Taha

Taha

Hi, I am Taha.

Blog posts

2026

And god said let there be dijkstra

18 minute read

Published:

Alright, in this work I want to explain the seminal work of Jittat Fakcharoenphol and Satish Rao in 2006, devising the algorithm of FR-Dijkstra, which is so insightful and meaningful.

Dynamic Approximate Nearest-Neighbor Search in Doubling Metrics

18 minute read

Published:

Well, in this blog post we are going to review several papers and understand the techniques that are utilized in order to tackle several problems related to areas such as nearest neighbor problems, the facility location problem, and so on.

Lewis Weights, Leverage Scores, and Whitening — Chapter 2

12 minute read

Published:

This is the second chapter in my series on Sparsifying Sums of Norms. Chapter 1 set up the general problem and briefly mentioned Lewis weights. In the second post of this series on sparsification of the norm, I want to turn to the $\ell_p$ row sampling paper. I’ll be spending a few posts on this one, because I think it is genuinely revealing on the subject it is worth pondering carefully and going through in some detail. The paper relies heavily on the works of Milman and Talagrand(for example), so I may cover those seminal papers too at some point. At a high level, this paper is a vast generalization of the Johnson–Lindenstrauss lemma, with a few extremely technical steps, but the overall theme is clear and elegant. I especially like how they used Talagrand’s result and intuition, and the way they brought in Lewis weights to pull everything together.

Probabilistic Tree Embeddings and Hierarchical Cut Decompositions

10 minute read

Published:

As part of my effort to understand the broader literature surrounding the areas I hope to work on in the future, I am studying both foundational papers and contemporary developments. Alongside my primary focus on the work of James R. Lee, I plan to read a number of folklore and classical papers that have shaped the modern theory of metric embeddings.

2025

Bukh’s Problem on Coloring Random Subgraphs

5 minute read

Published:

So, assume that we have a graph $G = (V, E)$. By $G_p$ for some $p \in (0,1)$ we mean a subgraph such that the probability of each edge appearing in that subgraph is independent with probability $p$.