Alejandro A. Schäffer

A polynomial time algorithm for finding the prime factors of cartesian-product graphs☆

Abstract We consider the computational complexity of recognizinf concerned cartesian product graphs. Sabidussi gives a non-algorithmic proof that the cartesian factorization is unique. He uses a tower of successively coarser equivalence relations on the edge set in which each prime factor of the graph is identified with an equivalence class in the coarsest of the relations. We first explore the structure and size of the relation at the base of the tower. Then we give a polynomial-time algorithm to compute the relations and to construct the prime factors of any connected graph. The bounds on the size of the relations are crucial to the runtime analysis of our algorithm.

Convex hulls of piecewise-smooth Jordan curves

Abstract Several linear-time algorithms for finding the convex hull of a simple polygon in the plane are known. This paper extends a linear-time algorithm for polygons to a linear-time algorithm for input that includes smooth, convex, curved edges, such as conic sections.

Fast parallel algorithms for chordal graphs

We present an NC algorithm for recognizing chordal graphs, and we present NC algorithms for finding the following objects on chordal graphs: all maximal cliques, an intersection graph representation, an optimal coloring, a perfect elimination scheme, a maximum independent set, a minimum clique cover, and the chromatic polynomial. The well known polynomial algorithms for these problems seem highly sequential, and therefore a different approach is needed to find parallel algorithms.

Finding the prime factors of strong direct product graphs in polynomial time

Abstract The strong direct product is one of several standard product operators on graphs. Dorfler and Imrich (1969) and, independently, McKenzie (1971) showed that any connected graph has a unique prime factorization under the strong direct product. We give a polynomial-time algorithm for finding this unique factorization, thus settling a question of Nesetřil.

Time bounds on fault‐tolerant broadcasting

Broadcasting is the process by which a message originated at one vertex is delivered to all other vertices of a network, subject to the restriction that a vertex may participate in only one message transfer during a given time unit. A k fault-tolerant broadcasting scheme is a calling scheme that gurantees the completion of the broadcast in the presence of up to k link failures. Let Tk(n) denote the minimum time required for k fault-tolerant broadcasting in an n-vertex network. Liestman [Networks15 (1985) 159–171] showed that for every n and k such that n − 2 ≥ k ≥ 1, Tk(n) ⩾ [log n]+k. This paper establishes a matching upper bound, showing that for such n and k, Tk(n) ϵ O (log n +k). In particular, we present various efficient broadcasting schemes achieving almost optimal multiplicative constants. Our best upper bound uses new partial results on a tree-packing problem that may be of independent interest.

The evolution of tumour phylogenetics: principles and practice

Rapid advances in high-throughput sequencing and a growing realization of the importance of evolutionary theory to cancer genomics have led to a proliferation of phylogenetic studies of tumour progression. These studies have yielded not only new insights but also a plethora of experimental approaches, sometimes reaching conflicting or poorly supported conclusions. Here, we consider this body of work in light of the key computational principles underpinning phylogenetic inference, with the goal of providing practical guidance on the design and analysis of scientifically rigorous tumour phylogeny studies. We survey the range of methods and tools available to the researcher, their key applications, and the various unsolved problems, closing with a perspective on the prospects and broader implications of this field.

Triangulating three-colored graphs in linear time and linear space

Kannan and Warnow [Triangulating Three-Colored Graphs, Proc. 2nd SODA, 1991, pp. 337–343 and SIAM J. Discrete Math., 5 (1992), pp. 249–258] describe an algorithm to decide whether a three-colored graph can be triangulated so that all the edges connect vertices of different colors. This problem is motivated by a problem in evolutionary biology. Kannan and Warnow have two implementation strategies for their algorithm: one uses slightly superlinear time, while the other uses linear time but quadratic space. We note that three-colored triangulatable graphs are always planar, and we use this fact to modify Kannan and Warnow’s algorithm to obtain an algorithm that uses both linear time and linear space.

Computing the bump number with techniques from two-processor scheduling

Let (X, <) be a partially ordered set. A linear extension x1, x2, ... has a bump whenever xi<xi+1, and it has a jump whenever xiand xi+1are incomparable. The problem of finding a linear erxtension that minimizes the number of jumps has been studied extensively; Pulleyblank shows that it is NP-complete in the general case. Fishburn and Gehrlein raise the question of finding a linear extension that minimizes the number of bumps. We show that the bump number problem is closely related to the well-studied problem of scheduling unit-time tasks with a precedence partial order on two identical processors. We point out that a variant of Gabows linear-time algorithm for the two-processor scheduling problem solves the bump number problem. Habib, Möhring, and Steiner have independently discovered a different polynomial-time algorithm to solve the bump number problem.

An implicit data structure for searching a multikey table in logarithmic time

Abstract A data structure is implicit if it uses no extra strorage beyond the space needed for the data and a constant number of parameters. We describe an implicit data structure for storing n k -key records, which supports searching for a record, under any key, in the asymptotically optimal search time O (1g n ). This is in sharp contrast to an Ω(n 1− 1 k ) lower bound which holds if all comparisons must be against the sought key value. The theoretical tools we develop also yield a more practical way to halve the number of memory references used in obvious non-implicit solutions to the multikey table problem.

Multiple matching of rectangular patterns

We describe the first .efiicient algorithm for simultaneously matching multiple rectangular patterns of varying sizes and aspect, ratios in a rectangular text. Efficient means significantly better asymptotically than known al,qorithrns that handle one height, width, or aspect ratio at a time. Our algorithm features an interesting use of multidimensional range searching, as well as new adaptations of several known techniques for two climensional string matching. We also extend our algorithm to a dynamic setting where the set of patterns can change over time.