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Basic Data Structures
Published in Charles J. Alpert, Dinesh P. Mehta, Sachin S. Sapatnekar, Handbook of Algorithms for Physical Design Automation, 2008
Even though an interval lies on a line that is a one-dimensional space, it is actually a two-dimensional datum because it has two independent parameters. An interval starting at a and ending at b is represented by [a, b]. It is not possible to have a total order over the set of intervals. The idea of the interval tree is to partition the set of intervals into three groups based on a given point x:intervals to the left of the point L(x), intervals to the right of the point R(x), and intervals overlapping with the point C(x). The subsets L(x) and R(x) of intervals can be recursively represented. The subset C(x) also needs to be organized for the queries. Even though C(x) could include all the intervals in the original set, organizing them is much simpler: they can be ordered both on their left points and on their right points. If the query point q < x, only the left points of C(x) need to be checked in increasing order; if q > x, only the right points of C(x) need to be checked in decreasing order. To balance L(x) and R(x), thus to have a short tree, it is desired to use the median of all the endpoints as x. Figure 4.2 shows an interval tree for a set of intervals, where the intervals in C(x) are organized in two lists according to their left and right points.
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Let R be a partial order on S. Then R is called a total order or linear order if ∀x,y∈S, either (x,y)∈R or (y,x)∈R. Definition 1.2.13 A set S together with a partial order is called a partially ordered set ( poset). A set S together with a total order is called a totally ordered set or a chain.
Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The set ℝ together with the natural order of ≤ is an example of a totally ordered set while the relation is a subset of (Example 3.1.1) is a partial order which is not a total order. A poset of primary interest in this treatise is provided by the next example.
On the measure of conflicts: an argumentation-based framework
Published in Journal of Applied Non-Classical Logics, 2018
A knowledge baseK is a finite set of propositional formulae. We further assume a distinguished enumeration for every subset of K, its canonical enumeration. Importantly, it only serves to provide a total order in which the formulae of any subset of K are to be conjoined to yield a formula logically equivalent to this subset. Therefore, no other constraint is imposed on K, particularly K is not expected to be consistent. It needs not even be the case that individual formulae in K are consistent. We let denote the classical consequence relation. We write to denote that K is inconsistent.
On a practical notion of Geoffrion proper optimality in multicriteria optimization
Published in Optimization, 2020
P. K. Shukla, J. Dutta, K. Deb, P. Kesarwani
The next example shows that solution for are also possible. Our analysis suggests that if exists then it is a singleton set for , irrespective of the number of criteria. In a multicriteria optimization, a singleton as an optimal set occurs if a total order is used as the underlying ordering relation. In our context, this would mean that, for , -Geoffrion proper optimality is equivalent to a total order. A general proof evades us so far.
Using global variance-based sensitivity analysis to prioritise bridge retrofits in a regional road network subject to seismic hazard
Published in Structure and Infrastructure Engineering, 2023
Gitanjali Bhattacharjee, Jack W. Baker
To reduce computational expense, a subset of 30 ground-motion intensity maps, denoted is initially selected from the set of 1992 ground-motion intensity maps using an optimisation method that minimises the difference between the annual exceedance curves of and the full set of maps (Miller & Baker, 2015). The total-order Sobol’ indices of each bridge in the set of interest are estimated using according to Algorithm 1. Another subset of 45 ground-motion intensity maps is then selected from the same portfolio of ground-motion maps using the same optimisation method and settings used to produce contains 19 maps also included in though they are weighted differently (i.e. have different annual occurrence rates wj) in each set. is used to test (1) the performance of various bridge retrofit selection strategies and (2) whether bridge Sobol’ indices computed on a smaller set of scenarios ( result in good performance on a larger set of scenarios. Both and produce the same ground-motion hazard at selected locations and the same distribution of numbers of damaged bridges – within a user-defined tolerance – as would the full set of maps, and are therefore hazard-consistent (Miller & Baker, 2015).