2 3
2 2 0 2 1 2 2 2 3 2 1 2 0 2 2 2 3
2 0 2 1 2 0 2 1 2 3 2 2 2 3 2 2 2
3 3 0 3 1 3 2 3 3 3 1 3 0 3 2 3 3




The 2D pattern formed by the
"mesh method" uses a logical combination of a 1D De Bruijn
sequence. The 1D sequence must have an "even" property for the torus to
have the De Bruin property. 
This torus with colored squares
as symbols. The bottom row and right column are
repeated at the bottom and
right, so every 2x2 pattern occurs exactly once. 
Two different cyclic rotations
of the same De Bruijn torus. 





0 0 1 1 0 2 1 2
2 0 3 1 3 2 3 3
0 0 0 1 0 0 0 1 0 2 0 3 0 3 0 3 0
0 0 0 0 1 0 2 0 2 0 0 0 1 0 2 0 3
1 0 1 1 1 0 1 1 1 2 1 3 1 3 1 3 1
1 1 0 1 1 1 2 1 2 1 0 1 1 1 2 1 3
0 0 0 1 0 0 0 1 0 2 0 3 0 3 0 3 0
2 2 0 2 1 2 2 2 2 2 0 2 1 2 2 2 3
1 0 1 1 1 0 1 1 1 2 1 3 1 3 1 3 1
2 2 0 2 1 2 2 2 2 2 0 2 1 2 2 2 3
2 0 2 1 2 0 2 1 2 2 2 3 2 3 2 3 2
0 0 0 0 1 0 2 0 2 0 0 0 1 0 2 0 3
3 0 3 1 3 0 3 1 3 2 3 3 3 3 3 3 3
1 1 0 1 1 1 2 1 2 1 0 1 1 1 2 1 3
3 0 3 1 3 0 3 1 3 2 3 3 3 3 3 3 3
2 2 0 2 1 2 2 2 2 2 0 2 1 2 2 2 3
3 0 3 1 3 0 3 1 3 2 3 3 3 3 3 3 3
3 3 0 3 1 3 2 3 2 3 0 3 1 3 2 3 3 



The De Bruijn sequence used here
is "odd", as it contains the substring 313,
so the mesh method does not
result in a De Bruijn torus. 
Not a De Bruijn torus, constructed by the mesh method with
a De Bruijn sequence, but one that is "odd". 











