De Bruijn sequence

The De Bruijn sequence for alphabet size k = 2 and substring length n = 2. In general there are many sequences for a particular n and k but in this example it is unique, up to cycling.

In combinatorial mathematics, a De Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a contiguous subsequence). Such a sequence is denoted by B(k, n) and has length kn, which is also the number of distinct substrings of length n on A; de Bruijn sequences are therefore optimally short. There are distinct De Bruijn sequences B(k, n).

The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn. According to him,[1] the existence of De Bruijn sequences for each order together with the above properties were first proved, for the case of alphabets with two elements, by Camille Flye Sainte-Marie in 1894,[2] whereas the generalization to larger alphabets is originally due to Tanja van Aardenne-Ehrenfest and himself.

In most applications, A = {0,1}.

History

The earliest known example of a De Bruijn sequence comes from Sanskrit prosody where, since the work of Pingala, each possible three-syllable pattern of long and short syllables is given a name, such as 'y' for short–long–long and 'm' for long–long–long. To remember these names, the mnemonic yamātārājabhānasalagām is used, in which each three-syllable pattern occurs starting at its name: 'yamātā' has a short–long–long pattern, 'mātārā' has a long–long–long pattern, and so on, until 'salagām' which has a short–short–long pattern. This mnemonic, equivalent to a De Bruijn sequence on binary 3-tuples, is of unknown antiquity, but is at least as old as C. P. Brown's 1869 book on Sanskrit prosody that mentions it and considers it "an ancient line, written by Pāṇini".[3][4][5][6][7]

In 1894, A. de Rivière raised the question in an issue of the French problem journal L'Intermédiaire des Mathématiciens, of the existence of a circular arrangement of length which contains all binary sequences of length . The problem was solved, along with the count , by C. Flye Sainte-Marie in the same year.[1] This was largely forgotten, and Martin (1934) proved the existence of such cycles for general alphabet size in place of 2, with an algorithm for constructing them. Finally, when in 1944 Kees Posthumus conjectured the count for binary sequences, de Bruijn proved the conjecture in 1946, through which the problem became well-known.[1]

Karl Popper independently describes these objects in his The Logic of Scientific Discovery (1934), calling them "shortest random-like sequences".[8]

Examples

Construction

The De Bruijn sequences can be constructed by taking a Hamiltonian path of an n-dimensional De Bruijn graph over k symbols (or equivalently, a Eulerian cycle of a (n − 1)-dimensional De Bruijn graph).[9]

An alternative construction involves concatenating together, in lexicographic order, all the Lyndon words whose length divides n.[10]

De Bruijn sequences can also be constructed using shift registers[11] or via finite fields.[12]

Example

A De Bruijn graph. Every four-digit sequence occurs exactly once if one traverses every edge exactly once and returns to one's starting point (an Eulerian cycle). Every three-digit sequence occurs exactly once if one visits every node exactly once (a Hamiltonian path).

Goal: to construct a B(2, 4) De Bruijn sequence of length 24 = 16 using Eulerian (n − 1 = 4 − 1 = 3) 3-D De Bruijn graph cycle.

Each edge in this 3-dimensional De Bruijn graph corresponds to a sequence of four digits: the three digits that label the vertex that the edge is leaving followed by the one that labels the edge. If one traverses the edge labeled 1 from 000, one arrives at 001, thereby indicating the presence of the subsequence 0001 in the De Bruijn sequence. To traverse each edge exactly once is to use each of the 16 four-digit sequences exactly once.

For example, suppose we follow the following Eulerian path through these nodes:

000, 000, 001, 011, 111, 111, 110, 101, 011,
110, 100, 001, 010, 101, 010, 100, 000.

