arXiv subjects

Meera Sitharam

Publications and source records attributed to Meera Sitharam.

At least 19 recordsLinked to original sources

arXiv2025

Nucleation-free independent graphs with implied nonedges

We give inductive constructions of independent graphs that contain implied nonedges but do not contain any non-trivial rigid subgraphs, or \emph{nucleations}: some of the constructions and proofs apply to 3-dimensional abstract rigidity matroids with their respective definitions of nucleations and implied nonedges. The first motivation for the inductive constructions of this paper, which generate an especially intractable class of flexible circuits, is to illuminate further obstacles to settling Graver's maximality conjecture that the 3-dimensional generic rigidity matroid is isomorphic to Whiteley's cofactor matroid (the unique maximal matroid in which all graphs isomorphic to $K_5$ are circuits). While none of the explicit examples we provide refutes the maximality conjecture (since their properties hold in both matroids) the construction schemes are useful regardless whether the conjecture is true or false, e.g. for constructing larger (counter)examples from smaller ones. The second motivation is to make progress towards a polynomial-time algorithm for deciding independence in the abovementioned maximal matroid. Nucleation-free graphs with implied nonedges, such as the families constructed in this paper, are the key obstacles that must be dealt with for improving the current state of the art.

math.CO↗

arXiv2024

Parallel Online Directed Acyclic Graph Exploration for Atlasing Soft-Matter Assembly Configuration Spaces

The paper formalizes a version of parallel online directed acyclic graph (DAG) exploration, general enough to be readily mapped to many computational scenarios. In both the offline and online versions, vertices are weighted with the work units required for their processing, at least one parent must be completely processed before a child is processed, and at any given time only one processor can work on any given vertex. The online version has the following additional natural restriction: only after a vertex is processed, are its required work units or its children known. Using the Actor Model of parallel computation, it is shown that a natural class of parallel online algorithms meets a simple competitive ratio bound. We demonstrate and focus on the problem's occurrence in the scenario of energy landscape roadmapping or atlasing under pair-potentials, a highly compute-and-storage intensive modeling component integral to diverse applications involving soft-matter assembly. The method is experimentally validated using a C++ Actor Framework (CAF) software implementation built atop EASAL (Efficient Atlasing and Search of Assembly Landscapes), a substantial opensource software suite, running on multiple CPU cores of the HiperGator supercomputer, demonstrating linear speedup results.

cs.DC↗

arXiv2024

Graphs with single interval Cayley configuration spaces in 3-dimensions

We prove a conjectured graph theoretic characterization of a geometric property of 3 dimensional linkages posed 15 years ago by Sitharam and Gao, motivated by their equivalent characterization for $d\le 2$ that does not generalize to $d\ge 3$. A linkage $(G,\ell)$ contains a finite simple undirected graph $G$ and a map $\ell$ that assigns squared Euclidean lengths to the edges of $G$. A \emph{$d$-realization} of $(G,\ell)$ is an assignment of points in $\mathbb{R}^d$ to the vertices of $G$ for which pairwise squared distances between points agree with $\ell$. For any positive integer $d \leq 3$, we characterize pairs $(G,f)$, where $f$ is a nonedge of $G$, such that, for any linkage $(G,\ell)$, the lengths attained by $f$ form a single interval - over the (typically a disconnected set of) $d$-realizations of $(G,\ell)$. Although related to the minor closed class of $d$-flattenable graphs, the class of pairs $(G,f)$ with the above property is not closed under edge deletions, has no obvious well quasi-ordering, and there are infinitely many minimal graph-nonedge pairs - with respect to edge contractions - in the complement class. Our characterization overcomes these obstacles, is based on the forbidden minors for $d$-flattenability for $d \leq 3$, and contributes to the theory of Cayley configurations with many applications. Helper results and corollaries provide new tools for reasoning about configuration spaces and completions of partial 3-tree linkages, (non)convexity of Euclidean measurement sets in $3$-dimensions, their projections, fibers and sections. Generalizations to higher dimensions and efficient algorithmic characterizations are conjectured.

cs.CG↗

arXiv2024

Best of two worlds: Cartesian sampling and volume computation for distance-constrained configuration spaces using Cayley coordinates

