arXiv subjects

John Peca-Medlin

Publications and source records attributed to John Peca-Medlin.

11 recordsLinked to original sources

arXiv2026

Burning rooted graph products

The burning number $b(G)$ of a graph $G$ is the minimum number of rounds required to burn all vertices when, at each discrete step, existing fires spread to neighboring vertices and one new fire may be ignited at an unburned vertex. This parameter measures the speed of influence propagation in a network and has been studied as a model for information diffusion and resource allocation in distributed systems. A central open problem, the Burning Number Conjecture (BNC), asserts that every graph on $n$ vertices can be burned in at most $\lceil \sqrt n\rceil$ rounds, a bound known to be sharp for paths and verified for several structured families of trees. We investigate rooted graph products, focusing on comb graphs obtained by attaching a path (a ``tooth'') to each vertex of a path (the ``spine''). Unlike classical symmetric graph products, rooted products introduce hierarchical bottlenecks: communication between local subnetworks must pass through designated root vertices, providing a natural model for hub-and-spoke or chain-of-command architectures. We prove that the BNC holds for all comb graphs and determine the precise asymptotic order of their burning number in every parameter regime, including exact formulas in the spine-dominant case that generalize the known formula for paths. Our approach is constructive, based on an explicit greedy algorithm that is optimal or near-optimal depending on the regime.

math.CO↗

arXiv2025

The Horton-Strahler number of butterfly trees

The Horton-Strahler (HS) number, a classical measure of branching complexity arising in hydrology and register allocation, is studied for butterfly trees, a recursive family of binary trees generated by block-merging operations. These trees arise as binary search trees of butterfly permutations, which form the $2$-Sylow subgroup of the symmetric group on $N = 2^n$ elements and appear in models of parallel computation and structured Gaussian elimination. For a single merging step applied to two independent Catalan trees with $m$ nodes, we show that HS$(\mathcal T_1 \oplus \mathcal T_2)/\log_2(2m) \to 1/2$ in probability, so the classical Catalan scaling is preserved under this restricted construction. In the simple butterfly model, where each level is formed from identical copies and encoded by an $n$-bit string $\mathbf{x}$, the HS number admits an exact representation as an additive functional of an explicit $8$-state Markov chain driven by iid bits $x_j \sim \mathrm{Bern}(p)$, and can be computed in $\mathcal O(n)$ time from $\mathbf{x}$. This yields a complete limit theory, including a strong law HS$(\mathcal T_n^B)/n \to \mu_p = pq/(1-pq)$ almost surely and a functional central limit theorem with variance $\sigma_p^2 = pq(1 - 3pq - 2p^2q^2)/(1-pq)^3$. For general butterfly trees, obtained by recursively merging independent subtrees, the increment depends on an expanding edge profile, and the process does not admit a finite-state reduction. We give an $\mathcal O(N)$ algorithm to compute the HS number directly from the $(N-1)$-bit encoding, characterize the zero-HS class, and combine exact enumeration for small $n$ with Monte Carlo simulations up to $n=25$, supporting HS$(\mathcal T_n^B)/n \to \alpha \approx 0.4450$ in probability for uniform butterfly trees, placing the general model strictly between the simple butterfly limit $1/3$ and the Catalan limit $1/2$.

math.PR↗

arXiv2025

Heights of butterfly trees

Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure motivated by parallel data architectures -- corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height $h_n$ of a random BST satisfies $h_n / \log n \to c^* \approx 4.311$, we show that block BSTs with $nm$ nodes and fixed external size $m$ satisfy $h_{n,m} / \log n \to c^* + h_m$ in distribution. We then study butterfly trees: BSTs with $N = 2^n$ nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of $S_2$), we give a full distributional description showing polynomial height growth: $\mathbb{E} h_n^{\operatorname{B}} = \Theta(N^\alpha)$ with $\alpha = \log_2(3/2) \approx 0.58496$. For nonsimple butterfly trees (from wreath products), we prove power-law bounds: $cN^\alpha\cdot (1 + o(1)) \le \mathbb{E} h_n^{\operatorname{B}} \le dN^\beta\cdot (1 + o(1))$, with $\beta \approx 0.913189$.

