Representing prefix and border tables: results on enumeration

For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table or equivalently border table. In this paper, we define the combinatorial class of prefix lists, namely a sequence of integers together with their size, and an injection ψ from the class of prefix tables to the class of prefix lists. We call a valid prefix list the image by ψ of a prefix table. In particular, we describe algorithms converting a prefix/border table to a prefix list and inverse linear algorithms from computing from a prefix list L = ψ(P) two words respectively in a minimal size alphabet and on a maximal size alphabet with P as prefix table. We then give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + ϕ)n (with $\varphi=\frac{1+\sqrt{5}}{2}$ the golden mean) and also present a corresponding lower bound.

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Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽

10.3390/g9030062 ◽

2018 ◽

Vol 9 (3) ◽

pp. 62 ◽

Cited By ~ 2

Author(s):

Anjali Singh ◽

Anjana Gupta

Keyword(s):

Linear Programming ◽

Lower Bound ◽

Real World ◽

Upper Bound ◽

Duality Theorem ◽

Matrix Game ◽

Matrix Games ◽

Linguistic Information ◽

The Real ◽

Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

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LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195901000559 ◽

2001 ◽

Vol 11 (04) ◽

pp. 401-421 ◽

Cited By ~ 2

Author(s):

ALEJANDRO LÓPEZ-ORTIZ ◽

SVEN SCHUIERER

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Real Line ◽

Upper Bound ◽

Competitive Ratio ◽

Search Strategy ◽

The Real ◽

Simple Polygons ◽

On Line ◽

Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

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BOUNDS ON THE MINIMAL SUMSET SIZE FUNCTION IN GROUPS

International Journal of Number Theory ◽

10.1142/s1793042107001085 ◽

2007 ◽

Vol 03 (04) ◽

pp. 503-511 ◽

Cited By ~ 2

Author(s):

SHALOM ELIAHOU ◽

MICHEL KERVAIRE

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Solvable Groups ◽

Upper Bounds ◽

Minimal Size ◽

Lower And Upper Bounds ◽

Numerical Function ◽

Size Function

In this paper, we give lower and upper bounds for the minimal size μG(r,s) of the sumset (or product set) of two finite subsets of given cardinalities r,s in a group G. Our upper bound holds for solvable groups, our lower bound for arbitrary groups. The results are expressed in terms of variants of the numerical function κG(r,s), a generalization of the Hopf–Stiefel function that, as shown in [6], exactly models μG(r,s) for G abelian.

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A Viable Approach to Mitigating Irreproducibility

Stats ◽

10.3390/stats4010015 ◽

2021 ◽

Vol 4 (1) ◽

pp. 205-215

Author(s):

David Trafimow ◽

Tonghui Wang ◽

Cong Wang

Keyword(s):

Upper Bound ◽

Multiple closed orbits for N-body-type problems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031968 ◽

1998 ◽

Vol 58 (1) ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Shiqing Zhang

Keyword(s):

Lower Bound ◽

Periodic Solutions ◽

Upper Bound ◽

Critical Value ◽

Body Type ◽

Fixed Energy ◽

Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

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Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416502047 ◽

2016 ◽

Vol 26 (12) ◽

pp. 1650204 ◽

Cited By ~ 6

Author(s):

Jihua Yang ◽

Liqin Zhao

Keyword(s):

Lower Bound ◽

Limit Cycle ◽

Hamiltonian Systems ◽

Limit Cycles ◽

Upper Bound ◽

Melnikov Function ◽

Piecewise Smooth ◽

First Order ◽

Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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Isolated minima of the product of n linear forms

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028048 ◽

1953 ◽

Vol 49 (1) ◽

pp. 59-62 ◽

Cited By ~ 6

Author(s):

E. S. Barnes

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Linear Forms ◽

Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

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The Algebraic Degree of Phase-Type Distributions

Journal of Applied Probability ◽

10.1017/s0021900200006963 ◽

2010 ◽

Vol 47 (03) ◽

pp. 611-629

Author(s):

Mark Fackrell ◽

Qi-Ming He ◽

Peter Taylor ◽

Hanqin Zhang

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Polynomial Algorithm ◽

Algebraic Degree ◽

Nonnegative Matrices ◽

Stieltjes Transform ◽

Special Cases ◽

Phase Type ◽

Phase Type Distributions ◽

The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽

10.1007/s00453-021-00856-1 ◽

2021 ◽

Author(s):

Seungbum Jo ◽

Rahul Lingala ◽

Srinivasa Rao Satti

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Upper Bound ◽

Total Order ◽

Two Dimensional ◽

Information Theoretic ◽

Cartesian Tree ◽

Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

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