Small Forbidden Configurations

Anstee, Richard P.; Griggs, Jerrold R.; Sali, Attila

doi:10.1007/bf03352989

Cited by 25 publications

(42 citation statements)

References 7 publications

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“…A very basic induction proof of this can be found in [3]. For my application I only need the very special case where k = 2 and m = 6, which gives s = figure 4.…”

Section: Proof: Assume That Nmentioning

confidence: 99%

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Reversible and Irreversible Information Networks

Riis

2007

IEEE Trans. Inform. Theory

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Abstract-It is shown that there exist information networks where messages can be sent (utilising Network Coding) more easily in one direction than in the opposite direction. This is valid even though each channel is assumed to have the same capacity in both directions.It is shown that irreversible information networks only have solutions that use non-linear Network Coding. I argue that this result is more surprising than might appear at first sight and that it follows using ideas resembeling the path integral in Quantum Mechanics. [7]. The idea can be illustrated by considering the "butterfly" network in figure 1. I. MAGIC IN INFORMATIONx y x+y x y r: y r: x figure 1 The task is to send the message x from the upper left corner to the lower right corner and to send the message y from the upper right corner to the lower left corner. We say the lower left (lower right) node requires x (requires y) and write this requirement as r : y (r : x). The messages x, y ∈ A are selected from some finite alphabet A. Assume that each information channel can carry at most one message at a time. If the messages x and y are sent simultaneously there is a bottleneck in the middle information channel. On the other hand if we, for example, organise A as a commutative group (A, +) and send x + y ∈ A through the middle channel, the messages x and y can easily be recovered at 'output' nodes at the bottom of the network (since y = (x + y) − x and x = (x + y) − y).It is often convenient to think about each message as a flow of elements from A. Viewed this way we can consider messages a and b as sequences . . . a −2 , a −1 , a 0 , a 1 , a 2 . . . The information network in figure 1 is an example of a multiple-unicast information network. In general a multiple-unicast information network N = (V, E; s 1 , t 1

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“…A very basic induction proof of this can be found in [3]. For my application I only need the very special case where k = 2 and m = 6, which gives s = figure 4.…”

Section: Proof: Assume That Nmentioning

confidence: 99%

“…Case (3) give no hope of reconstructing z j . Case (2) only make is possible to reconstruct z j if s i or s i + s k is a constant (i.e.…”

Section: Proof: Assume That Nmentioning

confidence: 99%

Reversible and Irreversible Information Networks

Riis

2007

IEEE Trans. Inform. Theory

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“…Together with previous results [5,6,3,4], this yields an attractive classification, which is proven in Section 7. For a (0,1)-matrix C let C denote the (0,1)-complement of C and for two matrices C, D on the same number of rows let C\D denote the matrix of those columns of C not in D.…”

Section: Corollary 12 Let K M Be Given and Let A Be A Simple M-rowmentioning

confidence: 62%

“…The second claim is clear for the former two cases. In the last case, consider the various columns α = α S = 0 in short supply on S. If there are rows p, q of S such that for each choice of α we have α| {p,q} = 1 1 then, permuting rows p, q to the top, we see that A| S contains a copy of F k (t), with the ordering as in (2). In particular since each α has column sum at least 2 this would be possible if there were only one such α.…”

Section: Proof Given S ∈ [M]mentioning

confidence: 98%

Linear algebra methods for forbidden configurations

Anstee

Fleming

2011

Combinatorica

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We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m, F ) as the maximum number of columns in a simple m-rowed matrix A for which no k × l submatrix of A is a row and column permutation of F . In set theory notation, F is a forbidden trace. For all k-rowed F (simple or nonsimple) Füredi has shown that forb(m, F ) is O(m k ). We are able to determine for which k-rowed F we have that forb(m, F ) is O(m k−1 ) and for which k-rowed F we have that forb(m, F ) is Θ(m k ).We need a bound for a particular choice of F . Define D12 to be the k ×(2 k −2 k−2 −1) (0,1)-matrix consisting of all nonzero columns on k rows that do not haveˆ1 1˜i n rows 1 and 2. Let 0 denote the column of k 0's. Define F k (t) to be the concatenation of 0 with t + 1 copies of D12. We are able to show that forb(m, F k (t)) is Θ(m k−1 ). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m, F ).

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“…We consider a new form of the standard induction [1] for forbidden configurations [2]. For a matrix A, let µ(x, A) denote the multiplicity of x as a column of A.…”

Section: New Inductionmentioning

confidence: 99%

Large forbidden configurations and design theory

Anstee

Sali

2016

Studia Scientiarum Mathematicarum Hungarica

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Let forb(m, F ) denote the maximum number of columns possible in a (0,1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F . We consider cases where the configuration F has a number of columns that grows with m. For a k × matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k+α ). Results of Keevash on the existence of designs provide constructions that provide asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

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Small Forbidden Configurations

Cited by 25 publications

References 7 publications

Reversible and Irreversible Information Networks

Reversible and Irreversible Information Networks

Linear algebra methods for forbidden configurations

Large forbidden configurations and design theory

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