A De Bruijn torus using
Truchet's tile set, the four rotations of a diagonal division of a
square. Any permutation of tiles, mirror, or cyclic rotation of the
torus is also a De Bruijn torus. 
The same torus after a cyclic
rotation. 
2D 2symbol, every 3x3
There are 512 (2^{9}) 2symbol 3x3 arrays.
There is no minimal square solution that tesselates a torus (because
512 is not the square of an integer), but there may be 2^{4} x 2^{5} (16
x 32), 2^{3} x 2^{6} (8 x 64), or 2^{2} x 2^{7}
(4 x 128) solutions.
I don't know any solutions for this, or
have any ideas for methods of constructing one  except for brute
force search of all combinations. But the brute force search space is
far too large.
2D 2symbol, ThueMorse 2x2s
Is there a toroidal array that contains all 2x2's
that can occur in a ThueMorse
pattern, with no repeats? All Larrangements don't occur, all
others do.
Corners tilings
[To Do: An example of a combinatorial system
with a second neighborhood constraint. ]
The pattern is formed from these rotations and inversions of a single
"corner motif":
The left half of the pattern is this arrangement:
This pattern has a nice set of symmetries and
antisymmetries [ To Do:
Describe in B/W matrix.]. But it doesn't include the fifth set [ To Do:
Show ].
Is there an square, or nearly square matrix of motifs that contains all
2x2 patterns in an arrangement where the white and black (a's and b's)
have the same relationship?
Entropy of De Bruijn patterns
In a random string, where every letter has an equal
probability of occurring at every location, the expected value of the
number of times each ntuple occurs is the same for any given n  a
random string is a normal
number. De Bruijn sequences have equal frequencies of ntuples for
one particular n, but not for all n. Tuples of length > n never
occur. Is it true for tuples of length < n? What is the information
entropy of De Bruijn strings, and how does it change with n? As n
increases, do De Bruin strings asymptotically approach normal numbers?
Are normal numbers De Bruin strings as n approaches infinity?
Consider the 2symbol every quadruple De
Bruijn sequences in the context of all 16 bit cyclic strings. If we
randomly pick a large number of 16 bit strings, the distribution of
substrings will asymptoptically approach the distribution of all 16 bit
strings. This distribution for each n will be binomial:
[To Do: Distributions for n = 1 to ~6 for all 16 bit strings, compared
with the De Bruijn strings.]
Programs and code
[To Do: Generalize, fix up and document these.]
References
A Meshing
Technique for de Bruijn Tori
Glenn Hulbert and Garth Isaak, Contemporary Math. 178 (1994)
153160.
On the de Bruijn Torus
Problem, Glenn Hulbert, Garth Isaak
Garth Isaak
(accessed 2/8/10)
"Perfect Maps: how determine your
location in n
dimensions
List the string 00011101 cyclically.
Each
triple
occurs
exactly
once.
This is known as a perfect map or de Bruijn cycle. Many questions
can be asked about higher dimensional versions of this and version
with an alphabet of size k (instead of 2). In particular
are the `obvious' necessary conditions sufficient? Kenny Patterson
in London has answered this completely in 2dimensions for k a prime
power. Many other interesting questions arise about related structures
called perfect factors and perfect multifactors which arise in the
study of these objects. For more information and references to other
work see these papers my recent paper (which is partially survey and
partially new unifying results and notation
Constructing Higher Dimensional Perfect Factors
(to appear in Aequationes Mathematicae)"
Research
problems on Gray codes and universal cycles
Brad Jackson, Brett Stevens, Glenn Hurlbert
Discrete Mathematics 309 (2009) 53415348
Open problems arising from the Workshop on Generalizations of de Bruijn
cycles and Gray
Codes at the Banff International Research Station in December, 2004.
PROBLEM 480. De Bruijn Tori
Ron Graham
Dept. of Comp. Sci. & Engineering, Univ. of CaliforniaSan Diego,
La Jolla, CA, USA
graham@ucsd.edu
We consider a 2dimensional generalization of de Bruijn cycles. An
RbyS kary array A embedded on a torus is a de Bruijn torus with
window size (m; n) if every mbyn array occurs once. For short, we say
that A is a kary (R,S,m,n)de Bruijn torus.
Question 1: Is it possible to construct a kary (R,S,m,n)de
Bruijn torus whenever R, S, m, n, k are positive integers with k >
1, RS = k^{mn}, R > m, and S > n?
Comment: This question has been answered affirmatively for "square''
tori (the case R = S and m = n). The binary case (k = 2) was solved by
Fan, Fan, Ma, and Siu [1] and the general case by Hurlbert and Isaak
[2]. The problem was also solved for m = n = 2 by Hurlbert, Mitchell,
and Paterson [3].
