arXiv subjects

Reyan Ahmed

Publications and source records attributed to Reyan Ahmed.

At least 19 recordsLinked to original sources

arXiv2026

Size Should Not Matter: Evaluating Network Visualizations with Stress

The normalized stress metric is widely used to assess graph drawing quality, measuring how closely distances between vertices in a layout match their graph-theoretic distances. This metric is a standard for both evaluation and optimization in many popular graph layout algorithms. However, normalized stress is highly sensitive to the scale (size) of the drawing, leading to potentially misleading comparisons between layouts produced by different algorithms. Uniformly scaling a layout can significantly alter stress values without changing the underlying structure, even to the extent that a clearly superior layout can appear to have a higher stress score than a random layout. Although this issue is recognized within the network visualization community, it is rarely addressed with sufficient detail in publications, resulting in critical calculation errors in recent studies. In this paper, we systematically examine various stress metrics used in the literature and demonstrate that commonly used metrics are affected by layout scale, undermining their reliability for comparison. We identify scale-invariant alternatives and propose scale-normalized stress for fair stress-based evaluation.

cs.CG↗

arXiv2026

Homoglyph-based Adversarial Perturbation of Introductory Computer Science Theory Problems

Different AI tools such as ChatGPT, Gemini, and Claude are becoming very popular. Although they are helpful for many day-to-day tasks, they can be used in unexpected ways. For example, the learning objectives of a course may not be achieved if students use these tools to solve their homework problems. This paper proposes a simple method to address this issue in the lazy student model. The method uses homoglyph-based adversarial perturbation to first modify the question without changing the semantic meaning of the question. Then a few characters are perturbed by their homoglyphs. Our experimental result shows the theoretical problems of introductory computer science courses can be effectively perturbed. We also propose an interactive tool to conveniently use our method.

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arXiv2025

Word2VecGD: Neural Graph Drawing with Cosine-Stress Optimization

We propose a novel graph visualization method leveraging random walk-based embeddings to replace costly graph-theoretical distance computations. Using word2vec-inspired embeddings, our approach captures both structural and semantic relationships efficiently. Instead of relying on exact shortest-path distances, we optimize layouts using cosine dissimilarities, significantly reducing computational overhead. Our framework integrates differentiable stress optimization with stochastic gradient descent (SGD), supporting multi-criteria layout objectives. Experimental results demonstrate that our method produces high-quality, semantically meaningful layouts while efficiently scaling to large graphs. Code available at: https://github.com/mlyann/graphv_nn

cs.CG↗

arXiv2025

Online Facility Assignments on Polygons

We study the online facility assignment problem on regular polygons, where all sides are of equal length. The influence of specific geometric settings has remained mostly unexplored, even though classical online facility assignment problems have mainly dealt with linear and general metric spaces. We fill this gap by considering the following four basic geometric settings: equilateral triangles, rectangles, regular $n$-polygons, and circles. The facilities are situated at fixed positions on the boundary, and customers appear sequentially on the boundary. A customer needs to be assigned immediately without any information about future customer arrivals. We study a natural greedy algorithm. First, we study an equilateral triangle with three facilities at its corners; customers can appear anywhere on the boundary. We then analyze regular $n$-sided polygons, obtaining a competitive ratio of $2n-1$, showing that the algorithm performance degrades linearly with the number of corner points for polygons. For the circular configuration, the competitive ratio is $2n-1$ when the distance between two adjacent facilities is the same. And the competitive ratios are $n^2-n+1$ and $2^n - 1$ for varying distances linearly and exponentially respectively. Each facility has a fixed capacity proportional to the geometric configuration, and customers appear only along the boundary edges. Our results also show that simpler geometric configurations have more efficient performance bounds and that spacing facilities uniformly apart prevent worst-case scenarios. The findings have many practical implications because large networks of facilities are best partitioned into smaller and geometrically simple pieces to guarantee good overall performance.

