arXiv subjects

Rezaul Chowdhury

Publications and source records attributed to Rezaul Chowdhury.

9 recordsLinked to original sources

arXiv2026

How many users have been here for a long time? Efficient solutions for counting long aggregated visits

This paper addresses the Counting Long Aggregated Visits problem, which is defined as follows. We are given $n$ users and $m$ regions, where each user spends some time visiting some regions. For a parameter $k$ and a query consisting of a subset of $r$ regions, the task is to count the number of distinct users whose aggregate time spent visiting the query regions is at least $k$. This problem is motivated by queries arising in the analysis of large-scale mobility datasets. We present several exact and approximate data structures for supporting counting long aggregated visits, as well as conditional and unconditional lower bounds. First, we describe an exact data structure that exhibits a space-time tradeoff, as well as efficient approximate solutions based on sampling and sketching techniques. We then study the problem in geometric settings where regions are points in $\mathbb{R}^d$ and queries are hyperrectangles, and derive exact data structures that achieve improved performance in these structured spaces.

cs.DS↗

arXiv2025

Vantage Point Selection Algorithms for Bottleneck Capacity Estimation

Motivated by the problem of estimating bottleneck capacities on the Internet, we formulate and study the problem of vantage point selection. We are given a graph $G=(V, E)$ whose edges $E$ have unknown capacity values that are to be discovered. Probes from a vantage point, i.e, a vertex $v \in V$, along shortest paths from $v$ to all other vertices, reveal bottleneck edge capacities along each path. Our goal is to select $k$ vantage points from $V$ that reveal the maximum number of bottleneck edge capacities. We consider both a non-adaptive setting where all $k$ vantage points are selected before any bottleneck capacity is revealed, and an adaptive setting where each vantage point selection instantly reveals bottleneck capacities along all shortest paths starting from that point. In the non-adaptive setting, by considering a relaxed model where edge capacities are drawn from a random permutation (which still leaves the problem of maximizing the expected number of revealed edges NP-hard), we are able to give a $1-1/e$ approximate algorithm. In the adaptive setting we work with the least permissive model where edge capacities are arbitrarily fixed but unknown. We compare with the best solution for the particular input instance (i.e. by enumerating all choices of $k$ tuples), and provide both lower bounds on instance optimal approximation algorithms and upper bounds for trees and planar graphs.

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arXiv2023

Fast American Option Pricing using Nonlinear Stencils

We study the binomial, trinomial, and Black-Scholes-Merton models of option pricing. We present fast parallel discrete-time finite-difference algorithms for American call option pricing under the binomial and trinomial models and American put option pricing under the Black-Scholes-Merton model. For $T$-step finite differences, each algorithm runs in $O(\left(T\log^2{T}\right)/p + T)$ time under a greedy scheduler on $p$ processing cores, which is a significant improvement over the $Θ({T^2}/{p}) + Ω(T\log{T})$ time taken by the corresponding state-of-the-art parallel algorithm. Even when run on a single core, the $O(T\log^2{T})$ time taken by our algorithms is asymptotically much smaller than the $Θ(T^2)$ running time of the fastest known serial algorithms. Implementations of our algorithms significantly outperform the fastest implementations of existing algorithms in practice, e.g., when run for $T \approx 1000$ steps on a 48-core machine, our algorithm for the binomial model runs at least $15\times$ faster than the fastest existing parallel program for the same model with the speed-up factor gradually reaching beyond $500\times$ for $T \approx 0.5 \times 10^6$. It saves more than 80\% energy when $T \approx 4000$, and more than 99\% energy for $T > 60,000$. Our option pricing algorithms can be viewed as solving a class of nonlinear 1D stencil (i.e., finite-difference) computation problems efficiently using the Fast Fourier Transform (FFT). To our knowledge, ours are the first algorithms to handle such stencils in $o(T^2)$ time. These contributions are of independent interest as stencil computations have a wide range of applications beyond quantitative finance.

cs.CE↗

arXiv2023

Cache-Oblivious Parallel Convex Hull in the Binary Forking Model

We present two cache-oblivious sorting-based convex hull algorithms in the Binary Forking Model. The first is an algorithm for a presorted set of points which achieves $O(n)$ work, $O(\log n)$ span, and $O(n/B)$ serial cache complexity, where $B$ is the cache line size. These are all optimal worst-case bounds for cache-oblivious algorithms in the Binary Forking Model. The second adapts Cole and Ramachandran's cache-oblivious sorting algorithm, matching its properties including achieving $O(n \log n)$ work, $O(\log n \log \log n)$ span, and $O(n/B \log_M n)$ serial cache complexity. Here $M$ is the size of the private cache.

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arXiv2022

An Optimal Level-synchronous Shared-memory Parallel BFS Algorithm with Optimal parallel Prefix-sum Algorithm and its Implications for Energy Consumption

We present a work-efficient parallel level-synchronous Breadth First Search (BFS) algorithm for shared-memory architectures which achieves the theoretical lower bound on parallel running time. The optimality holds regardless of the shape of the graph. We also demonstrate the implication of this optimality for the energy consumption of the program empirically. The key idea is never to use more processing cores than necessary to complete the work in any computation step efficiently. We keep the rest of the cores idle to save energy and to reduce other resource contentions (e.g., bandwidth, shared caches, etc). Our BFS does not use locks and atomic instructions and is easily extendible to shared-memory coprocessors.

cs.DC↗

arXiv2021

Fast Stencil Computations using Fast Fourier Transforms

Stencil computations are widely used to simulate the change of state of physical systems across a multidimensional grid over multiple timesteps. The state-of-the-art techniques in this area fall into three groups: cache-aware tiled looping algorithms, cache-oblivious divide-and-conquer trapezoidal algorithms, and Krylov subspace methods. In this paper, we present two efficient parallel algorithms for performing linear stencil computations. Current direct solvers in this domain are computationally inefficient, and Krylov methods require manual labor and mathematical training. We solve these problems for linear stencils by using DFT preconditioning on a Krylov method to achieve a direct solver which is both fast and general. Indeed, while all currently available algorithms for solving general linear stencils perform $Θ(NT)$ work, where $N$ is the size of the spatial grid and $T$ is the number of timesteps, our algorithms perform $o(NT)$ work. To the best of our knowledge, we give the first algorithms that use fast Fourier transforms to compute final grid data by evolving the initial data for many timesteps at once. Our algorithms handle both periodic and aperiodic boundary conditions, and achieve polynomially better performance bounds (i.e., computational complexity and parallel runtime) than all other existing solutions. Initial experimental results show that implementations of our algorithms that evolve grids of roughly $10^7$ cells for around $10^5$ timesteps run orders of magnitude faster than state-of-the-art implementations for periodic stencil problems, and 1.3$\times$ to 8.5$\times$ faster for aperiodic stencil problems.

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arXiv2020

Low-Depth Parallel Algorithms for the Binary-Forking Model without Atomics

The binary-forking model is a parallel computation model, formally defined by Blelloch et al. very recently, in which a thread can fork a concurrent child thread, recursively and asynchronously. The model incurs a cost of $Θ(\log n)$ to spawn or synchronize $n$ tasks or threads. The binary-forking model realistically captures the performance of parallel algorithms implemented using modern multithreaded programming languages on multicore shared-memory machines. In contrast, the widely studied theoretical PRAM model does not consider the cost of spawning and synchronizing threads, and as a result, algorithms achieving optimal performance bounds in the PRAM model may not be optimal in the binary-forking model. Often, algorithms need to be redesigned to achieve optimal performance bounds in the binary-forking model and the non-constant synchronization cost makes the task challenging. Though the binary-forking model allows the use of atomic {\em test-and-set} (TS) instructions to reduce some synchronization overhead, assuming the availability of such instructions puts a stronger requirement on the hardware and may limit the portability of the algorithms using them. In this paper, we avoid the use of locks and atomic instructions in our algorithms except possibly inside the join operation which is implemented by the runtime system. In this paper, we design efficient parallel algorithms in the binary-forking model without atomics for three fundamental problems: Strassen's (and Strassen-like) matrix multiplication (MM), comparison-based sorting, and the Fast Fourier Transform (FFT). All our results improve over known results for the corresponding problem in the binary-forking model both with and without atomics.

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arXiv2020

A Computational Model for Tensor Core Units

To respond to the need of efficient training and inference of deep neural networks, a plethora of domain-specific hardware architectures have been introduced, such as Google Tensor Processing Units and NVIDIA Tensor Cores. A common feature of these architectures is a hardware circuit for efficiently computing a dense matrix multiplication of a given small size. In order to broaden the class of algorithms that exploit these systems, we propose a computational model, named the TCU model, that captures the ability to natively multiply small matrices. We then use the TCU model for designing fast algorithms for several problems, including matrix operations (dense and sparse multiplication, Gaussian Elimination), graph algorithms (transitive closure, all pairs shortest distances), Discrete Fourier Transform, stencil computations, integer multiplication, and polynomial evaluation. We finally highlight a relation between the TCU model and the external memory model.

cs.DS↗

arXiv2019

Data Races and the Discrete Resource-time Tradeoff Problem with Resource Reuse over Paths

A determinacy race occurs if two or more logically parallel instructions access the same memory location and at least one of them tries to modify its content. Races often lead to nondeterministic and incorrect program behavior. A data race is a special case of a determinacy race which can be eliminated by associating a mutual-exclusion lock or allowing atomic accesses to the memory location. However, such solutions can reduce parallelism by serializing all accesses to that location. For associative and commutative updates, reducers allow parallel race-free updates at the expense of using some extra space. We ask the following question. Given a fixed budget of extra space to mitigate the cost of races in a parallel program, which memory locations should be assigned reducers and how should the space be distributed among the reducers in order to minimize the overall running time? We argue that the races can be captured by a directed acyclic graph (DAG), with nodes representing memory cells and arcs representing read-write dependencies between cells. We then formulate our optimization problem on DAGs. We concentrate on a variation of this problem where space reuse among reducers is allowed by routing extra space along a source to sink path of the DAG and using it in the construction of reducers along the path. We consider two reducers and the corresponding duration functions (i.e., reduction time as a function of space budget). We generalize our race-avoiding space-time tradeoff problem to a discrete resource-time tradeoff problem with general non-increasing duration functions and resource reuse over paths. For general DAGs, the offline problem is strongly NP-hard under all three duration functions, and we give approximation algorithms. We also prove hardness of approximation for the general resource-time tradeoff problem and give a pseudo-polynomial time algorithm for series-parallel DAGs.

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