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Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays
 Theoretical Computer Science
"... Steiner triple systems are well studied combinatorial designs that have been shown to possess properties desirable for the construction of multiple erasure codes in RAID architectures. The ordering of the columns in the parity check matrices of these codes affects system performance. Combinatoria ..."
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Steiner triple systems are well studied combinatorial designs that have been shown to possess properties desirable for the construction of multiple erasure codes in RAID architectures. The ordering of the columns in the parity check matrices of these codes affects system performance. Combinatorial problems involved in the generation of good and bad column orderings are defined, and examined for small numbers of accesses to consecutive data blocks in the disk array. 1 Background A Steiner triple system is an ordered pair (S; T ) where S is a finite set of points or symbols and T is a set of 3element subsets of S called triples, such that each pair of distinct elements of S occurs together in exactly one triple of T . The order of a Steiner triple system (S; T ) is the size of the set S, denoted jSj. A Steiner triple system of order v is often written as STS(v). An STS(v) exists if and only if v 1; 3 (mod 6) (see [6], for example). We can relax the requirement that every pair ...
On nesting of path designs
 J. Comb. Math. Comb. Comp
"... Let h ≥ 1. For each admissible v, we exhibit a nested balanced path design H(v, 2h + 1, 1). For each admissible odd v, we exhibit a nested balanced path design H(v, 2h, 1). For every v ≡ 4 (mod 6), v ≥ 10, we exhibit a nested balanced path design H(v, 4, 1) except possibly if v ∈ {16, 52, 70}. For e ..."
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Let h ≥ 1. For each admissible v, we exhibit a nested balanced path design H(v, 2h + 1, 1). For each admissible odd v, we exhibit a nested balanced path design H(v, 2h, 1). For every v ≡ 4 (mod 6), v ≥ 10, we exhibit a nested balanced path design H(v, 4, 1) except possibly if v ∈ {16, 52, 70}. For each v ≡ 0 (mod 4h), v ≥ 4h, we exhibit a nested path design P (v, 2h+ 1, 1). For each v ≡ 0 (mod 4h − 2), v ≥ 4h − 2, we exhibit a nested path design P (v, 2h, 1). For every v ≡ 3 (mod 6), v ≥ 9, we exhibit a nested path design P (v, 4, 1) except possibly if v = 39. 1
Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks
 Electronic J. Combin
"... Two Steiner triple systems (X,A)and(X,B) are said to intersect in m pairwise disjoint blocks if A # B = m and all blocks in A#B are pairwise disjoint. For each v, we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pair ..."
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Two Steiner triple systems (X,A)and(X,B) are said to intersect in m pairwise disjoint blocks if A # B = m and all blocks in A#B are pairwise disjoint. For each v, we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pairwise disjoint blocks. 1
Grooming for twoperiod optical networks
 Networks
"... Minimizing the number of adddrop multiplexers (ADMs) in a unidirectional SONET ring can be formulated as a graph decomposition problem. When traffic requirements are uniform and alltoall, groomings that minimize the number of ADMs (equivalently, the drop cost) have been characterized for grooming ..."
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Minimizing the number of adddrop multiplexers (ADMs) in a unidirectional SONET ring can be formulated as a graph decomposition problem. When traffic requirements are uniform and alltoall, groomings that minimize the number of ADMs (equivalently, the drop cost) have been characterized for grooming ratio at most six. However, when two different traffic requirements are supported, these solutions do not ensure optimality. In twoperiod optical networks, n vertices are required to support a grooming ratio of C in the first time period, while in the second time period a grooming ratio of C ′, C ′ < C, is required for v ≤ n vertices. This allows the twoperiod grooming problem to be expressed as an optimization problem on graph decompositions of Kn that embed graph decompositions of Kv for v ≤ n. Using this formulation, optimal twoperiod groomings are found for small grooming ratios using techniques from the theory of graphs and designs.
Almost Resolvable 4Cycle Systems
"... A 4cycle system of order n is said to be almost resolvable provided its 4cycles can be partitioned into (n − 1)/2 almost parallel classes ( = (n − 1)/4 vertex disjoint 4cycles) and a half parallel class ( = (n−1)/8 vertex disjoint 4cycles.) We construct an almost resolvable 4cycle system of eve ..."
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A 4cycle system of order n is said to be almost resolvable provided its 4cycles can be partitioned into (n − 1)/2 almost parallel classes ( = (n − 1)/4 vertex disjoint 4cycles) and a half parallel class ( = (n−1)/8 vertex disjoint 4cycles.) We construct an almost resolvable 4cycle system of every order n ≡ 1 (mod 8) except 9 (for which no such system exists) and possibly 33, 41 and 57. 1
Completing some spectra for 2perfect cycle systems
 Australasian Journal of Combinatorics
, 1993
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Minimum Weights of Point Codes of Steiner Triple Systems
 J Statist Plann Inference
, 2001
"... The point code of a Steiner triple system always contains vectors of weight equal to the replication number. The existence of Steiner triple systems whose point code has minimum weight less than the replication number is examined. In particular, the possible minimum weights are determined. 1 Backgro ..."
