Feodor F. Dragan

Distance Approximating Trees for Chordal and Dually Chordal Graphs

In this paper we show that, for each chordal graphG, there is a treeTsuch thatTis a spanning tree of the squareG2ofGand, for every two vertices, the distance between them inTis not larger than the distance inGplus 2. Moreover, we prove that, ifGis a strongly chordal graph or even a dually chordal graph, then there exists a spanning treeTofGthat is an additive 3-spanner as well as a multiplicative 4-spanner ofG. In all cases the treeTcan be computed in linear time.

The algorithmic use of hypertree structure and maximum neighbourhood orderings

The use of (generalized) tree structure in graphs is one of the main topics in the field of efficient graph algorithms. The well-known partial κ-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are known to be important also in connection with relational database schemes where hypergraphs with tree structure (acyclic hypergraphs) and their elimination orderings (perfect elimination orderings for chordal graphs, Graham-reduction for acyclic hypergraphs) are studied.

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points <i>u,v,w,x</i>, the two larger of the sums <i>d</i>(<i>u,v</i>)+<i>d</i>(<i>w,x</i>), <i>d</i>(<i>u,w</i>)+<i>d</i>(<i>v,x</i>), <i>d</i>(<i>u,x</i>)+<i>d</i>(<i>v,w</i>) differ by at most 2δ. Given a finite set <i>S</i> of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of <i>S</i> with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for <i>S</i> with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs <i>G</i>=(<i>V,E</i>) with uniformly bounded degrees of vertices, the exact center of <i>S</i> can be computed in linear time <i>O</i>(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs <i>G</i> on <i>n</i> vertices with an additive error <i>O</i>(δlog₂ <i>n</i>). This construction has an additive error comparable with that given by Gromov for <i>n</i>-point δ-hyperbolic spaces, but can be implemented in <i>O</i>(|E|) time (instead of <i>O</i>(<i>n</i>²)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.

New Graph Classes of Bounded Clique-Width

Abstract The clique-width of graphs is a major new concept with respect to the efficiency of graph algorithms; it is known that every problem expressible in a certain kind of Monadic Second Order Logic called LinEMSOL(τ1,L) by Courcelle et al. is linear-time solvable on any graph class with bounded clique-width for which a k-expression for the input graph can be constructed in linear time. The notion of clique-width extends the one of treewidth since bounded treewidth implies bounded clique-width. We give a complete classification of all graph classes defined by forbidden one-vertex extensions of the P4 (i.e., the path with four vertices a,b,c,d and three edges ab,bc,cd) with respect to bounded clique-width. Our results extend and improve recent structural and complexity results in a systematic way.

A linear‐time algorithm for connected r‐domination and Steiner tree on distance‐hereditary graphs

A distance-hereditary graph is a connected graph in which every induced path is isometric, i.e., the distance of any two vertices in an induced path equals their distance in the graph. We present a linear time labeling algorithm for the minimum cardinality connected r-dominating set and Steiner tree problems on distance-hereditary graphs.

Summarizing transactional databases with overlapped hyperrectangles

Transactional data are ubiquitous. Several methods, including frequent itemset mining and co-clustering, have been proposed to analyze transactional databases. In this work, we propose a new research problem to succinctly summarize transactional databases. Solving this problem requires linking the high level structure of the database to a potentially huge number of frequent itemsets. We formulate this problem as a set covering problem using overlapped hyperrectangles (a concept generally regarded as tile according to some existing papers); we then prove that this problem and its several variations are NP-hard, and we further reveal its relationship with the compact representation of a directed bipartite graph. We develop an approximation algorithm Hyper which can achieve a logarithmic approximation ratio in polynomial time. We propose a pruning strategy that can significantly speed up the processing of our algorithm, and we also propose an efficient algorithm Hyper+ to further summarize the set of hyperrectangles by allowing false positive conditions. Additionally, we show that hyperrectangles generated by our algorithms can be properly visualized. A detailed study using both real and synthetic datasets shows the effectiveness and efficiency of our approaches in summarizing transactional databases.

Provably good global buffering using an available buffer block plan

To implement high-performance global interconnect without impacting the performance of existing blocks, the use of buffer blocks is increasingly popular in structured-custom and block-based ASIC/SOC methodologies. Recent works by Cong et al. (1999) and Tang and Wong (2000) give algorithms to solve the buffer block planning problem. In this paper we address the problem of how to perform buffering of global nets given an existing buffer block plan. Assuming that global nets have been already decomposed into two-pin connections, we give a provably good algorithm based on a recent approach of Garg and Konemann (1998) and Fleischer (1999). Our method routes connections using available buffer blocks, such that required upper and lower bounds on buffer intervals-as well as wirelength upper bounds per connection-are satisfied. Our model allows more than one buffer to be inserted into any given connection. In addition, our algorithm observes buffer parity constraints, i.e., it will choose to use an inverter or a buffer (=co-located pair of inverters) according to source and destination signal parity. The algorithm outperforms previous approaches and has been validated on top-level layouts extracted from a recent high-end microprocessor design.

Convexity and HHD-Free Graphs

It is well known that chordal graphs can be characterized via m-convexity. In this paper we introduce the notion of m3-convexity (a relaxation of m-convexity) which is closely related to semisimplicial ordering of graphs. We present new characterizations of HHD-free graphs via m3-convexity and obtain some results known from [B. Jamison and S. Olariu, Adv. Appl. Math., 9 (1988), pp. 364--376] as corollaries. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all m3-convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHD-free graph contains a r-dominating clique with respect to a given vertex radius function r.

Dominating Cliques in Distance-Hereditary Graphs

A graph is distance-hereditary if and only if each cycle on five or more vertices has at least two crossing chords. We present linear time algorithms for the minimum r-dominating clique and maximum strict r-packing set problems on distance-hereditary graphs. Some related problems such as diameter, radius, central vertex, r-dominating by cliques and r-dominant clique are investigated too.

Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs

We present simple methods for approximating the diameters, radii, and centers of finite sets in δ-hyperbolic geodesic spaces and graphs. We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n) comparable with that given by M. Gromov.