Omnistructures Anant Godbole, ETSU 2011 Cumberland Conference, - - PowerPoint PPT Presentation

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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

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This is joint work with Sunil Abraham, Katie Banks, Greg Brockman, Mike Deren, Cihan Eroglu, Stephanie Sapp, and Nicholas Triantafillou.

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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}.

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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.

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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.

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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

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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.

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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}.

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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:

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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

lim

k→∞ P(Sequence is k − omni) =

0, if r < aH(1..a), or 1, if r > aH(1..a) where H(1..a) = a

1 1 i ≈ ln a.

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Missing Words

◮ If a = 2, H(1..a) = 3, so that with (smaller) larger than 3k

letters the sequence is very (un)likely to be k omni.

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Missing Words

◮ If a = 2, H(1..a) = 3, so that with (smaller) larger than 3k

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.

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Missing Words

◮ If a = 2, H(1..a) = 3, so that with (smaller) larger than 3k

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 < 3k and if

E(M) is small if n > 3k

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Missing Words

◮ If a = 2, H(1..a) = 3, so that with (smaller) larger than 3k

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 < 3k and if

E(M) is small if n > 3k

◮ This is not true, however!

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Threshold for E(M) is different

◮ With

D(a, r) = (a − 1)r−1rr ar−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.

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Threshold for E(M) is different

◮ With

D(a, r) = (a − 1)r−1rr ar−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 2H(1..2) = 3 for a = 2. However, again for a = 2, we can show that D(2, r) = 1 when r ≈ 4.403.

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Threshold for E(M) is different

◮ With

D(a, r) = (a − 1)r−1rr ar−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 2H(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 3k

and 4.403k, sequences are omni with high probability, and yet the expected number of missing sequences is huge.

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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 ak2 k × k matrices over A.

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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 ak2 k × k matrices over A.

◮ The smallest size of an omnimosaic is defined to be ω(k, a)

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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 ak2 k × k matrices over A.

◮ The smallest size of an omnimosaic is defined to be ω(k, a) ◮ the example

    1 1 1 1 1 1 1 1     shows that ω(2, 2) = 4.

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Universal Graphs

◮ If a = 2 we quickly see that an omnimosaic is a bipartite

graph with n elements in each color class so that each possible k × k bipartite graph occurs as an induced subgraph (with isomorphic graphs counting as separate cases).

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Universal Graphs

◮ If a = 2 we quickly see that an omnimosaic is a bipartite

graph with n elements in each color class so that each possible k × k bipartite graph occurs as an induced subgraph (with isomorphic graphs counting as separate cases).

◮ In this respect this work continues along the lines of the vast

body of work on Universal Graphs done by Moon, Chung and her colleagues, and more recent authors such as Alstrup, Butler and Frieze. A good historical account that includes many more references is the paper by Alstrup. Studied are graphs that are induced universal for all graph isomorphisms; graphs that are universal or induced universal for families of graphs; and random graphs. In the nomenclature of the above authors, when a = 2, omnimosaics would likely be termed as “bipartite induced universal graphs.” Moreover,

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The work of Brightwell, Alon, etc

◮ In two papers arising from a rather unlikely problem (graph

• rientations), Brightwell and Kohayakawa; and Alon, Pach,

and Solymosi essentially solve the problem for non-bipartite graphs (a=2).

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The work of Brightwell, Alon, etc

◮ In two papers arising from a rather unlikely problem (graph

• rientations), Brightwell and Kohayakawa; and Alon, Pach,

and Solymosi essentially solve the problem for non-bipartite graphs (a=2).

◮ Our main results contain constructions that do better than

Moon’s; use general values of a; and yield a slightly sharper result than Alon et al’s.

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The work of Brightwell, Alon, etc

◮ In two papers arising from a rather unlikely problem (graph

• rientations), Brightwell and Kohayakawa; and Alon, Pach,

and Solymosi essentially solve the problem for non-bipartite graphs (a=2).

◮ Our main results contain constructions that do better than

Moon’s; use general values of a; and yield a slightly sharper result than Alon et al’s.

◮ ω(2, 3) = 6

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The work of Brightwell, Alon, etc

◮ In two papers arising from a rather unlikely problem (graph

• rientations), Brightwell and Kohayakawa; and Alon, Pach,

and Solymosi essentially solve the problem for non-bipartite graphs (a=2).

◮ Our main results contain constructions that do better than

Moon’s; use general values of a; and yield a slightly sharper result than Alon et al’s.

◮ ω(2, 3) = 6 ◮

kak/2 e ≤ ω(k, a) ≤ kak/2 e (1 + o(1)) for a well-specified function o(1) that tends to zero as k → ∞ and which may be taken to be 2 log k/k.

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Lower Bound

◮ The lower bound in the result is trivial and we dispose it off

right away.

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Lower Bound

◮ The lower bound in the result is trivial and we dispose it off

right away.

◮ There are

n

k

2 k × k sub-matrices of an n × n matrix; these are to cover all ak2 possibilities so, by the pigeonhole principle we must have

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Lower Bound

◮ The lower bound in the result is trivial and we dispose it off

right away.

◮ There are

n

k

2 k × k sub-matrices of an n × n matrix; these are to cover all ak2 possibilities so, by the pigeonhole principle we must have

◮ n k

2 ≥ ak2.

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Lower Bound

◮ The lower bound in the result is trivial and we dispose it off

right away.