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The point code of a Steiner triple system always contains vectors of weight equal to the replication number. The existence of Steiner triple systems whose point code has minimum weight less than the replication number is examined. In particular, the possible minimum weights are determined. 1 Background Let V be a finite set of v elements, and let G = fG 1 ; G 2 ; : : : ; G s g be a partition of V into subsets called groups. Let B be a collection of subsets of V called blocks, and set K = fjBj : B 2 Bg, the set of block sizes. If (V; B) has the property that every pair of elements either appears in exactly one block or in exactly one group it is a group divisible design, and is denoted by KGDD. When K = fkg, the notation kGDD is typically used. The type of the GDD is z t 1 1 z t 2 2 z t 3 3 \Delta \Delta \Delta z t s s when the number of groups of size z i is t i . A Steiner triple system of order v, or STS(v), is a 3GDD of type 1 v . Let (V; B) be an STS(v). There are b...
Faithful Enclosing Of Triple Systems: Doubling The Index
, 1991
"... . A triple system of order v 3 and index is faithfully enclosed in a triple system of order w v and index ¯ when the triples induced on some v elements of the triple system of order w are precisely those from the triple system of order v. When = ¯, faithful enclosing is embedding; when = 0, ..."
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. A triple system of order v 3 and index is faithfully enclosed in a triple system of order w v and index ¯ when the triples induced on some v elements of the triple system of order w are precisely those from the triple system of order v. When = ¯, faithful enclosing is embedding; when = 0, faithful enclosing asks for an independent set of size v in a triple system of order w. When ¯ = 2, we prove that a faithful enclosing of a triple system of order v and index into a triple system of order w and index ¯ exists if and only if w d 3v\Gamma1 2 e, ¯ j 0 (mod gcd(w \Gamma 2; 6)), and (v; w) 62 f(3; 5); (5; 7)g. 1. Background and necessary conditions A triple system of order v and index , denoted TS(v; ), is a pair (V; B). V is a set of v elements, and B is a collection of 3element subsets of V called triples or blocks. Every 2subset of V appears in precisely of the triples of B. A triple system is simple if it has no repeated blocks. Let T 1 = (V; B) be a TS(v; ) and T ...
On the PBDClosure of Sets Containing 3
 Discrete Appl. Math
, 1997
"... We describe a method used to prove nonexistence of pairwise balanced designs. We determine the exact closure of all subsets K of the set f3; 4; : : : ; 22g with K " f11; 12; : : : ; 22g 6= ; and 3 2 K. 1 Introduction Let K be a set of positive integers. Then a pairwise balanced design (PBD[v; ..."
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We describe a method used to prove nonexistence of pairwise balanced designs. We determine the exact closure of all subsets K of the set f3; 4; : : : ; 22g with K " f11; 12; : : : ; 22g 6= ; and 3 2 K. 1 Introduction Let K be a set of positive integers. Then a pairwise balanced design (PBD[v; K]) of order v with block sizes from K is a pair (V; B), where V is a finite set (the point set) of cardinality v and B is a family of subsets (called blocks) of V which satisfy the following properties: (i) if B 2 B, then jBj 2 K; (ii) every pair of distinct elements of V occurs in exactly one block of B. A set S of positive integers is said to be PBDclosed if the existence of a PBD[v; S] implies that v belongs to S. Let K be a set of positive integers and let B(K) = fv j 9PBD[v; K]g. Then B(K) is a PBDclosed set called the closure of K. In [4] H.D.Gronau, R.Mullin and Ch.Pietsch determined the complete closure of all subsets of the set f3; 4; : : : ; 10g which include 3. In this paper...
Author manuscript, published in "SIAM Journal on Discrete Mathematics (2011)" DOI: 10.1137/100806035 THE αARBORICITY OF COMPLETE UNIFORM HYPERGRAPHS
"... Abstract. αAcyclicity is an important notion in database theory. The αarboricity of a hypergraph H is the minimum number of αacyclic hypergraphs that partition the edge set of H. The αarboricity of the complete 3uniform hypergraph is determined completely. ..."
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Abstract. αAcyclicity is an important notion in database theory. The αarboricity of a hypergraph H is the minimum number of αacyclic hypergraphs that partition the edge set of H. The αarboricity of the complete 3uniform hypergraph is determined completely.