Digraphs that have at most one walk of a given length with the same endpoints

Let Θ(n,k) be the set of digraphs of order n that have at most one walk of length k with the same endpoints. Let θ(n,k) be the maximum number of arcs of a digraph in Θ(n,k). We prove that if n≥5 and k≥n−1 then θ(n,k)=n(n−1)/2 and this maximum number is attained at D if and only if D is a transitive tournament. θ(n,n−2) and θ(n,n−3) are also determined.     

Space Efficient Suffix Trees

We give first the representation of a suffix tree that uses n lg n + O(n) bits of space and supports searching for a pattern string in the given text (from a fixed size alphabet) in O(m) time, where n is the size of the text and m is the length of the pattern. The structure is quite simple and answers a question raised by Muthukrishnan (in Proceedings of the FST and TCS Preconference Workshop on Randomization, 1997, pp. 23–27). Previous compact representations of suffix trees had either a higher lower order term in space and had some expectation assumption or required more time for searching. When the size of the alphabet k is not viewed as a constant, this structure can be modified to use the same space but take O(m lg k) time for string searching or to use an additional O(n lg k) bits but take the same O(m) time for searching. We then give several index structures for binary texts, with less space including

• a structure that uses a suffix array (lg  bits) and an additional () bits,

• an indexing structure that takes bits of space, and

• an ( lg ) bit structure which answers in () time, the decision question of whether a given pattern of length occurs in the text.

Each of these structures uses a different technique, either in the storage scheme or in the search algorithm, in reducing the space requirement. The first one uses a suffix array, a sparse suffix tree, and a table structure. Finding all the occurrences of a pattern using this structure takes O(m + s) time, where s is the number of occurrences of the pattern in the text. The second structure constructs a sparse suffix tree for all the suffixes that start with the bit that occurs more times in the given binary text. The last structure uses an iterative algorithm to search for the pattern. This structure is the first o(n lg n) bit index to support the decision version of indexing queries in time linear in the length of the pattern. But this does not support the general indexing queries where we want to find the position of the occurrence of the pattern.Our main contribution is the development of techniques to use the succinct tree representation through balanced parentheses for suffix trees.     

The order of hypotraceable oriented graphs

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented graphs of order n for every n≥8 except possibly for n=9,11 and that is the only one of order less than 8.Furthermore, we determine all the hypotraceable oriented graphs of order 8 and explain the relevance of these results to the problem of determining, for given k≥2, the maximum order of nontraceable oriented digraphs each of whose induced subdigraphs of order k is traceable.     

Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1.     

Average-case results on heapsort

A new algorithm for rearranging a heap is presented and analysed in the average case. The average case upper bound for deleting the maximum element of a random heap is improved, and is shown to be less than [logn]+0.299+M(n) comparisons, *) whereM(n) is between 0 and 1. It is also shown that a heap can be constructed using 1.650n+O(logn) comparisons with this algorithm, the best result for any algorithm which does not use any extra space. The expected time to sortn elements is argued to be less thann logn+0.670n+O(logn), while simulation result points at an average case ofn log n+0.4n which will make it the fastest in-place sorting algorithm. The same technique is used to show that the average number of comparisons when deleting the maximum element of a heap using Williams' algorithm for rearrangement is 2([logn]–1.299+L(n)) whereL(n) also is between 0 and 1, and the average cost for Floyd-Williams Heapsort is at least 2nlogn–3.27n, counting only comparisons. An analysis of the number of interchanges when deleting the maximum element of a random heap, which is the same for both algorithms, is also presented.     

Open irredundance and maximum degree in graphs

A necessary and sufficient condition for an open irredundant set of vertices of a graph to be maximal is obtained. This result is used to show that the smallest cardinality amongst the maximal open irredundant sets in an n-vertex isolate-free graph with maximum degree Δ is at least n(3Δ−1)/(2Δ³−5Δ²+8Δ−1) for Δ≥5, n/8 for Δ=4 and 2n/11 for Δ=3. The bounds are the best possible.     

New text indexing functionalities of the compressed suffix arrays

New text indexing functionalities of the compressed suffix arrays are proposed. The compressed suffix array proposed by Grossi and Vitter is a space-efficient data structure for text indexing. It occupies only O(n) bits for a text of length n; however it also uses the text itself that occupies bits for the alphabet . In this paper we modify the data structure so that pattern matching can be done without any access to the text. In addition to the original functions of the compressed suffix array, we add new operations search, decompress and inverse to the compressed suffix arrays. We show that the new index can find occ occurrences of any substring P of the text in O(|P|logn+occlog^εn) time for any fixed 1ε>0 without access to the text. The index also can decompress a part of the text of length m in O(m+log^εn) time. For a text of length n on an alphabet such that , our new index occupies only bits where is the order-0 entropy of the text. Especially for ε=1 the size is bits. Therefore the index will be smaller than the text, which means we can perform fast queries from compressed texts.     

Lempel–Ziv Factorization Using Less Time & Space

For 30 years the Lempel–Ziv factorization LZ _x of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZ _x was based on Θ(n)-time (or, depending on the measure used, O(n log n)-time) processing of the suffix tree ST _x of x. Recently Abouelhoda et al. proposed an efficient Lempel–Ziv factorization algorithm based on an “enhanced” suffix array – that is, a suffix array SA _x together with supporting data structures, principally an “interval tree”. In this paper we introduce a collection of fast space-efficient algorithms for LZ factorization, also based on suffix arrays, that in theory as well as in many practical circumstances are superior to those previously proposed; one family out of this collection achieves true Θ(n)-time alphabet-independent processing in the worst case by avoiding tree structures altogether. The work of the first and third authors was supported in part by grants from the Natural Sciences & Engineering Research Council of Canada.     

