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Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence ..."
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Cited by 86 (11 self)
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Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NPcomplete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NPcomplete. This problem is als...
A Register Allocation Framework Based on Hierarchical Cyclic Interval Graphs
 In International Workshop on Compiler Construction, Paderdorn
, 1993
"... In this paper, we propose the use of cyclic interval graphs as an alternative representation for register allocation. The "thickness" of the cyclic interval graph captures the notion of overlap between live ranges of variables relative to each particular point of time in the program execution. We de ..."
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Cited by 56 (12 self)
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In this paper, we propose the use of cyclic interval graphs as an alternative representation for register allocation. The "thickness" of the cyclic interval graph captures the notion of overlap between live ranges of variables relative to each particular point of time in the program execution. We demonstrate that cyclic interval graphs provide a feasible and effective representation that accurately captures the periodic nature of live ranges found in loops. A new heuristic algorithm for minimum register allocation, the fat cover algorithm, has been developed and implemented to exploit such program structure. In addition, a new spilling algorithm is proposed that makes use of the extra information available in the interval graph representation. These two algorithms work together to provide a twophase register allocation process that does not require iteration of the spilling or coloring phases. We extend the notion of cyclic interval graphs to hierarchical cyclic interval graphs and we...
Efficient Algorithms for the Domination Problems on Interval and CircularArc Graphs
 SIAM J. Comput
, 1998
"... Abstract. This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorit ..."
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Cited by 9 (1 self)
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Abstract. This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorithms run in time O(n) orO(n log log n) where n is the number of vertices. The results are then extended to solve the same problems on circulararc graphs in O(n + m) time where m is the number of edges of the input graph.
Satisfiability Problems on Intervals and Unit Intervals
 Theoretical Computer Science
, 1997
"... For an interval graph with some additional order constraints between pairs of nonintersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the order constraints. Previous algorithms for this problem (known also as seriation with side constrain ..."
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Cited by 5 (1 self)
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For an interval graph with some additional order constraints between pairs of nonintersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the order constraints. Previous algorithms for this problem (known also as seriation with side constraints) required quadratic time. This problem contains as subproblems interval graph and interval order recognition. On the other hand, it is a special case of the interval satisfiability problem, which is concerned with the realizability of a set of intervals along a line, subject to precedence and intersection constraints. We study such problems for all possible restrictions on the types of constraints, when all intervals must have the same length. We give efficient algorithms for several restrictions of the problem, and show the NPcompleteness of another restriction. 1 Introduction Two intervals x; y on the real line may either intersect or one of them is completely to the left of the othe...
Recognizing and representing proper interval graphs in parallel using merging and sorting
, 2006
"... We present a parallel algorithm for recognizing and representing a proper interval graph in O(log 2 n) time with O(m + n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from w ..."
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Cited by 3 (0 self)
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We present a parallel algorithm for recognizing and representing a proper interval graph in O(log 2 n) time with O(m + n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from which an interval representation is easily obtained. It is simple, uses no complex data structures, and extends ideas from an optimal sequential algorithm for recognizing and representing a proper interval graph [DHH96]. 1
Representing Digraphs Using Intervals Or Circular Arcs
"... Containment and overlap representations of digraphs are studied, with the following results. The interval containment digraphs are the digraphs of Ferrers dimension 2, and the circulararc containment digraphs are the complements of circulararc intersection digraphs. A poset is an interval containm ..."
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Cited by 1 (1 self)
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Containment and overlap representations of digraphs are studied, with the following results. The interval containment digraphs are the digraphs of Ferrers dimension 2, and the circulararc containment digraphs are the complements of circulararc intersection digraphs. A poset is an interval containment poset if and only if its comparability digraph is an interval containment digraph, and a graph is an interval graph if and only if the corresponding symmetric digraph with loops is an interval digraph. In an appropriate model of overlap representation using intervals, the unit rightoverlap interval digraphs are precisely the unit interval digraphs, and the adjacency matrices of rightoverlap interval digraphs have a simple structural characterization bounding their Ferrers dimension by 3. Keywords: digraph, intersection representation, containment, overlap, unit interval, Ferrers dimension Running head: REPRESENTATIONS OF DIGRAPHS y Research supported in part by NSA/MSP Grant MDA9049...
Difference Graphs
, 2000
"... Intersection and measured intersection graphs are quite common in the literature. In this paper we introduce the analogous concept of measured difference graphs: Given an arbitrary hypergraph {H1 , ..., Hn}, let us associate to it a graph on vertex set [n] = 2, ..., n} in which (i, j) is an edge i# ..."
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Cited by 1 (0 self)
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Intersection and measured intersection graphs are quite common in the literature. In this paper we introduce the analogous concept of measured difference graphs: Given an arbitrary hypergraph {H1 , ..., Hn}, let us associate to it a graph on vertex set [n] = 2, ..., n} in which (i, j) is an edge i# the corresponding sets H i and H j are "sufficiently different". More precisely, given an integer threshold k, we consider three definitions, according to which (i, j) is an edge i# (1) 2k, (2) max{H i k, and (3) min{H i k.
Interval Graphs with Side Constraints
, 1995
"... We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When ..."
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We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a linear time algorithm. Extant algorithms for this problem (known also as seriation with side constraints) required quadratic time. This problem contains as subproblems interval graph and interval order recognition. When the constraints are bounds on distances between endpoints, and the graph admits a unique clique order, we show that the problem is polynomial. The special case of this problem where the constraints are bounds on interval length is shown to be linearly equivalent to deciding if a system of difference inequalities is feasible. However, we show that even when the lengths of all intervals ar...