Outline Collaborators One Dimension Two Dimensions Three Dimensions Omnistructures Anant Godbole, ETSU 2011 Cumberland Conference, Louisville May 11, 2011 Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Collaborators One Dimension Two Dimensions Three Dimensions Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions This is joint work with Sunil Abraham, Katie Banks, Greg Brockman, Mike Deren, Cihan Eroglu, Stephanie Sapp, and Nicholas Triantafillou. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions deBruijn’s Theorem ◮ The sequence 11101000, when wrapped around, is an efficient compressed way to list the eight sequences in { 0 , 1 } 3 as contiguous substrings. It is an example of a universal cycle, or U-cycle, of three letter words on the binary alphabet { 0 , 1 } . Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions deBruijn’s Theorem ◮ The sequence 11101000, when wrapped around, is an efficient compressed way to list the eight sequences in { 0 , 1 } 3 as contiguous substrings. It is an example of a universal cycle, or U-cycle, of three letter words on the binary alphabet { 0 , 1 } . ◮ The first result on U-cycles is due to deBruijn, who showed that U-cycles exist for k -letter words on an n letter alphabet, no matter what the values of k , n are. These cycles are called deBruijn cycles. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions deBruijn’s Theorem ◮ The sequence 11101000, when wrapped around, is an efficient compressed way to list the eight sequences in { 0 , 1 } 3 as contiguous substrings. It is an example of a universal cycle, or U-cycle, of three letter words on the binary alphabet { 0 , 1 } . ◮ The first result on U-cycles is due to deBruijn, who showed that U-cycles exist for k -letter words on an n letter alphabet, no matter what the values of k , n are. These cycles are called deBruijn cycles. ◮ The classical sample space from Statistics { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } uses 24 characters, but the U-cycle does the job in 8 characters. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions deBruijn’s Theorem ◮ The sequence 11101000, when wrapped around, is an efficient compressed way to list the eight sequences in { 0 , 1 } 3 as contiguous substrings. It is an example of a universal cycle, or U-cycle, of three letter words on the binary alphabet { 0 , 1 } . ◮ The first result on U-cycles is due to deBruijn, who showed that U-cycles exist for k -letter words on an n letter alphabet, no matter what the values of k , n are. These cycles are called deBruijn cycles. ◮ The classical sample space from Statistics { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } uses 24 characters, but the U-cycle does the job in 8 characters. ◮ If the eight words are to appear as subsequences rather than strings, however, we need just 6 bits: HTHTHT Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Omnibus sequences, or Omnisequences ◮ Leo Tolstoy’s novel War and Peace has the following property: it contains this paragraph as a subsequence: Ignoring punctuation and special fonts, if one were to write just the letters and spaces that appear in the book as a string, then there would be a subsequence of that string that is identical to the string of letters and spaces in this paragraph. The full property is more general – War and Peace contains as a subsequence any possible string of up to nine hundred fifty letters and spaces such as the first 950 characters of President Obama’s Inaugural Address, as well as a string of 950 “ q ”s. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Omnibus sequences, or Omnisequences ◮ Leo Tolstoy’s novel War and Peace has the following property: it contains this paragraph as a subsequence: Ignoring punctuation and special fonts, if one were to write just the letters and spaces that appear in the book as a string, then there would be a subsequence of that string that is identical to the string of letters and spaces in this paragraph. The full property is more general – War and Peace contains as a subsequence any possible string of up to nine hundred fifty letters and spaces such as the first 950 characters of President Obama’s Inaugural Address, as well as a string of 950 “ q ”s. ◮ War and Peace is thus a tome that is nine hundred fifty-omnibus (or omni) over the twenty seven character alphabet { a , b , c , . . . , z , SPACE } . Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Average Case Behavior ◮ In general, to get a minimal k -omni sequence over the alphabet { 1 , 2 , . . . , a } , we can just write the alphabet back to back k times. Average case behavior is more important. Roll an a sided die n times: Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Average Case Behavior ◮ In general, to get a minimal k -omni sequence over the alphabet { 1 , 2 , . . . , a } , we can just write the alphabet back to back k times. Average case behavior is more important. Roll an a sided die n times: ◮ Theorem Let r > 0 be a constant, and fix a ≥ 2 , n = rk, where n , k are both integers. Then � 0 , if r < aH (1 .. a ) , or k →∞ P ( Sequence is k − omni ) = lim 1 , if r > aH (1 .. a ) where H (1 .. a ) = � a 1 i ≈ ln a. 1 Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Missing Words ◮ If a = 2, H (1 .. a ) = 3, so that with (smaller) larger than 3 k letters the sequence is very (un)likely to be k omni. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Missing Words ◮ If a = 2, H (1 .. a ) = 3, so that with (smaller) larger than 3 k letters the sequence is very (un)likely to be k omni. ◮ Let M be the number of missing k -letter words in an n string. The sequence is omni iff M = 0. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Missing Words ◮ If a = 2, H (1 .. a ) = 3, so that with (smaller) larger than 3 k letters the sequence is very (un)likely to be k omni. ◮ Let M be the number of missing k -letter words in an n string. The sequence is omni iff M = 0. ◮ Thus it is natural to ask if E ( M ) is large if n < 3 k and if E ( M ) is small if n > 3 k Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Missing Words ◮ If a = 2, H (1 .. a ) = 3, so that with (smaller) larger than 3 k letters the sequence is very (un)likely to be k omni. ◮ Let M be the number of missing k -letter words in an n string. The sequence is omni iff M = 0. ◮ Thus it is natural to ask if E ( M ) is large if n < 3 k and if E ( M ) is small if n > 3 k ◮ This is not true, however! Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Threshold for E ( M ) is different ◮ With ( a − 1) r − 1 r r D ( a , r ) = a r − 1 ( r − 1) r − 1 , and a fixed, E ( M ) → 0 as k → ∞ if D ( a , r ) ≤ 1, and E ( M ) → ∞ ( k → ∞ ) if D ( a , r ) > 1. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Threshold for E ( M ) is different ◮ With ( a − 1) r − 1 r r D ( a , r ) = a r − 1 ( r − 1) r − 1 , and a fixed, E ( M ) → 0 as k → ∞ if D ( a , r ) ≤ 1, and E ( M ) → ∞ ( k → ∞ ) if D ( a , r ) > 1. ◮ Recall, e.g., that for k -omni strings, the threshold ratio (prior to which the probability of a string being k -omni was 0, beyond which it was 1) is 2 H (1 .. 2) = 3 for a = 2. However, again for a = 2, we can show that D (2 , r ) = 1 when r ≈ 4 . 403. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Threshold for E ( M ) is different ◮ With ( a − 1) r − 1 r r D ( a , r ) = a r − 1 ( r − 1) r − 1 , and a fixed, E ( M ) → 0 as k → ∞ if D ( a , r ) ≤ 1, and E ( M ) → ∞ ( k → ∞ ) if D ( a , r ) > 1. ◮ Recall, e.g., that for k -omni strings, the threshold ratio (prior to which the probability of a string being k -omni was 0, beyond which it was 1) is 2 H (1 .. 2) = 3 for a = 2. However, again for a = 2, we can show that D (2 , r ) = 1 when r ≈ 4 . 403. ◮ What is going on? It appears that for values of n between 3 k and 4 . 403 k , sequences are omni with high probability, and yet the expected number of missing sequences is huge. Anant Godbole, ETSU Omnistructures
Outline Collaborators One Dimension Two Dimensions Three Dimensions Omnimosaics ◮ An omnimosaic O ( n , k , a ) is defined to be an n × n matrix, with entries from the set A = { 1 , 2 , . . . , a } , that contains, as a submatrix, each of the a k 2 k × k matrices over A . Anant Godbole, ETSU Omnistructures
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