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arXiv2025

Subsetwise and Multi-Level Additive Spanners with Lightness Guarantees

An \emph{additive +$βW$ spanner} of an edge weighted graph $G=(V,E)$ is a subgraph $H$ of $G$ such that for every pair of vertices $u$ and $v$, $d_{H}(u,v) \le d_G(u,v) + βW$, where $d_G(u,v)$ is the shortest path length from $u$ to $v$ in $G$. While additive spanners are very well studied in the literature, spanners that are both additive and lightweight have been introduced more recently [Ahmed et al., WG 2021]. Here the \emph{lightness} is the ratio of the spanner weight to the weight of a minimum spanning tree of $G$. In this paper, we examine the widely known subsetwise setting when the distance conditions need to hold only among the pairs of a given subset $S$. We generalize the concept of lightness to subset-lightness using a Steiner tree and provide polynomial-time algorithms to compute subsetwise additive $+εW$ spanner and $+(4+ε) W$ spanner with $O_ε(|S|)$ and $O_ε(|V_H|^{1/3} |S|^{1/3})$ subset-lightness, respectively, where $ε$ is an arbitrary positive constant. We next examine a multi-level version of spanners that often arises in network visualization and modeling the quality of service requirements in communication networks. The goal here is to compute a nested sequence of spanners with the minimum total edge weight. We provide an $e$-approximation algorithm to compute multi-level spanners assuming that an oracle is given to compute single-level spanners, improving a previously known 4-approximation [Ahmed et al., IWOCA 2023].

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arXiv2023

Graph Sparsifications using Neural Network Assisted Monte Carlo Tree Search

Graph neural networks have been successful for machine learning, as well as for combinatorial and graph problems such as the Subgraph Isomorphism Problem and the Traveling Salesman Problem. We describe an approach for computing graph sparsifiers by combining a graph neural network and Monte Carlo Tree Search. We first train a graph neural network that takes as input a partial solution and proposes a new node to be added as output. This neural network is then used in a Monte Carlo search to compute a sparsifier. The proposed method consistently outperforms several standard approximation algorithms on different types of graphs and often finds the optimal solution.

cs.LG↗

arXiv2023

A Scalable Method for Readable Tree Layouts

Large tree structures are ubiquitous and real-world relational datasets often have information associated with nodes (e.g., labels or other attributes) and edges (e.g., weights or distances) that need to be communicated to the viewers. Yet, scalable, easy to read tree layouts are difficult to achieve. We consider tree layouts to be readable if they meet some basic requirements: node labels should not overlap, edges should not cross, edge lengths should be preserved, and the output should be compact. There are many algorithms for drawing trees, although very few take node labels or edge lengths into account, and none optimizes all requirements above. With this in mind, we propose a new scalable method for readable tree layouts. The algorithm guarantees that the layout has no edge crossings and no label overlaps, and optimizes one of the remaining aspects: desired edge lengths and compactness. We evaluate the performance of the new algorithm by comparison with related earlier approaches using several real-world datasets, ranging from a few thousand nodes to hundreds of thousands of nodes. Tree layout algorithms can be used to visualize large general graphs, by extracting a hierarchy of progressively larger trees. We illustrate this functionality by presenting several map-like visualizations generated by the new tree layout algorithm.

cs.CG↗

arXiv2023

Nearly Optimal Steiner Trees using Graph Neural Network Assisted Monte Carlo Tree Search

Graph neural networks are useful for learning problems, as well as for combinatorial and graph problems such as the Subgraph Isomorphism Problem and the Traveling Salesman Problem. We describe an approach for computing Steiner Trees by combining a graph neural network and Monte Carlo Tree Search. We first train a graph neural network that takes as input a partial solution and proposes a new node to be added as output. This neural network is then used in a Monte Carlo search to compute a Steiner tree. The proposed method consistently outperforms the standard 2-approximation algorithm on many different types of graphs and often finds the optimal solution.

cs.LG↗

arXiv2023

Splitting Vertices in 2-Layer Graph Drawings

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are \NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.

cs.CG↗

arXiv2023

Multi-Priority Graph Sparsification

A \emph{sparsification} of a given graph $G$ is a sparser graph (typically a subgraph) which aims to approximate or preserve some property of $G$. Examples of sparsifications include but are not limited to spanning trees, Steiner trees, spanners, emulators, and distance preservers. Each vertex has the same priority in all of these problems. However, real-world graphs typically assign different ``priorities'' or ``levels'' to different vertices, in which higher-priority vertices require higher-quality connectivity between them. Multi-priority variants of the Steiner tree problem have been studied in prior literature but this generalization is much less studied for other sparsification problems. In this paper, we define a generalized multi-priority problem and present a rounding-up approach that can be used for a variety of graph sparsifications. Our analysis provides a systematic way to compute approximate solutions to multi-priority variants of a wide range of graph sparsification problems given access to a single-priority subroutine.

