129,465 research outputs found
Operations on (ordered) interval sets
Intervals play an important role in various kinds of database-applications in practice, for example in historical, spatial, and temporal databases. As a consequence, there is a practical need for a clear and proper treatment of various useful operations on intervals and interval sets in a database context. However, the semantics of some important operations on interval sets are not always treated or not treated very clearly in the literature; e.g., often they are defined in an algorithmic rather than a declarative manner. Moreover, implementation proposals are often not as straightforward as they could be. This paper presents a declarative treatment of various operations on interval sets, also introducing some new notions (such as ordered interval sets, their visible points, and their surface). Then the paper formally ?links? such (mathematical) intervals to their database representations. Finally the paper provides straightforward translations from these formal database representations to standard SQL, without the need for SQL extensions.
Biased quantitative measurement of interval ordered homothetic preferences
We represent interval ordered homothetic preferences with a quantitative homothetic utility function and a multiplicative bias. When preferences are weakly ordered (i.e. when indifference is transitive), such a bias equals 1. When indifference is intransitive, the biasing factor is a positive function smaller than 1 and measures a threshold of indifference. We show that the bias is constant if and only if preferences are semiordered, and we identify conditions ensuring a linear utility function. We illustrate our approach with indifference sets on a two dimensional commodity space.Weak order, semiorder, interval order, intransitive indifference, independence, homothetic, representation, linear utility
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
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