The suffix binary search tree and suffix AVL tree

Suffix trees and suffix arrays are classical data structures that are used to represent the set of suffixes of a given string, and thereby facilitate the efficient solution of various string processing problems—in particular on-line string searching. Here we investigate the potential of suitably adapted binary search trees as competitors in this context. The suffix binary search tree (SBST) and its balanced counterpart, the suffix AVL-tree, are conceptually simple, relatively easy to implement, and offer time and space efficiency to rival suffix trees and suffix arrays, with distinct advantages in some circumstances—for instance in cases where only a subset of the suffixes need be represented.

Construction of a suffix BST for an n-long string can be achieved in O(nh) time, where h is the height of the tree. In the case of a suffix AVL-tree this will be O(nlogn) in the worst case. Searching for an m-long substring requires O(m+l) time, where l is the length of the search path. In the suffix AVL-tree this is O(m+logn) in the worst case. The space requirements are linear in n, generally intermediate between those for a suffix tree and a suffix array.

Empirical evidence, illustrating the competitiveness of suffix BSTs, is presented.     

Almost resolvable maximum packings of complete graphs with 5-cycles

Let X be the vertex set of K_nA k-cycle packing of K_n is a triple (X,C,L), where C is a collection of edge disjoint k-cycles of K_n and L is the collection of edges of K_n not belonging to any of the k-cycles in C. A k-cycle packing (X,C,L) is called resolvable if C can be partitioned into almost parallel classes. A resolvable maximum k-cycle packing of K_n, denoted by k-RMCP(n), is a resolvable k-cycle packing of K_n, (X,C,L), in which the number of almost parallel classes is as large as possible. Let D(n, k) denote the number of almost parallel classes in a k-RMCP(n). D(n, k) for k = 3, 4 has been decided. When n ≡ k (mod 2k) and k ≡ 1 (mod 2) or n ≡ 1 (mod 2k) and k ∈{6, 8, 10, 14}∪{m: 5≤m≤49, m ≡ 1 (mod 2)}, D(n, k) also has been decided with few possible exceptions. In this paper, we shall decide D(n, 5) for all values of n≥5.     

On ramsey numbers for large disjoint unions of graphs

Let be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r(G, H) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from . Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and mn,

r(mK_k, nK_l)=(k − 1)m+ln+r(K_k−1, K_l−1)−2.

     

Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.     

Embedding rectangular matrices in hadamard matrices

It is shown that any m×n±1 matrix may be embedded in a Hadamard matrix of order kl, where k and l are the least orders greater than or equal to m and nrespectively in which Hadamard matrices exist.     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.     

On convex segments in a triangulation

A segment (=1-cell) of a planar triangulation σ is convex if it is common to two triangles (2-cells) whose union is a convex set. We determine the maximal number of convex segments of a triangulation over all triangulations σ having n boundary vertices and m inner vertices (n3,m0).     

Some peculiar boundary phenomena for extremes of rth nearest neighbor links

Let D_n,r denote the largest rth nearest neighbor link for n points drawn independently and uniformly from the unit d-cube C_d. We show that according as r < d or r>d, the limiting behavior of D_n,r, as n → ∞, is determined by the two-dimensional ‘faces’ respectively one-dimensional ‘edges’ of the boundary of C_d. If d = r, a ‘balance’ between faces and edges occurs. In case of a d-dimensional sphere (instead of a cube) the boundary dominates the asymptotic behavior of D_n,r if d 3 or if d = 2, r 3.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Semi-planar Steiner loops of cardinality 2n