Volume calculation of configurational spaces acts as a vital part in configurational entropy calculation, which contributes towards calculating free energy landscape for molecular systems. In this article, we present our sampling-based volume computation method using distance-based Cayley coordinate, mitigating drawbacks: our method guarantees that the sampling procedure stays in lower-dimensional coordinate space (instead of higher-dimensional Cartesian space) throughout the whole process; and our mapping function, utilizing Cayley parameterization, can be applied in both directions with low computational cost. Our method uniformly samples and computes a discrete volume measure of a Cartesian configuration space of point sets satisfying systems of distance inequality constraints. The systems belong to a large natural class whose feasible configuration spaces are effectively lower dimensional subsets of high dimensional ambient space. Their topological complexity makes discrete volume computation challenging, yet necessary in several application scenarios including free energy calculation in soft matter assembly modeling. The algorithm runs in linear time and empirically sub-linear space in the number of grid hypercubes (used to define the discrete volume measure) \textit{that intersect} the configuration space. In other words, the number of wasted grid cube visits is insignificant compared to prevailing methods typically based on gradient descent. Specifically, the traversal stays within the feasible configuration space by viewing it as a branched covering, using a recent theory of Cayley or distance coordinates to convexify the base space, and by employing a space-efficient, frontier hypercube traversal data structure. A software implementation and comparison with existing methods is provided.

cs.CG↗

arXiv2024

Angular constraints on planar frameworks

Consider a collection of points in the plane and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on the set of direction constraints between the points is equivalent to the algebraic matroid on the set of distances between the points. This is the well-studied generic 2-dimensional rigidity matroid of a graph. This article studies a higher-level construction built on the slope data: an angle constraint system obtained by prescribing relationships between pairs of slopes. The central question we analyze is: when is an angle system rigid, in the sense that every nontrivial motion alters one of the fixed angles? We formulate the problem in matricial terms for certain edge-colored graphs, finding precise necessary conditions for when such edge-colored graphs are rigid, and a combinatorial characterization of generic rigidity for a special case. We also prove the validity of an equivalent formulation of the angle matroid as the algebraic matroid of a field extension.

math.CO↗

arXiv2024

Flip Paths Between Lattice Triangulations

We present a $O(n^{\frac{3}{2}})$-time algorithm for the \emph{shortest (diagonal) flip path problem} for \emph{lattice} triangulations with $n$ points, improving over previous $O(n^2)$-time algorithms. For a large, natural class of inputs, our bound is tight in the sense that our algorithm runs in time linear in the number of flips in the output flip path. Our results rely on an independently interesting structural elucidation of shortest flip paths as the linear orderings of a unique partially ordered set, called a \emph{minimum flip plan}, constructed by a novel use of Farey sequences from elementary number theory. Flip paths between general (not necessarily lattice) triangulations have been studied in the combinatorial setting for nearly a century. In the Euclidean geometric setting, finding a shortest flip path between two triangulations is NP-complete. However, for lattice triangulations, which are studied as spin systems, there are known $O\left(n^2\right)$-time algorithms to find shortest flip paths. These algorithms, as well as ours, apply to \emph{constrained} flip paths that ensure a set of \emph{constraint} edges are present in every triangulation along the path. Implications for determining simultaneously flippable edges, i.e. finding optimal simultaneous flip paths between lattice triangulations, and for counting lattice triangulations are discussed.

cs.CG↗

arXiv2023

Maximum likelihood thresholds via graph rigidity

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{ö}s-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{ö}s-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most $4$ and describe the threshold probability for the MLT to switch from $3$ to $4$. We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.

math.CO↗

arXiv2022

Parallel Exploration of Directed Acyclic Graphs using the Actor Model

In this paper we describe a generic scheme for the parallel exploration of directed acyclic graphs starting from one or more `roots' of the graph. Our scheme is designed for graphs with the following properties, (i) discovering neighbors at any node requires a non-trivial amount of computation, it is not a simple lookup; (ii) once a node is processed, all its neighbors are discovered; (iii) each node can be discovered through multiple paths, but should only be processed once. Several computational problems can be reduced to traversing such graphs, where the goal is to explore the graph and build a traversal roadmap. As a proof of concept for the effectiveness of our scheme at achieving speedup due to parallelism, we implement the scheme for the parallel exploration of assembly landscape using the EASAL methodology.

