Leonidas Palios

Decomposition Algorithms in Geometry

Decomposing complex shapes into simpler components has always been a focus of attention in computational geometry. The reason is obvious: most geometric algorithms perform more efficiently and are easier to implement and debug if the objects have simple shapes. For example, mesh-generation is a standard staple of the finite-element method; partitioning polygons or polyhedra into convex pieces or simplices is a typical preprocessing step in automated design, robotics, and pattern recognition. In computer graphics, decompositions of two-dimensional scenes are used in contour filling, hit detection, clipping and windowing; polyhedra are decomposed into smaller parts to perform hidden surface removal and ray-tracing.

Detecting Holes and Antiholes in Graphs

AbstractIn this paper we study the problems of detecting holes and antiholes in general undirected graphs, and we present algorithms for these problems. For an input graph G on n vertices and m edges, our algorithms run in O(n + m2) time and require O(n m) space; we thus provide a solution to the open problem posed by Hayward et al. asking for an O(n4)-time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is the use of the depth-first-search traversal on appropriate auxiliary graphs in which moving between any two adjacent vertices is equivalent to walking along a P4 (i.e., a chordless path on four vertices) of the input graph or on its complement, respectively. The approach can be generalized so that for a fixed constant k ≥ 5 we obtain an O(nk-1)-time algorithm for the detection of a hole (antihole resp.) on at least k vertices. Additionally, we describe a different approach which allows us to detect antiholes in graphs that do not contain chordless cycles on five vertices in O(n + m2) time requiring O(n + m) space. Again, for a fixed constant k ≥ 6, the approach can be extended to yield O(nk-2)-time and O(n2)-space algorithms for detecting holes (antiholes resp.) on at least k vertices in graphs which do not contain holes (antiholes resp.) on k - 1 vertices. Our algorithms are simple and can be easily used in practice. Finally, we also show how our detection algorithms can be augmented so that they return a hole or an antihole whenever such a structure is detected in the input graph; the augmentation takes O(n + m) time and space.

Adding an edge in a cograph

In this paper, we establish structural properties of cographs which enable us to present an algorithm which, for a cograph G and a non-edge xy (i.e., two non-adjacent vertices x and y) of G, finds the minimum number of edges that need to be added to the edge set of G such that the resulting graph is a cograph and contains the edge xy. The motivation for this problem comes from algorithms for the dynamic recognition and online maintenance of graphs; the proposed algorithm could be a suitable addition to the algorithm of Shamir and Sharan [13] for the online maintenance of cographs. The proposed algorithm runs in time linear in the size of the input graph and requires linear space.

An Optimal Parallel Co-Connectivity Algorithm

Abstract In this paper we consider the problem of computing the connected components of the complement of a given graph. We describe a simple sequential algorithm for this problem, which works on the input graph and not on its complement, and which for a graph on n vertices and m edges runs in optimal O(n+m) time. Moreover, unlike previous linear co-connectivity algorithms, this algorithm admits efficient parallelization, leading to an optimal O(log n)-time and O((n+m)log n)-processor algorithm on the EREW PRAM model of computation. It is worth noting that, for the related problem of computing the connected components of a graph, no optimal deterministic parallel algorithm is currently available. The co-connectivity algorithms find applications in a number of problems. In fact, we also include a parallel recognition algorithm for weakly triangulated graphs, which takes advantage of the parallel co-connectivity algorithm and achieves an O(log2 n) time complexity using O((n+m2) log n) processors on the EREW PRAM model of computation.

An efficient shape-based approach to image retrieval

Abstract We consider the problem of finding the best match for a given query shape among candidate shapes stored in a shape base. This is central to a wide range of applications, such as, digital libraries, digital film databases, environmental sciences, and satellite image repositories. We present an efficient matching algorithm built around a novel similarity criterion and based on shape normalization about the shapes diameter, which reduces the effects of noise or limited accuracy during the shape extraction procedure. Our matching algorithm works by gradually “fattening” the query shape until the best match is discovered. The algorithm exhibits poly-logarithmic time behavior assuming uniform distribution of the shape vertices in the locus of their normalized positions.

Algorithms for P 4 -Comparability Graph Recognition and Acyclic P 4 -Transitive Orientation

Abstract We consider two problems pertaining to P4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P4-comparability graph and the problem of producing an acyclic P4-transitive orientation of a P4-comparability graph. These problems have been considered by Hoàng and Reed who described O(n4)- and O(n5)-time algorithms for their solution, respectively, where n is the number of vertices of the input graph. Faster algorithms have recently been presented by Raschle and Simon, and by Nikolopoulos and Palios; the time complexity of these algorithms for either problem is O(n + m2), where m is the number of edges of the graph. In this paper we describe O(n m)-time and O(n + m)-space algorithms for the recognition and the acyclic P4-transitive orientation problems on P4-comparability graphs. The algorithms rely on properties of the P4-components of a graph, which we establish, and on the efficient construction of the P4-components by means of the BFS-trees of the complement of the graph rooted at each of its vertices, without however explicitly computing the complement. Both algorithms are simple and use simple data structures.

Recognition and Orientation Algorithms for P4-Comparability Graphs

We consider two problems pertaining to P 4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P 4-comparability graph and the problem of producing an acyclic P 4-transitive orientation of a P 4-comparability graph. These problems have been considered by Hoang and Reed who described O(n 4) and O(n 5)-time algorithms for their solution respectively, where n is the number of vertices of the given graph. Recently, Raschle and Simon described O(n + m 2)-time algorithms for these problems, where m is the number of edges of the graph. In this paper, we describe difierent O(n + m 2)-time algorithms for the recognition and the acyclic P 4-transitive orientation problems on P 4- comparability graphs. Instrumental in these algorithms are structural relationships of the P 4-components of a graph, which we establish and which are interesting in their own right. Our algorithms are simple, use simple data structures, and have the advantage over those of Raschle and Simon in that they are non-recursive, require linear space and admit effcient parallelization.

Maximizing the number of spanning trees in Kn -complements of asteroidal graphs

In this paper we introduce the class of graphs whose complements are asteroidal (star-like) graphs and derive closed formulas for the number of spanning trees of its members. The proposed results extend previous results for the classes of the multi-star and multi-complete/star graphs. Additionally, we prove maximization theorems that enable us to characterize the graphs whose complements are asteroidal graphs and possess a maximum number of spanning trees.

A fully dynamic algorithm for the recognition of P 4 -sparse graphs

We consider the dynamic recognition problem for the class of P4-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4-sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish conditions for each modification to yield a P4-sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P4-sparse graphs. Moreover, by maintaining the children of each node of the modular decomposition tree in a binomial heap, we can handle vertex deletions in O(log n) time, at the expense of needing O(log n) time for each edge modification and O(d log n) time for the addition of a vertex adjacent to d vertices.

Recognizing HHDS-free graphs

In this paper, we consider the recognition problem on the HHDS-free graphs, a class of homogeneously orderable graphs, and we show that it has polynomial time complexity. In particular, we describe a simple O(n2m)-time algorithm which determines whether a graph G on n vertices and m edges is HHDS-free. To the best of our knowledge, this is the first polynomial-time algorithm for recognizing this class of graphs.