◮ There are

n

k

2 k × k sub-matrices of an n × n matrix; these are to cover all ak2 possibilities so, by the pigeonhole principle we must have

◮ n k

2 ≥ ak2.

◮ Now, since, by a na¨

ıve application of Stirling’s formula n k

• ≤

ne k k ,

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Lower Bound

◮ The lower bound in the result is trivial and we dispose it off

right away.

◮ There are

n

k

2 k × k sub-matrices of an n × n matrix; these are to cover all ak2 possibilities so, by the pigeonhole principle we must have

◮ n k

2 ≥ ak2.

◮ Now, since, by a na¨

ıve application of Stirling’s formula n k

• ≤

ne k k ,

◮ We see that we must have

n ≥ kak/2 e in order for the array to form an omnimosaic.

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Proof Technique

◮ Fill each pixel in the n × n array independently and uniformly

{1, 2, . . . , a} with probability 1/a.

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Proof Technique

◮ Fill each pixel in the n × n array independently and uniformly

{1, 2, . . . , a} with probability 1/a.

◮ Let M be the number of “missing” k × k matrices, i.e.

matrices that cannot be found as a submatrix of the random n × n array. Then

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Proof Technique

◮ Fill each pixel in the n × n array independently and uniformly

{1, 2, . . . , a} with probability 1/a.

◮ Let M be the number of “missing” k × k matrices, i.e.

matrices that cannot be found as a submatrix of the random n × n array. Then

◮

M =

ak2

• j=1

Ij, where Ij = 1 (or Ij = 0) according as the jth matrix is missing (or present) as a submatrix.

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◮ We will show that

P(array is not an omnimosaic) = P(M ≥ 1) ≤ E(M) =

ak2

• j=1

P(Ij = 1) < 1 (or → 0) if n ≥ kak/2

e

(1 + o(1)), where we have used Markov’s inequality and linearity of expectation, and where the last claim will be a consequence of Suen’s inequality.

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◮ We will show that

P(array is not an omnimosaic) = P(M ≥ 1) ≤ E(M) =

ak2

• j=1

P(Ij = 1) < 1 (or → 0) if n ≥ kak/2

e

(1 + o(1)), where we have used Markov’s inequality and linearity of expectation, and where the last claim will be a consequence of Suen’s inequality.

◮ We have a construction with n = kak/2

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Graphs

◮ What if we seek to construct a k-universal induced graph with

at most a − 1 edges between vertices?

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Graphs

◮ What if we seek to construct a k-universal induced graph with

at most a − 1 edges between vertices?

◮ Pigeonhole yields

n

k

• ≥ a(k

2), or Anant Godbole, ETSU Omnistructures

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Graphs

◮ What if we seek to construct a k-universal induced graph with

at most a − 1 edges between vertices?

◮ Pigeonhole yields

n

k

• ≥ a(k

2), or

◮

n ≥ ka(k−1)/2 e

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Graphs

◮ What if we seek to construct a k-universal induced graph with

at most a − 1 edges between vertices?

◮ Pigeonhole yields

n

k

• ≥ a(k

2), or

◮

n ≥ ka(k−1)/2 e

◮ Theorem:

n ≤ ka(k−1)/2 e (1 + o(1))

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Omnisculptures

◮ What if we seek to construct a three dimensional omnimosaic,

i.e., an omnisculpture, which is an n × n × n array that contains all k × k × k a-colored cubical blocks?

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Omnisculptures

◮ What if we seek to construct a three dimensional omnimosaic,

i.e., an omnisculpture, which is an n × n × n array that contains all k × k × k a-colored cubical blocks?

◮ Pigeonhole yields

n

k

3 ≥ ak3, or

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Omnisculptures

◮ What if we seek to construct a three dimensional omnimosaic,

i.e., an omnisculpture, which is an n × n × n array that contains all k × k × k a-colored cubical blocks?

◮ Pigeonhole yields

n

k

3 ≥ ak3, or

◮

n ≥ kak2/3 e

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Omnisculptures

◮ What if we seek to construct a three dimensional omnimosaic,

i.e., an omnisculpture, which is an n × n × n array that contains all k × k × k a-colored cubical blocks?

◮ Pigeonhole yields

n

k

3 ≥ ak3, or

◮

n ≥ kak2/3 e

◮ Theorem:

n ≤ kak2/3 e (1 + o(1))

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3-Uniform Hypergraphs

◮ What if we seek to construct a k-universal induced

three-uniform hypergraph with a possible edge colors?

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3-Uniform Hypergraphs

◮ What if we seek to construct a k-universal induced

three-uniform hypergraph with a possible edge colors?

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3-Uniform Hypergraphs

◮ What if we seek to construct a k-universal induced

three-uniform hypergraph with a possible edge colors?

◮ ◮ Pigeonhole yields

n

k

• ≥ a(k

3), or Anant Godbole, ETSU Omnistructures

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3-Uniform Hypergraphs

◮ What if we seek to construct a k-universal induced

three-uniform hypergraph with a possible edge colors?

◮ ◮ Pigeonhole yields

n

k

• ≥ a(k

3), or

◮

n ≥ ka(k−1)(k−2)/6 e

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3-Uniform Hypergraphs

◮ What if we seek to construct a k-universal induced

three-uniform hypergraph with a possible edge colors?

◮ ◮ Pigeonhole yields

n

k

• ≥ a(k

3), or

◮

n ≥ ka(k−1)(k−2)/6 e

◮ Theorem:

n ≤ ka(k−1)(k−2)/6 e (1 + o(1))

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