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arXiv2022

An FPT Algorithm for Bipartite Vertex Splitting

Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on one of the two parallel horizontal lines and the relationships are represented by edges (simple curves) between the two lines connecting the corresponding vertices. One of the common objectives in such drawings is to minimize the number of crossings this, however, is computationally expensive and may still result in drawings with so many crossings that they affect the readability of the drawing. We consider a recent approach to remove crossings in such visualizations by splitting vertices, where the goal is to find the minimum number of vertices to be split to obtain a planar drawing. We show that determining whether a planar drawing exists after splitting at most $k$ vertices is fixed parameter tractable in $k$.

cs.CG↗

arXiv2022

Visualizing Evolving Trees

Evolving trees arise in many real-life scenarios from computer file systems and dynamic call graphs, to fake news propagation and disease spread. Most layout algorithms for static trees do not work well in an evolving setting (e.g., they are not designed to be stable between time steps). Dynamic graph layout algorithms are better suited to this task, although they often introduce unnecessary edge crossings. With this in mind we propose two methods for visualizing evolving trees that guarantee no edge crossings, while optimizing (1) desired edge length realization, (2) layout compactness, and (3) stability. We evaluate the two new methods, along with five prior approaches (three static and two dynamic), on real-world datasets using quantitative metrics: stress, desired edge length realization, layout compactness, stability, and running time. The new methods are fully functional and available on github.

cs.CG↗

arXiv2021

Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, $(SGD)^2$

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, $(SGD)^2$, that can handle multiple readability criteria. $(SGD)^2$ can optimize any criterion that can be described by a differentiable function. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., node resolution, angular resolution, aspect ratio). The approach is scalable and can handle large graphs. A variation of the underlying approach can also be used to optimize many desirable properties in planar graphs, while maintaining planarity. Finally, we provide quantitative and qualitative evidence of the effectiveness of $(SGD)^2$: we analyze the interactions between criteria, measure the quality of layouts generated from $(SGD)^2$ as well as the runtime behavior, and analyze the impact of sample sizes. The source code is available on github and we also provide an interactive demo for small graphs.

cs.CG↗

arXiv2021

Computing Steiner Trees using Graph Neural Networks

Graph neural networks have been successful in many learning problems and real-world applications. A recent line of research explores the power of graph neural networks to solve combinatorial and graph algorithmic problems such as subgraph isomorphism, detecting cliques, and the traveling salesman problem. However, many NP-complete problems are as of yet unexplored using this method. In this paper, we tackle the Steiner Tree Problem. We employ four learning frameworks to compute low cost Steiner trees: feed-forward neural networks, graph neural networks, graph convolutional networks, and a graph attention model. We use these frameworks in two fundamentally different ways: 1) to train the models to learn the actual Steiner tree nodes, 2) to train the model to learn good Steiner point candidates to be connected to the constructed tree using a shortest path in a greedy fashion. We illustrate the robustness of our heuristics on several random graph generation models as well as the SteinLib data library. Our finding suggests that the out-of-the-box application of GNN methods does worse than the classic 2-approximation method. However, when combined with a greedy shortest path construction, it even does slightly better than the 2-approximation algorithm. This result sheds light on the fundamental capabilities and limitations of graph learning techniques on classical NP-complete problems.