These are the output sequences of length k:

0 0 0 0
_ 0 0 0 1
_ _ 0 0 1 1

This corresponds to the following De Bruijn sequence:

0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1

The eight vertices appear in the sequence in the following way:

      {0  0  0} 0  1  1  1  1  0  1  1  0  0  1  0  1
       0 {0  0  0} 1  1  1  1  0  1  1  0  0  1  0  1
       0  0 {0  0  1} 1  1  1  0  1  1  0  0  1  0  1
       0  0  0 {0  1  1} 1  1  0  1  1  0  0  1  0  1
       0  0  0  0 {1  1  1} 1  0  1  1  0  0  1  0  1
       0  0  0  0  1 {1  1  1} 0  1  1  0  0  1  0  1
       0  0  0  0  1  1 {1  1  0} 1  1  0  0  1  0  1
       0  0  0  0  1  1  1 {1  0  1} 1  0  0  1  0  1
       0  0  0  0  1  1  1  1 {0  1  1} 0  0  1  0  1
       0  0  0  0  1  1  1  1  0 {1  1  0} 0  1  0  1
       0  0  0  0  1  1  1  1  0  1 {1  0  0} 1  0  1
       0  0  0  0  1  1  1  1  0  1  1 {0  0  1} 0  1
       0  0  0  0  1  1  1  1  0  1  1  0 {0  1  0} 1
       0  0  0  0  1  1  1  1  0  1  1  0  0 {1  0  1}
   ... 0} 0  0  0  1  1  1  1  0  1  1  0  0  1 {0  1 ...
   ... 0  0} 0  0  1  1  1  1  0  1  1  0  0  1  0 {1 ...

...and then we return to the starting point. Each of the eight 3-digit sequences (corresponding to the eight vertices) appears exactly twice, and each of the sixteen 4-digit sequences (corresponding to the 16 edges) appears exactly once.

Algorithm

The following Python code calculates a De Bruijn sequence, given k and n, based on an algorithm from Frank Ruskey's Combinatorial Generation.[13]

def de_bruijn(k, n):
    """
    De Bruijn sequence for alphabet k
    and subsequences of length n.
    """
    try:
        # let's see if k can be cast to an integer;
        # if so, make our alphabet a list
        _ = int(k)
        alphabet = list(map(str, range(k)))

    except (ValueError, TypeError):
        alphabet = k
        k = len(k)

    a = [0] * k * n
    sequence = []

    def db(t, p):
        if t > n:
            if n % p == 0:
                sequence.extend(a[1:p + 1])
        else:
            a[t] = a[t - p]
            db(t + 1, p)
            for j in range(a[t - p] + 1, k):
                a[t] = j
                db(t + 1, t)
    db(1, 1)
    return "".join(alphabet[i] for i in sequence)

print(de_bruijn(2, 3))
print(de_bruijn("abcd", 2))

which prints

00010111
aabacadbbcbdccdd

Note that these sequences are understood to "wrap around" in a cycle. For example, the first sequence contains 110 and 100 in this fashion.

Uses

The sequence can be used to shorten a brute-force attack on a PIN-like code lock that does not have an "enter" key and accepts the last n digits entered. For example, a digital door lock with a 4-digit code would have B(10, 4) solutions, with length 10000. Therefore, only at most 10000 + 3 = 10003 (as the solutions are cyclic) presses are needed to open the lock. Trying all codes separately would require 4 × 10000 = 40000 presses.

The symbols of a De Bruijn sequence written around a circular object (such as a wheel of a robot) can be used to identify its angle by examining the n consecutive symbols facing a fixed point. This angle-encoding problem is known as the "rotating drum problem".[14] Gray codes can be used as similar rotary positional encoding mechanisms.

De Bruijn cycles are of general use in neuroscience and psychology experiments that examine the effect of stimulus order upon neural systems,[15] and can be specially crafted for use with functional magnetic resonance imaging.[16]

A De Bruijn sequence can be used to quickly find the index of the LSB or MSB in a word using bitwise operations.[17][18] An example of returning the index of the least significant bit from a 32 bit unsigned integer is given below using bit manipulation and multiplication.

unsigned int v;   
int r;           
static const int MultiplyDeBruijnBitPosition[32] = 
{
  0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8, 
  31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9
};
r = MultiplyDeBruijnBitPosition[((uint32_t)((v & -v) * 0x077CB531U)) >> 27];

The index of the LSB in v is stored in r and if v has no set bits the operation returns 0. The constant, 0x077CB531U, in the expression is a De Bruijn sequence.

f-fold De Bruijn sequences

f-fold n-ary De Bruijn sequence' is an extension of the notion n-ary De Bruijn sequence, such that the sequence of the length contains every possible subsequence of the length n exactly f times. For example, for the cyclic sequences 11100010 and 11101000 are two-fold 2-ary binary De Bruijn sequences. The number of two-fold De Bruijn sequences, for is , the other known numbers[19] are , , and .

