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Aidin Niaparast

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Data Structures and Algorithms Discrete Mathematics Machine Learning math.CO Social and Information Networks

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preprint2026arXiv

Incremental Strongly Connected Components with Predictions

Algorithms with predictions is a growing area that aims to leverage machine-learned predictions to design faster beyond-worst-case algorithms. In this paper, we use this framework to design a learned data structure for the incremental strongly connected components (SCC) problem. In this problem, the $n$ vertices of a graph are known a priori and the $m$ directed edges arrive over time. The goal is to efficiently maintain the strongly connected components of the graph after each insert. Our algorithm receives a possibly erroneous prediction of the edge sequence and uses it to precompute partial solutions to support fast inserts. We show that our algorithm achieves nearly optimal bounds with good predictions and its performance smoothly degrades with the prediction error. We also implement our data structure and perform experiments on real datasets. Our empirical results show that the theory is predictive of practical runtime improvements.

preprint2022arXiv

Timeliness Through Telephones: Approximating Information Freshness in Vector Clock Models

We consider an information dissemination problem where the root of an undirected graph constantly updates its information. The goal is to keep every other node in the graph about the root as freshly informed as possible. Our synchronous information spreading model uses telephone calls at each time step, in which any node can call at most one neighbor, thus forming a matching over which information is transmitted at each step. We introduce two problems in minimizing two natural objectives (Maximum and Average) of the latency of the root's information at all nodes in the network. After deriving a simple reduction from the maximum rooted latency problem to the well-studied minimum broadcast time problem, we focus on the average rooted latency version. We introduce a natural problem of finding a finite schedule that minimizes the average broadcast time from a root. We show that any average rooted latency induces a solution to this average broadcast problem within a constant factor and conversely, this average broadcast time is within a logarithmic factor of the average rooted latency. Then, by approximating the average broadcast time problem via rounding a time-indexed linear programming relaxation, we obtain a logarithmic approximation to the average latency problem. Surprisingly, we show that using the average broadcast time for average rooted latency introduces this necessary logarithmic factor overhead even in trees. We overcome this hurdle and give a 40-approximation for trees. For this, we design an algorithm to find near-optimal locally-periodic schedules in trees where each vertex receives information from its parent in regular intervals. On the other side, we show how such well-behaved schedules approximate the optimal schedule within a constant factor.

preprint2021arXiv

On a question of Haemers regarding vectors in the nullspace of Seidel matrices

In 2011, Haemers asked the following question: If $S$ is the Seidel matrix of a graph of order $n$ and $S$ is singular, does there exist an eigenvector of $S$ corresponding to $0$ which has only $\pm 1$ elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number $N$, there exists a graph whose Seidel matrix $S$ is singular such that for any integer vector in the nullspace of $S$, the absolute value of any entry in this vector is more than $N$. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question.