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Consecutive interval query and dynamic programming on intervals
Discrete Applied Mathematics, 1998
Given a set of n points (nodes) on a line and a set of m weighted intervals defined on the nodes, we consider a particular dynamic programming (DP) problem on these intervals. If the weight function of the DP has convex or concave property, we can solve this DP problem efficiently by using matrix searching in Monge matrices, together with a new query data structure, which we call the consecutive query structure. We invoke our algorithm to obtain fast algorithms for sequential partition of a graph and for maximum K-clique of an interval graph.
2009
Interval graph is a very important subclass of intersection graphs and perfect graphs. It has many applications in different real life situations. The problems on interval graph are solved by using different data structures among them interval tree is very useful. During last decade this data structure is used to solve many problems on interval graphs due to its nice properties. Some of its important properties are presented here. Here we introduced some problems on interval graphs which are solved by using the data structure interval tree. A brief review of interval graph is also given here.
Interval Computations: Introduction, Uses, and Resources
1996
Interval analysis is a broad Þeld in which rigorous mathematics is associated with with scientiÞc computing. A number,of researchers worldwide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established mathematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientiÞc and engineering applications
Overlap interval partition join
Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data, 2014
Each tuple in a valid-time relation includes an interval attribute T that represents the tuple's valid time. The overlap join between two valid-time relations determines all pairs of tuples with overlapping intervals. Although overlap joins are common, existing partitioning and indexing schemes are inefficient if the data includes long-lived tuples or if intervals intersect partition boundaries. We propose Overlap Interval Partitioning (OIP), a new partitioning approach for data with an interval. OIP divides the time range of a relation into k base granules and defines overlapping partitions for sequences of contiguous granules. OIP is the first partitioning method for interval data that gives a constant clustering guarantee: the difference in duration between the interval of a tuple and the interval of its partition is independent of the duration of the tuple's interval. We offer a detailed analysis of the average false hit ratio and the average number of partition accesses for queries with overlap predicates, and we prove that the average false hit ratio is independent of the number of short-and long-lived tuples. To compute the overlap join, we propose the Overlap Interval Partition Join (OIPJOIN), which uses OIP to partition the input relations on-the-fly. Only the tuples from overlapping partitions have to be joined to compute the result. We analytically derive the optimal number of granules, k, for partitioning the two input relations, from the size of the data, the cost of CPU operations, and the cost of main memory or disk IOs. Our experiments confirm the analytical results and show that the OIPJOIN outperforms state-ofthe-art techniques for the overlap join.
An interval extension based on occurrence grouping
Computing, 2011
In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. This paper presents a so-called "occurrence grouping" interval extension [ f ] og of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f og that is monotonic w.r.t. two subsets x a and x b of the occurrences of x: f og is increasing w.r.t. x a and decreasing w.r.t. x b. [ f ] og is the interval extension by monotonicity of f og and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based overestimate of the image diameter of [ f ] og. Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.
On Representing an Interval Graph Using the Minimum Number of Interval Lengths
The interval count problem is that of determining the smallest num-ber of interval lengths required to represent an interval model of a given interval graph or interval order. Despite the large number of studies about interval graphs and interval orders, few results on the interval count problem exist in fact. We provide a short survey about the interval count and related problems.
Interval processing with the UB-Tree
Proceedings International Database Engineering and Applications Symposium
Advanced data warehouses and web databases have set the demand for processing large sets of time ranges, quality classes, fuzzy data, personalized data and extended objects. Since, all of these data types can be mapped to intervals, interval indexing can dramatically speed up or even be an enabling technology for these new applications. We introduce a method for managing intervals by indexing the dual space with the UB-Tree. We show that our method is an effective and efficient solution, benefitting from all good characteristics of the UB-Tree, i.e., concurrency control, worst case guarantees for insertion, deletion and update as well as efficient query processing. Our technique can easily be integrated into an RDBMS engine providing the UB-Tree as access method. We also show that our technique is superior and more flexible to previously suggested techniques.
The interval searching algorithm for broadcast communications of Gallager, Tsybakov and Mikhailov is analyzed. We present ultimate characterizations of the burst response of the algorithm, that is. when the number of collided packers becomes large. Three quantities are of interest: the conflict resolution interval (CRI), the fraction of the resolved interval (RI) and the number of resolved packets (RP). If n is the multiplicity of a conflict, then it is proved that the m-th moments of eRr, RI and RP are o(log"' n), 0 (n-m) and 0 (1) respectively. In addition, for the first two moments of the parameters precise asymptotic approximations are presented. The methodology proposed in this paper is, in particular, applicable to asymptotic analysis of other interval searching algorithms, and in general, to other tree-type data structure algorithms.
Interval Tree and its Applications1
Advanced Modeling and Optimization, 2009
Abstract. Interval graph is a very important subclass of intersection graphs and perfect graphs. It has many applications in different real life situations. The problems on interval graph are solved by using different data structures among them interval tree is very useful. ...