De Bruijn torus

Main article: De Bruijn torus

A De Bruijn torus is a toroidal array with the property that every k-ary m-by-n matrix occurs exactly once.

Such a pattern can be used for two-dimensional positional encoding in a fashion analogous to that described above for rotary encoding. Position can be determined by examining the m-by-n matrix directly adjacent to the sensor, and calculating its position on the De Bruijn torus.

De Bruijn decoding

Computing the position of a particular unique tuple or matrix in a De Bruijn sequence or torus is known as the De Bruijn Decoding Problem. Efficient O(n log n) decoding algorithms exists for special, recursively constructed sequences[20] and extend to the two dimensional case.[21] De Bruijn decoding is of interest, e.g., in cases where large sequences or tori are used for positional encoding.

See also

Notes

  1. 1 2 3 De Bruijn (1975).
  2. ↑ Flye Sainte-Marie, C. (1894), "Solution to question nr. 48", L'Intermédiaire des Mathématiciens, 1: 107–110
  3. ↑ C. P. Brown, 1869, Sanskrit Prosody and Numerical Symbols Explained, p. 28
  4. ↑ Subhash Kak, 2000, Yamātārājabhānasalagāṃ an interesting combinatoric sūtra, Indian Journal of History of Science, 35.2 (2000), 123–127.
  5. ↑ Rachel W. Hall. Math for poets and drummers. Math Horizons 15 (2008) 10–11.
  6. ↑ Donald Ervin Knuth (2006). The Art of Computer Programming, Fascicle 4: Generating All Trees – History of Combinatorial Generation. Addison–Wesley. p. 50. ISBN 978-0-321-33570-8.
  7. ↑ Stein, Sherman K. (1963), "Yamátárájabhánasalagám", The Man-made Universe: An Introduction to the Spirit of Mathematics, pp. 110–118. Reprinted in Wardhaugh, Benjamin, ed. (2012), A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton Univ. Press, pp. 139–144.
  8. ↑ Karl Popper (2002) [1934]. The logic of scientific discovery. Routledge. p. 294. ISBN 978-0-415-27843-0.
  9. ↑ Klein, Andreas (2013), Stream Ciphers, Springer, p. 59, ISBN 9781447150794.
  10. ↑ According to Berstel & Perrin (2007), the sequence generated in this way was first described (with a different generation method) by Martin (1934), and the connection between it and Lyndon words was observed by Fredricksen & Maiorana (1978).
  11. ↑ Goresky, Mark; Klapper, Andrew (2012), "8.2.5 Shift register generation of de Bruijn sequences", Algebraic Shift Register Sequences, Cambridge University Press, pp. 174–175, ISBN 9781107014992.
  12. ↑ Ralston, Anthony (1982), "de Bruijn sequences—a model example of the interaction of discrete mathematics and computer science", Mathematics Magazine, 55 (3): 131–143, doi:10.2307/2690079, MR 653429. See in particular "the finite field approach", pp. 136–139.
  13. ↑ "De Bruijn sequences". Sage. Retrieved 3 November 2016.
  14. ↑ van Lint, J. H.; Wilson, Richard Michael (2001), A Course in Combinatorics, Cambridge University Press, p. 71, ISBN 9780521006019.
  15. ↑ GK Aguirre, MG Mattar, L Magis-Weinberg. (2011) "de Bruijn cycles for neural decoding".. NeuroImage 56: 1293–1300
  16. ↑ "De Bruijn cycle generator".
  17. ↑ Anderson, Sean Eron (1997–2009). "Bit Twiddling Hacks". Stanford University. Retrieved 2009-02-12.
  18. ↑ Busch, Philip (2009). "Computing Trailing Zeros HOWTO". Retrieved 2015-01-29.
  19. ↑ Osipov (2016).
  20. ↑ Tuliani (2001).
  21. ↑ Hurlbert & Isaak (1993).

References

External links

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