On extremal sort sequences

A sort sequence S_n is a sequence of all unordered pairs of indices inI _n = {1, 2, ..., n}. With a sort sequenceS_n = (s₁ ,s₂ ,...,s_{( 2ⁿ )} )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(S_n). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence S_n except that the comparisons whose outcomes can be inferred from the results of the preceding comparisons are not performed. A sort sequence is said to be extremal if it maximizes a given objective function. First we consider the extremal sort sequences with respect to the objective function ω(S_n) — the expected number of active predictions inS _n. We study ω-extremal sort sequences in terms of their prediction vectors. Then we consider the objective function Ω(S_n) — the minimum number of active predictions in S_n over all input orderings.     

An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order

We prove that each OBDD (ordered binary decision diagram) for the middle bit of n-bit integer multiplication for one of the variable orders which so far achieve the smallest OBDD sizes with respect to asymptotic order of growth, namely the pairwise ascending order x₀,y₀,…,x_n−1,y_n−1, requires a size of Ω(2^(6/5)n). This is asymptotically optimal due to a bound of the same order by Amano and Maruoka (2007) [1].     

The complexity of finding generalized paths in tournaments

Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation υ₁υ₂…υ_n of the vertices, such that the results of the games υ₁υ₂, υ₂υ₃,…, υ_n−1υ_n match a prescribed pattern. If the pattern requires all arcs to go forwrd, i.e., υ₁ → υ₂, υ₂ → υ₃,…, υ_n−1 → υ_n, and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cn lg n, and we make the observation that O(n lg n) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-forward-backward, etc., admits an algorithm for which O(n) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn lg^αn for any α between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 4].     

Higher order representations of the Robbins–Monro process

For quasi-linear regression functions, the Robbins–Monro process X_n is decomposed in a sum of a linear form and a quadratic form both defined in the observation errors. Under regularity conditions, the remainder term is of order O(n^−3/2) with respect to the L_p-norm. If a cubic form is added, the remainder term can be improved up to an order of O(n⁻²). As a corollary the expectation of X_n is expanded up to an error of order O(n⁻²). This is used to correct the bias of X_n up to an error of order O(n^−3/2 log n).     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

The characterisation of the smallest two fold blocking sets in PG(<Emphasis Type="Italic">n</Emphasis>, 2)

We classify the smallest two fold blocking sets with respect to the (n−k)-spaces in PG(n, 2). We show that they either consist of two disjoint k-dimensional subspaces or are equal to a (k + 1)-dimensional space minus one point.     

Sorting in one round

Given an ordered set ofn elements whose order is not known to us, it is shown that by making slightly more thancn ^3/2 simultaneous comparison almost all comparisons can be deduced by direct implications. It is also shown that this result is essentially best possible, and that ifn is large, almost any choice ofcn ^3/2 comparisons will yield almost all comparisons by direct implications.     

A note on properties of locally finitely convergent sequences with respect to median filter

Let F_k be a mapping from R^Z to R^Z, satisfying that for x∈R^Z and n∈Z, F_k(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for k∈{2,3}, if there exists n₀∈Z such that x is locally finitely convergent with respect to F_k on {n₀,…,n₀+k−1}, then x is finitely convergent with respect to F_k. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true.     

Dirac-type conditions for hamiltonian paths and cycles in 3-uniform hypergraphs

A hamiltonian path (cycle) in an n-vertex 3-uniform hypergraph is a (cyclic) ordering of the vertices in which every three consecutive vertices form an edge. For large n, we prove an analog of the celebrated Dirac theorem for graphs: there exists n₀ such that every n-vertex 3-uniform hypergraph H, n?n₀, in which each pair of vertices belongs to at least n/2−1 (⌊n/2⌋) edges, contains a hamiltonian path (cycle, respectively). Both results are easily seen to be optimal.     

On the girth of extremal graphs without shortest cycles

Let EX(ν;{C₃,…,C_n}) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n+1 cycle? We prove that the diameter of G is at most n−1, and present several results concerning the above question: the girth of G is g=n+1 if (i) ν≥n+5, diameter equal to n−1 and minimum degree at least 3; (ii) ν≥12, ν∉{15,80,170} and n=6. Moreover, if ν=15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν≥2n−3 and n≥7 the girth is at most 2n−5. We also show that the answer to the question is negative for ν≤n+1+⌊(n−2)/2⌋.     

The structure of weighing matrices having large weights

We examine the structure of weighing matricesW(n, w), wherew=n–2,n–3,n–4, obtaining analogues of some useful results known for the casen–1. In this setting we find some natural applications for the theory ofsigned groups and orthogonal matrices with entries from signed groups, as developed in [3]. We construct some new series of Hadamard matrices from weighing matrices, including the following:W(n, n–2) implies an Hadamard matrix of order2n ifn0 mod 4 and order 4n otherwise;W(n, n–3) implies an Hadamard matrix of order 8n; in certain cases,W(n, n–4) implies an Hadamard matrix of order 16n. We explicitly derive 117 new Hadamard matrices of order 2 ^t p, p<4000, the smallest of which is of order 2³·419.Supported by an NSERC grant     

Some Best Constants in the Landau Inequality on a Finite Interval

An additive form of the Landau inequality forf∈Wⁿ_∞[−1, 1],is proved for 0<c?(cos(π/2n))⁻², 1?m?n−1, with equality forf(x)=T_n(1+(x−1)/c), 1?c?(cos(π/2n))⁻², whereT_nis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,for ‖f⁽ⁿ⁾‖?σ ‖f‖, 2ⁿ⁻¹n! cos²ⁿ(π/2n)?σ?2ⁿ⁻¹n!, 1?m?n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown.