arXiv subjects

Chirag Falor

Publications and source records attributed to Chirag Falor.

4 recordsLinked to original sources

arXiv2025

How fast are algorithms reducing the demands on memory? A survey of progress in space complexity

Algorithm research focuses primarily on how many operations processors need to do (time complexity). But for many problems, both the runtime and energy used are dominated by memory accesses. In this paper, we present the first broad survey of how algorithmic progress has improved memory usage (space complexity). We analyze 118 of the most important algorithm problems in computer science, reviewing the 800+ algorithms used to solve them. Our results show that space complexity has become much more important in recent years as worries have arisen about memory access bottle-necking performance (the ``memory wall''). In 20% of cases we find that space complexity improvements for large problems (n=1 billion) outpaced improvements in DRAM access speed, suggesting that for these problems algorithmic progress played a larger role than hardware progress in minimizing memory access delays. Increasingly, we also see the emergence of algorithmic Pareto frontiers, where getting better asymptotic time complexity for a problem requires getting worse asymptotic space complexity, and vice-versa. This tension implies that programmers will increasingly need to consider multiple algorithmic options to understand which is best for their particular problem. To help theorists and practitioners alike consider these trade-offs, we have created a reference for them at https://algorithm-wiki.csail.mit.edu.

cs.DS↗

arXiv2025

A convergent sum-of-squares hierarchy for compiled nonlocal games

We continue the line of work initiated by Kalai et al. (STOC '23), studying "compiled" nonlocal games played between a classical verifier and a single quantum prover, with cryptography simulating the spatial separation between the players. The central open question in this area is to understand the soundness of this compiler against quantum strategies, and apart from results for specific games, all that is known is the recent "qualitative" result of Kulpe et al. (STOC '25) showing that the success probability of a quantum prover in the compiled game is bounded by the game's quantum commuting-operator value in the limit as the cryptographic security parameter goes to infinity. In this work, we make progress towards a quantitative understanding of quantum soundness for general games, by giving a concrete framework to bound the quantum value of compiled nonlocal games. Building on the result of Kulpe et al. together with the notion of "nice" sum-of-squares certificates, introduced by Natarajan and Zhang (FOCS '23) to bound the value of the compiled CHSH game, we extend the niceness framework and construct a hierarchy of semidefinite programs that searches exclusively over nice certificates. We show that this hierarchy converges to the optimal quantum value of the game. Additionally, we present a transformation to make any degree-1 sum-of-squares certificate nice. This approach provides a systematic method to reproduce all known bounds for special classes of games together with Kulpe et al.'s bound for general games from the same framework.

quant-ph↗

arXiv2023

A Collapsible Polynomial Hierarchy for Promise Problems

The polynomial hierarchy has been widely studied in classical complexity theory. In this paper, we will generalize some commonly known results about the polynomial hierarchy to a version of the hierarchy extended to promise problems. This paper proposes new definitions of existential and universal operators for classes of promise problems. Applying these to BQP, we recover the hierarchy proposed by Gharibian et al. (MFCS 2018). Moreover, using our definition, we give an easy proof of the collapse of this hierarchy under a Karp-Lipton-like scenario, which was an open question for the original definition of Gharibian et al.

cs.CC↗

arXiv2022

The Quadruple Image Configurations of Asymptotically Circular Gravitational Lenses

The quadruple image configurations of gravitational lenses with vanishing ellipticity are examined. Even though such lenses asymptotically approach circularity, the configurations are stable if the position of the source relative to the vanishing diamond caustic is held constant. The configurations are the solutions of a quartic equation, an "Asymptotically Circular Lens Equation" (ACLE), parameterized by a single complex quantity. Several alternative parameterizations are examined. Relative magnifications of the images are derived. When a non-vanishing quadrupole, in the form of an external shear (XS), is added to the singular isothermal sphere (SIS), its configurations emerge naturally as stretched and squeezed versions of the circular configurations. And as the SIS+XS model is a good first approximation for most quadruply lensed quasars, their configurations likewise have only 2+1 salient dimensions. The asymptotically circular configurations can easily be adapted to the problem of Solar System "occultation flashes."

gr-qc↗