As with universal cycles, one can also consider de Bruijn tori for many
other combinatorial structures.
Question 2: For which positive integers R, S, m, n, k such that k >
mn, R >= m, S >= n and RS = [Product from i = 0 to
k1]{mni) do there exist de Bruijn tori of the mnpermutations of a
kset?
Question 3: For which positive integers R, S, m, n, k such
that k > mn, R >= m, S >= n and RS = [k
choose mn] do there exist de Bruijn tori of the mnsubsets of a kset?
References
[1] C.T. Fan, S.M. Fan, S.L. Ma, and M.K. Siu, On de Bruijn arrays, Ars
Combin. 19A (1985), 205213
[2] G. Hurlbert and G. Isaak, On the de Bruijn torus problem, J.
Combin. Th. (A) 64 (1993), 5062.
[3] G. Hurlbert, C. Mitchell, and K. Paterson, On the existence of de
Bruijn tori with 2 x 2 windows, J. Combin. Th. (A) 76 (1996), 21330.
Automorphisms
of subwordposets
P. Ligeti and P. Sziklai
Discrete
Mathematics Volume 305, Issues 13, 6 December 2005,
Pages 372378
In this paper the automorphism group of two posets, D^{k,n}
and B_{m,n}
is determined. D^{k,n}
is the poset of DNA strands of length at most n, built up with k
complement pairs of letters, and partially ordered by the subsequence
relation. B_{m,n}
is the set of all subsequences of the word u_{m,n}=a_{1}…a_{n}
defined over the alphabet {0,1,…,(m1)},
where . The automorphism group of B_{m,n}
was known already (see G. Burosch, H.D.O.F. Gronau, J.M. Laborde, The
automorphism group of the subsequence poset B_{m,n},
Order 16 (2) (1999) 179–194 (2000)), here a short proof is
presented as an illustration of the method used in the first part.
Keywords: Subword;
Poset; Automorphism
A038219
A maximally unpredictable sequence.
The sequence starts 0,1,0 and
continues according to the following
rule: find the longest sequence at the end that has occurred at least
once previously. If there are more than one previous occurrences select
the last one. The next digit of the sequence is the opposite of the one
following the previous occurrence.
A079101
A repetitionresistant sequence.
a(n)
= 0 or 1, chosen so as to maximize the number of different
subsequences that are formed.
a(n+1)=1
if and only if (a(1),a(2),...,a(n),0), but not
(a(1),a(2),...,a(n),1), has greater length of longest repeated segment
than (a(1),a(2),...,a(n)) has.
In
Feb, 2003, Alejandro Dau solved Problem 3 on the Unsolved Problems
and Rewards website, thus establishing that every binary word occurs
infinitely many times in this sequence.
Klaus Sutmer remarks (Jun 26 2006) that this sequence is very similar
to the EhrenfeuchtMycielski sequence A007061. Both
sequences have every finite binary word as a factor; in fact,
essentially the same proof works for both sequences.
Cumulative
distribution function for order 7 de Bruijn weight classes
Mayhew,
G.L.
Aerospace
conference, 2009 IEEE
Order
n de Bruijn sequences are the period 2^{n} binary
sequences from nstage feedback shift registers. The de Bruijn
sequences have good randomness and complexity properties. The quantity
of de Bruijn sequences in a weight class of the order n generating
functions is an unsolved NP complete problem. Weight class
distributions for small n have been obtained by exhaustive searches.
This paper uses cumulative distribution function to obtain a high
resolution projection of the quantity of de Bruijn sequences in each
order 7 weight class. The weight class probability mass function is a
shifted Binomial probability mass function which in the limit is
accurately represented as a Normal probability density function scaled
by a Beta probability density function. The order 7 weight class
cumulative distribution function can be modeled as a weighted sum of
two Normal cumulative distribution functions.
From de Bruijn sequence (Mathworld, accessed 3/1/2010)
The lexicographic sequence of
Lyndon
words of lengths
divisible
by n gives the lexicographically smallest de Bruijn sequence (Ruskey).
de Bruijn sequences can be generated by feedback shift registers
(Golomb 1967; Ronse 1984; Skiena 1990, p. 196).
de Bruijn, N. G. "A Combinatorial Problem." Koninklijke
Nederlandse
Akademie v. Wetenschappen 49, 758764, 1946.
Golomb, S. W. Shift Register Sequences. San Francisco, CA:
HoldenDay, 1967.
Good, I. J. "Normal Recurring Decimals." J. London Math. Soc.
21,
167172, 1946.
Knuth, D. E. "Oriented Subtrees of an Arc Digraph." J. Combin.
Th. 3, 309314, 1967.
Ronse, C. Feedback Shift Registers. Berlin:
SpringerVerlag, 1984.
Ruskey, F. "Information on Necklaces, Lyndon Words, de Bruijn
Sequences."
http://www.theory.csc.uvic.ca/~cos/inf/neck/NecklaceInfo.html.
Skiena, S. Implementing Discrete Mathematics: Combinatorics and
Graph Theory with Mathematica. Reading, MA:
AddisonWesley, pp. 195196, 1990.

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