It is well known that there is a planar sloop of cardinality n for each n≡2 or 4 (mod 6) (Math. Z. 111 (1969) 289–300). A semi-planar sloop is a simple sloop in which each triangle either generates the whole sloop or the 8-element sloop. In fact, Quackenbush (Canad. J. Math. 28 (1976) 1187–1198) has stated that there should be such semi-planar sloops. In this paper, we construct a semi-planar sloop of cardinality 2n for each n≡2 or 4 (mod 6). Consequently, we may say that there is a semi-planar sloop that is not planar of cardinality m for each m>16 and m≡4 or 8 (mod 12). Moreover, Quackenbush (Canad. J. Math. 28 (1976) 1187–1198) has proved that each finite simple planar sloop generates a variety, which covers the smallest non-trivial subvariety (the variety of all Boolean sloops) of the lattice of the subvarieties of all sloops. Similarly, it is easy to show that each finite semi-planar sloop generates another variety, which also covers the variety of all Boolean sloops. Furthermore, for any finite simple sloop of cardinality n, the author (Beiträge Algebra Geom. 43 (2) (2002) 325–331) has constructed a subdirectly irreducible sloop of cardinality 2n and containing as the only proper normal subsloop. Accordingly, if is a semi-planar sloop, then the variety generated by properly contains the subvariety .     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {W_nj, 1jn, n1} and {X_nj, 1jn, n1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.     

Bit-parallel approximate string matching algorithms with transposition

Using bit-parallelism has resulted in fast and practical algorithms for approximate string matching under Levenshtein edit distance, which permits a single edit operation to insert, delete or substitute a character. Depending on the parameters of the search, currently the fastest non-filtering algorithms in practice are the O(km/wn) algorithm of Wu and Manber, the O((k+2)(m−k)/wn) algorithm of Baeza-Yates and Navarro, and the O(m/wn) algorithm of Myers, where m is the pattern length, n is the text length, k is the error threshold and w is the computer word size. In this paper we discuss a uniform way of modifying each of these algorithms to permit also a fourth type of edit operation: transposing two adjacent characters in the pattern. This type of edit distance is also known as Damerau edit distance. In the end we also present an experimental comparison of the resulting algorithms.     

Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3kn, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3kn. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.     

Approximating geometric bottleneck shortest paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π/(3cos(π/6)) times the Euclidean distance |pq| between p and q, and all of whose edges have length at most |pq|.     

A note on a property of a weak self-similar perfect set

Let S be a compact, weak self-similar perfect set based on a system of weak contractions f_j, j=1,…,m each of which is characterized by a variable contraction coefficient _j(l) as d(f_j(x),f_j(y)) _j(l)d(x,y), d(x,y)<l, l>0. If the relation ∑^m_j=1j(l₀)<1 holds at at least one point l₀, then every nonempty compact metric space is a continuous image of the set S.     

Bondage number of strong product of two paths

The bondage number b(G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m≥2 and n≥2, b(P_m?P_n) = 7 - r(m) - r(n) if (r(m), r(n)) = (1, 1) or (3, 3), 6 - r(m) - r(n) otherwise, where r(t) is a function of positive integer t, defined as r(t) = 1 if t ≡ 1 (mod 3), r(t) = 2 if t ≡ 2 (mod 3), and r(t) = 3 if t ≡ 0 (mod 3).     

ContributionsOn optimal orientations of cartesian products of graphs (I)

For a graph G, let D(G) be the family of strong orientations of G, and define [ovbar|d] (G) = min[d(D) vb D ] D(G), where d(D) denotes the diameter of the digraph D. Let G × H denote the cartesian product of the graphs G and H. In this paper, we determine completely the values of and , except , where K_n, P_n and C_n denote the complete graph, path and cycle of order n, respectively.     

Degrees in a digraph whose nodes are graphs

By an f-graph we mean an unlabeled graph having no vertex of degree greater than f. Let D(n, f) denote the digraph whose node set is the set of f-graphs of order n and such that there is an arc from the node corresponding to graph H to the node corresponding to the graph K if and only if K is obtainable from H by the addition of a single edge. In earlier work, algorithms were developed which produce exact results about the structure of D(n, f), nevertheless many open problems remain. For example, the computation of the order and size of D(n, f) for a number of values of n and f have been obtained. Formulas for the order and size for f = 2 have also been derived. However, no closed form formulas have been determined for the order and size of D(n, f) for any value of f. Here we focus on questions concerning the degrees of the nodes in D(n,n − 1) and comment on related questions for D(n,f) for 2 f < n − 1.