cs.DC↗

arXiv2022

Computing maximum likelihood thresholds using graph rigidity

math.CO↗

arXiv2021

The Maximum Matroid of a Graph

The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a {\it maximum matroid for a graph} $G$ is introduced, and the existence and uniqueness of the maximum matroid for any graph $G$ is proved. The maximum matroid for $K_3$ is shown to be the cycle (or graphic) matroid. This result is pursued in two directions - to determine the maximum matroid for the $m$-cycle $C_m$ and to determine the maximum matroid for the complete graph $K_m$. The maximum matroid for $K_4$ is the matroid whose bases are the Laman graphs, related to structural rigidity of frameworks in the plane. The maximum matroid for $K_5$ is related to a famous 153 year old open problem of J. C. Maxwell.

math.CO↗

arXiv2021

Realizations of Isostatic Material Frameworks

This paper studies the set of equivalent realizations of isostatic frameworks in two dimensions, and algorithms for finding all such realizations. We show that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. We enumerate the complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions and study the impact of boundary length on the emergence of these realizations. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints; alternative methods - based on constraint reduction and distance-based covering map or Cayley parameterization of the search space - are presented. The application of these methods is studied on atomic clusters, a model two-dimensional glasses, and jamming.

cond-mat.dis-nn↗

arXiv2020

Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity

We describe a novel geometric methodology for analyzing free-energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent, and provides illustrations of its unique capabilities. An atlas is a labeled partition of the assembly landscape into a topological roadmap of maximal, contiguous, nearly-equipotential-energy conformational regions or macrostates, together with their neighborhood relationships. The new methodology decouples the roadmap generation from sampling and produces: (1) a query-able atlas of local potential energy minima, their basin structure, energy barriers, and neighboring basins; (2) paths between a specified pair of basins; and (3) approximations of relative path lengths, basin volumes (configurational entropy), and path probabilities. Results demonstrating the core algorithm's capabilities have been generated by a resource-light, opensource software implementation EASAL. EASAL atlases several hundred thousand macrostates in minutes on a standard laptop. Subsequent path and basin computations each take seconds. The core algorithm's correctness, time complexity, and efficiency-accuracy tradeoffs are formally guaranteed using modern geometric constraint systems. The methodology further links geometric variables of the input assembling units to a type of intuitive topological bar-code of the output atlas, which in turn determine stable assembled structures and kinetics. This succinct input-output relationship facilitates reverse analysis, and control towards design. We use the novel convex Cayley (distance-based) parametrization that is unique to assembly, as opposed to folding. Sampling microstates with macrostate-specific Cayley parameters avoids gradient-descent search used by all prevailing methods. This increases sampling efficiency, significantly reduces the number of repeated and discarded samples.

cs.CG↗

arXiv2020

Rapid prediction of crucial hotspot interactions for icosahedral viral capsid self-assembly by energy landscape atlasing validated by mutagenesis

Icosahedral viruses have their infectious genome encapsulated by a shell assembled by a multiscale process, starting from an integer multiple of 60 viral capsid or coat protein (VP) monomers. We predict and validate inter-atomic hotspot interactions between VP monomers that are important for the assembly of 3 icosahedral viral capsids: Adeno Associated Virus serotype 2 (AAV2) and Minute Virus of Mice (MVM), both T=1 single stranded DNA viruses, and Bromo Mosaic Virus (BMV), a T=3 single stranded RNA virus. Experimental validation is by in-vitro, site-directed mutagenesis data found in literature. We combine ab-initio predictions at two scales: at the interface-scale, we predict the importance (cruciality) of an interaction for successful subassembly across each interface between VP monomers; and at the capsid-scale, we predict the cruciality of an interface for successful capsid assembly. At the interface-scale, we measure cruciality by changes in the capsid free-energy landscape partition function when an interaction is removed. The partition function computation uses atlases of interface subassembly landscapes, rapidly generated by a novel geometric method and curated opensource software EASAL (efficient atlasing and search of assembly landscapes). At the capsid-scale, cruciality of an interface for successful assembly of the capsid is based on combinatorial entropy. Our study goes from resource-light, multiscale computational predictions of crucial hotspot inter-atomic interactions to validation using data on site-directed mutagenesis' effect on capsid assembly. By reliably and rapidly narrowing down target interactions, (no more than 1.5 hours per interface on a laptop with Intel Core i5-2500K 3.2Ghz CPU and 8GB of RAM) our predictions can inform and reduce time-consuming in-vitro and in-vivo experiments, or more computationally intensive in-silico analyses.