cs.LG↗

arXiv2021

Weighted Additive Spanners

A \emph{spanner} of a graph $G$ is a subgraph $H$ that approximately preserves shortest path distances in $G$. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured \emph{multiplicatively}. In this work, we investigate whether one can similarly extend constructions of spanners with purely \emph{additive} error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic $+2$ and $+4$ unweighted spanners (both all-pairs and pairwise) to $+2W$ and $+4W$ weighted spanners, where $W$ is the maximum edge weight. Specifically, we show that a weighted graph $G$ contains all-pairs (pairwise) $+2W$ and $+4W$ weighted spanners of size $O(n^{3/2})$ and $\widetilde{O}(n^{7/5})$ ($O(np^{1/3})$ and $O(np^{2/7})$) respectively. For a technical reason, the $+6$ unweighted spanner becomes a $+8W$ weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that $G$ contains all-pairs (pairwise) $+8W$ weighted spanners of size $O(n^{4/3})$ ($O(np^{1/4})$).

cs.DM↗

arXiv2021

On additive spanners in weighted graphs with local error

An \emph{additive $+β$ spanner} of a graph $G$ is a subgraph which preserves distances up to an additive $+β$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al.\ 2019 and 2020, Ahmed et al.\ 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al.\ 2020] provided constructions of sparse spanners with \emph{global} error $β= cW$, where $W$ is the maximum edge weight in $G$ and $c$ is constant. We improve these to \emph{local} error by giving spanners with additive error $+cW(s,t)$ for each vertex pair $(s,t)$, where $W(s, t)$ is the maximum edge weight along the shortest $s$--$t$ path in $G$. These include pairwise $+(2+\eps)W(\cdot,\cdot)$ and $+(6+\eps) W(\cdot, \cdot)$ spanners over vertex pairs $\Pc \subseteq V \times V$ on $O_{\eps}(n|\Pc|^{1/3})$ and $O_{\eps}(n|\Pc|^{1/4})$ edges for all $\eps > 0$, which extend previously known unweighted results up to $\eps$ dependence, as well as an all-pairs $+4W(\cdot,\cdot)$ spanner on $\widetilde{O}(n^{7/5})$ edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its \emph{lightness}, defined as the total edge weight of the spanner divided by the weight of an MST of $G$. We provide a $+\eps W(\cdot,\cdot)$ spanner with $O_{\eps}(n)$ lightness, and a $+(4+\eps) W(\cdot,\cdot)$ spanner with $O_{\eps}(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

cs.DM↗

arXiv2021

Multi-level Weighted Additive Spanners

Given a graph $G = (V,E)$, a subgraph $H$ is an \emph{additive $+β$ spanner} if $\dist_H(u,v) \le \dist_G(u,v) + β$ for all $u, v \in V$. A \emph{pairwise spanner} is a spanner for which the above inequality only must hold for specific pairs $P \subseteq V \times V$ given on input, and when the pairs have the structure $P = S \times S$ for some subset $S \subseteq V$, it is specifically called a \emph{subsetwise spanner}. Spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise spanner in weighted graphs, where the vertices in $S$ possess varying level, priority, or quality of service (QoS) requirements, and the goal is to compute a nested sequence of spanners with the minimum number of total edges. We first generalize the $+2$ subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several other algorithms for weighted additive spanners, both in terms of runtime and sparsity of output spanner, when applied at each level of the multi-level problem. Spanner sparsity is compared to the sparsest possible spanner satisfying the given error budget, obtained using an integer programming formulation of the problem. We run our experiments with respect to input graphs generated by several different random graph generators: Erdős--Rényi, Watts--Strogatz, Barabási--Albert, and random geometric models. By analyzing our experimental results we developed a new technique of changing an initialization parameter value that provides better performance in practice.

cs.DM↗

arXiv2020

Graph Drawing via Gradient Descent, $(GD)^2$

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple readability criteria. $(GD)^2$ can optimize any criterion that can be described by a smooth function. If the criterion cannot be captured by a smooth function, a non-smooth function for the criterion is combined with another smooth function, or auto-differentiation tools are used for the optimization. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., vertex resolution, angular resolution, aspect ratio). We provide quantitative and qualitative evidence of the effectiveness of $(GD)^2$ with experimental data and a functional prototype: \url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}.

cs.DS↗