math.PR↗

arXiv2025

Pivot probabilities and norm effects in Gaussian elimination for $β$-ensembles

We analyze pivot probabilities in Gaussian elimination with partial pivoting (GEPP) for $2 \times 2$ random matrix ensembles. For GUE matrices, we resolve a previously reported discrepancy between theoretical predictions and empirical observations by deriving the exact pivot probability under standard LAPACK-style implementations. We further show that Dumitriu-Edelman tridiagonal $β$-ensembles agree with the earlier theoretical expectations.

math.PR↗

arXiv2024

On the longest increasing subsequence and number of cycles of butterfly permutations

One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix $A$ and storing the permutation matrix factor $P$ from the resulting GEPP factorization $PA=LU$. We are interested in exploring properties of random butterfly permutations, which are generated using GEPP on specific random butterfly matrices. Our paper highlights new connections among random matrix theory, numerical linear algebra, group actions of rooted trees, and random permutations. We address the questions of the longest increasing subsequence (LIS) and number of cycles for particular uniform butterfly permutations, with full distributional descriptions and limit theorems for simple butterfly permutations. We also establish scaling limit results and limit theorems for nonsimple butterfly permutations, which include certain $p$-Sylow subgroups of the symmetric group of $N=p^n$ elements for prime $p$. For the LIS, we establish power law bounds on the expected LIS of the form $N^{α_p}$ and $N^{β_p}$ where $\frac12 < α_p < β_p < 1$ for each $p$ with $α_p = 1 - o_p(1)$, showing distinction from the typical $O(N^{1/2})$ expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations). For the number of cycles scaled by $(2-1/p)^n$, we establish a full CLT to a new limiting distribution depending on $p$ with positive support we introduce that is uniquely determined by its positive moments that satisfy explicit recursive formulas; this thus determines a CLT for the number of cycles for any uniform $p$-Sylow subgroup of $S_{p^n}$.

math.PR↗

arXiv2024

Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices

The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the 1960s, while he was only able to establish an upper bound for the GE with complete pivoting (GECP) growth problem. The GECP growth problem has seen a spike in recent interest, culminating in improved lower and upper bounds established by Bisain, Edelman, and Urschel in 2023, but still remains far from being fully resolved. Due to the complex dynamics governing the location of GECP pivots, analysis of GECP growth for particular input matrices often estimates the actual growth rather than computes the growth exactly. We present a class of dense random butterfly matrices for which we can compute the exact GECP growth. We extend previous results that established exact growth computations for butterfly matrices when using GEPP and GE with rook pivoting (GERP) to now also include GECP for structured subclasses of inputs. Moreover, we present a new method to construct random Hadamard matrices using butterfly matrices.

math.NA↗

arXiv2024

Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting

Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.

math.NA↗

arXiv2024

Random matrix statistics and safety rest areas on interstates in the United States

We analyze physical spacings between locations of safety rest areas on interstates in the United States. We show normalized safety rest area spacings on major interstates exhibit Wigner surmise statistics, which align with the eigenvalue spacings for the Gaussian Unitary Ensemble from random matrix theory as well as the one-dimensional gas interactions via the Coulomb potential. We identify economic and geographic regional traits at the state level that exhibit Poissonian statistics, which become more pronounced with increased geographical obstacles in interstate travel. Other regional filters (e.g., historical or political) produced results that did not diverge substantially from the overall Wigner surmise model.

physics.soc-ph↗

arXiv2023

Total positivity and spectral theory for Toeplitz Hessenberg matrix ensembles

In this paper we define and lay the groundwork for studying a novel matrix ensemble: totally positive Hessenberg Toeplitz operators, denoted TPHT. This is the intersection of two ensembles that have been significantly explored: totally positive Hessenberg matrices (TPH) and Hessenberg Toeplitz matrices (HT). TPHT has a rich linear algebraic and spectral structure that we describe. Along the way we find some previously unnoticed connections between certain Toeplitz normal forms for matrices and Lie theoretic interpretations. We also numerically study the spectral asymptotics of TPH matrices via the TPHT ensemble and use this to open a study of TPHT with random symbols.

math.SP↗

arXiv2023

Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices

Gaussian elimination with partial pivoting (GEPP) is a widely used method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $α$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $α$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.

math.NA↗

arXiv2023

Growth factors of random butterfly matrices and the stability of avoiding pivoting

Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.

math.NA↗