q-bio.BM↗

arXiv2020

Bounds on the Jensen Gap, and Implications for Mean-Concentrated Distributions

This paper gives upper and lower bounds on the gap in Jensen's inequality, i.e., the difference between the expected value of a function of a random variable and the value of the function at the expected value of the random variable. The bounds depend only on growth properties of the function and specific moments of the random variable. The bounds are particularly useful for distributions that are concentrated around the mean, a commonly occurring scenario such as the average of i.i.d. samples and in statistical mechanics.

math.PR↗

arXiv2020

Fast and Flexible Geometric Method For Enhancing MC Sampling of Compact Configurations For Protein Docking Problem

EASAL (Efficient Atlasing and Sampling of Assembly Landscapes) is a geometric method for sampling and computing integrals over the potential energy landscape of small molecular assemblies. EASAL's efficiency arises from the fact that small assembly landscapes permit the use of so-called Cayley (inter-atomic distance based) parameters for geometric representation and sampling of the assembly configuration space regions; this results in their isolation, convexification, customized sampling and systematic traversal using a comprehensive topological roadmap. We define custom-designed measurements to investigate and compare various sampling characteristics of EASAL and the traditional Monte Carlo (MC) sampling, including (i) sampling speed, (ii) efficiency and accuracy of uniform grid coverage, (iii) accuracy of weighted coverage at covering low energy regions, (iv) ability to localize sampling to macrostates, and (v) flexibility in sampling distributions. In particular, we compare the sampling characteristics of EASAL and MC in sampling the assembly landscape of 2 trans-membrane helices, with short-range pair-potentials. We demonstrate that EASAL provides a reasonable coverage of crucial but narrow regions of the energy landscape of low effective dimension, with much fewer samples and computational resources than MC sampling. Promising avenues for combining the complementary advantages of the two methods are discussed.

physics.chem-ph↗

arXiv2020

Plane-Activated Mapped Microstructure

Querying and interacting with models of massive material micro-structure requires localized on-demand generation of the micro-structure since the full-scale storing and retrieving is cost prohibitive. When the micro-structure is efficiently represented as the image of a canonical structure under a non-linear space deformation to allow it to conform to curved shape, the additional challenge is to relate the query of the mapped micro-structure back to its canonical structure. This paper presents an efficient algorithm to pull back a mapped micro-structure to a partition of the canonical domain structure into boxes and only activates boxes whose image is likely intersected by a plane. The active boxes are organized into a forest whose trees are traversed depth first to generate mapped micro-structure only of the active boxes. The traversal supports, for example, 3D print slice generation in additive manufacturing.

cs.GR↗

arXiv2018

Efficient Atlasing and Search of Configuration Spaces of Point-Sets Constrained by Distance Intervals

For configurations of point-sets that are pairwise constrained by distance intervals, the EASAL software implements a suite of algorithms that characterize the structure and geometric properties of the configuration space. The algorithms generate, describe and explore these configuration spaces using generic rigidity properties, classical results for stratification of semi-algebraic sets, and new results for efficient sampling by convex parametrization. The paper reviews the key theoretical underpinnings, major algorithms and their implementation. The paper outlines the main applications such as the computation of free energy and kinetics of assembly of supramolecular structures or of clusters in colloidal and soft materials. In addition, the paper surveys select experimental results and comparisons.

cs.CG↗

arXiv2018

Corner-Sharing Tetrahedra for Modeling Micro-Structure

State-of-the-art representations of volumetric multi-scale shape and structure can be classified into three broad categories: continuous, continuous-from-discrete, and discrete representations. We propose modeling micro-structure with a class of discrete Corner-Sharing Tetrahedra (CoSTs). CoSTs can represent bar-joint, tensegrity, line-incidence, and similar constraint systems that capture local physical constraints and global multi-scale properties for design and analysis. The paper develops a palette of simple geometry processing operations on CoSTs including graph manipulation, hierarchical refinement, randomization, and generating associated continuous representations.

cs.CG↗