Parameterized Leaf Power Recognition via Embedding into Graph Products

The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T. Recognition of the k-leaf power graphs for k >= 6 is still an open problem. In this paper, we provide an algorithm for this problem for sparse leaf power graphs. Our result shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we describe how to embed the leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. As a result, we can use methods based on monadic second-order logic (MSO_2) to recognize the existence of a leaf power as a subgraph of the product graph

Computing Optimal Leaf Roots of Chordal Cographs in Linear Time

A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G if and only if the distance between x and y in T is at most k. Such a tree T is called a k-leaf root of G. The computational problem of constructing a k-leaf root for a given graph G and an integer k, if any, is motivated by the challenge from computational biology to reconstruct phylogenetic trees. For fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time. In this paper, we propose to study optimal leaf roots of graphs G, that is, the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots is more justified as they yield more probable phylogenetic trees. Lafond's result does not imply polynomial-time computability of optimal leaf roots, because, even for optimal k-leaf roots, k may (exponentially) depend on the size of G. This paper presents a linear-time construction of optimal leaf roots for chordal cographs (also known as trivially perfect graphs). Additionally, it highlights the importance of the parity of the parameter k and provides a deeper insight into the differences between optimal k-leaf roots of even versus odd k. Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect leaf power, chordal cographComment: 22 pages, 2 figures, full version of the FCT 2023 pape

Infrastructure for Performance Monitoring and Analysis of Systems and Applications

The growth of High Performance Computer (HPC) systems increases the complexity with respect to understanding resource utilization, system management, and performance issues. HPC performance monitoring tools need to collect information at both the application and system levels to yield a complete performance picture. Existing approaches limit the abilities of the users to do meaningful analysis on actionable timescale. Efficient infrastructures are required to support largescale systems performance data analysis for both run-time troubleshooting and post-run processing modes. In this dissertation, we present methods to fill these gaps in the infrastructure for HPC performance monitoring and analysis. First, we enhance the architecture of a monitoring system to integrate streaming analysis capabilities at arbitrary locations within its data collection, transport, and aggregation facilities. Next, we present an approach to streaming collection of application performance data. We integrate these methods with a monitoring system used on large-scale computational platforms. Finally, we present a new approach for constructing durable transactional linked data structures that takes advantage of byte-addressable non-volatile memory technologies. Transactional data structures are building blocks of in-memory databases that are used by HPC monitoring systems to store and retrieve data efficiently. We evaluate the presented approaches on a series of case studies. The experiment results demonstrate the impact of our tools, while keeping the overhead in an acceptable margin