Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

On the Cubicity of Interval Graphs

A k-cube (or ``a unit cube in k dimensions'') is defined as the Cartesian product R-1 x . . . x R-k where R-i (for 1 <= i <= k) is an interval of the form [a(i), a(i) + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i. e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Delta, cub(G) <= inverted right perpendicular log(2) Delta inverted left perpendicular + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to inverted right perpendicular log(2) Delta inverted left perpendicular.

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K-4, then box(G) = 2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G) = 2 unless G is isomorphic to K4 (in which case its boxicity is 1).

Compiling Techniques for Coarse Grained Runtime Reconfigurable Architectures

In this paper we develop compilation techniques for the realization of applications described in a High Level Language (HLL) onto a Runtime Reconfigurable Architecture. The compiler determines Hyper Operations (HyperOps) that are subgraphs of a data flow graph (of an application) and comprise elementary operations that have strong producer-consumer relationship. These HyperOps are hosted on computation structures that are provisioned on demand at runtime. We also report compiler optimizations that collectively reduce the overheads of data-driven computations in runtime reconfigurable architectures. On an average, HyperOps offer a 44% reduction in total execution time and a 18% reduction in management overheads as compared to using basic blocks as coarse grained operations. We show that HyperOps formed using our compiler are suitable to support data flow software pipelining.

A measure of logical complexity of programs

The research in software science has so far been concentrated on three measures of program complexity: (a) software effort; (b) cyclomatic complexity; and (c) program knots. In this paper we propose a measure of the logical complexity of programs in terms of the variable dependency of sequence of computations, inductive effort in writing loops and complexity of data structures. The proposed complexity mensure is described with the aid of a graph which exhibits diagrammatically the dependence of a computation at a node upon the computation of other (earlier) nodes. Complexity measures of several example programs have been computed and the related issues have been discussed. The paper also describes the role played by data structures in deciding the program complexity.

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

Multiresolution Area-based Fractal Dimension Estimation of Signals Applied to EEG Data

In this paper, we present an approach to estimate fractal complexity of discrete time signal waveforms based on computation of area bounded by sample points of the signal at different time resolutions. The slope of best straight line fit to the graph of log(A(rk)A / rk(2)) versus log(l/rk) is estimated, where A(rk) is the area computed at different time resolutions and rk time resolutions at which the area have been computed. The slope quantifies complexity of the signal and it is taken as an estimate of the fractal dimension (FD). The proposed approach is used to estimate the fractal dimension of parametric fractal signals with known fractal dimensions and the method has given accurate results. The estimation accuracy of the method is compared with that of Higuchi's and Sevcik's methods. The proposed method has given more accurate results when compared with that of Sevcik's method and the results are comparable to that of the Higuchi's method. The practical application of the complexity measure in detecting change in complexity of signals is discussed using real sleep electroencephalogram recordings from eight different subjects. The FD-based approach has shown good performance in discriminating different stages of sleep.

WISENOM. A formal organic chemical nomenclature system

A formal chemical nomenclature system WISENOM based on a context-free grammar and graph coding is described. The system is unique, unambiguous, easily pronounceable, encodable, and decodable for organic compounds. Being a formal system, every name is provable as a theorem or derivable as a terminal sentence by using the basic axioms and rewrite rules. The syntax in Backus-Naur form, examples of name derivations, and the corresponding derivation trees are provided. Encoding procedures to convert connectivity tables to WISENOM, parsing, and decoding are described.

Acyclic Edge Coloring of Graphs with Maximum Degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a'(G) <= Delta+2, where Delta=Delta(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Delta(G)<= 4, with the additional restriction that m <= 2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m <= 2n, when Delta(G)<= 4. It follows that for any graph G if Delta(G)<= 4, then a'(G) <= 7.

In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains

The test based on comparison of the characteristic coefficients of the adjancency matrices of the corresponding graphs for detection of isomorphism in kinematic chains has been shown to fail in the case of two pairs of ten-link, simple-jointed chains, one pair corresponding to single-freedom chains and the other pair corresponding to three-freedom chains. An assessment of the merits and demerits of available methods for detection of isomorphism in graphs and kinematic chains is presented, keeping in view the suitability of the methods for use in computerized structural synthesis of kinematic chains. A new test based on the characteristic coefficients of the “degree” matrix of the corresponding graph is proposed for detection of isomorphism in kinematic chains. The new test is found to be successful in the case of a number of examples of graphs where the test based on characteristic coefficients of adjancency matrix fails. It has also been found to be successful in distinguishing the structures of all known simple-jointed kinematic chains in the categories of (a) single-freedom chains with up to 10 links, (b) two-freedom chains with up to 9 links and (c) three-freedom chains with up to 10 links.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

An (O)over-tilde(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V, E). The expected running time of our algorithm is (O) over tilde (mc) where vertical bar E vertical bar = m and c is the maximum u-v edge connectivity, where u, v is an element of V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n - 1; so the expected run-ning time of our algorithm for simple unweighted graphs is (O) over tilde (mn). All the algorithms currently known for constructing a Gomory-Hu tree [8, 9] use n - 1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest (O) over tilde (n(20/9)) max flow algorithm due to Karger and Levine[11] yields the current best running time of (O) over tilde (n(20/9)n) for Gomory-Hu tree construction on simple unweighted graphs with m edges and n vertices. Thus we present the first (O) over tilde (mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs. We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S subset of V can be reused for computing a minimum Steiner cut for certain Steiner sets S' subset of S.

On the Arrangement of Cliques in Chordal Graphs with respect to the Cuts

A cut (A, B) (where B = V - A) in a graph G = (V, E) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y is an element of B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with kappa(G) + vertices (where kappa(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 <= i <= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(G) + 1 - i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Omega(k(2)), where kappa(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.

Comprehensive Path-sensitive Data-flow Analysis

Data-flow analysis is an integral part of any aggressive optimizing compiler. We propose a framework for improving the precision of data-flow analysis in the presence of complex control-flow. W initially perform data-flow analysis to determine those control-flow merges which cause the loss in data-flow analysis precision. The control-flow graph of the program is then restructured such that performing data-flow analysis on the resulting restructured graph gives more precise results. The proposed framework is both simple, involving the familiar notion of product automata, and also general, since it is applicable to any forward data-flow analysis. Apart from proving that our restructuring process is correct, we also show that restructuring is effective in that it necessarily leads to more optimization opportunities. Furthermore, the framework handles the trade-off between the increase in data-flow precision and the code size increase inherent in the restructuring. We show that determining an optimal restructuring is NP-hard, and propose and evaluate a greedy strategy. The framework has been implemented in the Scale research compiler, and instantiated for the specific problem of Constant Propagation. On the SPECINT 2000 benchmark suite we observe an average speedup of 4% in the running times over Wegman-Zadeck conditional constant propagation algorithm and 2% over a purely path profile guided approach.

A necessary and sufficient condition for acyclic graphs

This letter gives a new necessary and sufficient condition to determine whether a directed graph is acyclic.