With some combinatorial constructions one can guarantee that at least more than. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Our main focus is on sublineartime algorithms for combinatorial problems, especially for graph problems and optimization problems in metric spaces. Combinatorial sublineartime fourier algorithms article pdf available in foundations of computational mathematics 103. Fast reliable algorithms for matrices with structure advances in design and control. Specifically, it is used in gps synchronization, spectrum sensing and analogtodigital converters the fast fourier transform fft plays an indispensable role on many scientific domains, especially on signal processing.
Applied and computational harmonic analysis 34 1, 5782, 20. Bibliography open problems in sublinear algorithms. Let us denote the optimal mterm fourier representation of a signal a of length n by ropt. As a reference for runtime comparisons, we also plot the time taken by fftw3 on the same machine.
In 11, pawar and ramchandran present an algorithm based on aliasing and the peeling decoder for computing the fourier transform of a signal with noisy observations for the case when the fourier transform is. In this paper we develop the first known deterministic sublineartime sparse fourier transform algorithm which is guaranteed to produce. However, for most natural problems the algorithm must use randomization. In this model, we assume we have access to samples from a discrete distribution p on an nelement set. Sublinear time algorithms as stated in lecture, the idea behind these algorithms is to relax our notions of correctness and examine a small, random sample of the input in order to run quickly on massive data sets. Sublinear time algorithms for the sparse recovery problem. The fft reduces the runtime complexity of calculating the discrete fourier transform dft of a length n array from the naive o n 2 to o n log.
Sparse fft for functions with short frequency support dolomites. Sublineartime algorithms computer science department. Foundations of computational mathematics 10, 303338. Its runtime is sublinear in the input length and scales quadratically in the sparsity k. Siam journal on computing society for industrial and. An implementation of the sparse fft algorithms discovered by mark iwen and described in gopher fast fourier transform gfft browse files at joinlogin. However, with the advent of big data era, the fft still. The resulting advances show that for sparse data i. The fourier algorithms developed in 32 were obtained by utilizing modi. Some applications of coding theory in computational complexity luca trevisan. Improved time bounds for nearoptimal sparse fourier. For sublinear space algorithms, we assume that the turing machine has a readonly tape, a sublinear working space, and a writeonly output tape. Seth gilbert august 1631, 2018 abstract today, we will focus on sublinear time graph algorithms. At the time of its introduction in the mid1960s, it dramatically increased the.
A deterministic sparse fft algorithm for vectors with. The goal of this wiki is to collate a set of open problems in sublinear algorithms and to track progress that is made on these problems. In paola flocchini and leszek gasieniec, editors, structural information and communication complexity, th international colloquium, sirocco 2006, chester, uk, july 25, 2006, proceedings, volume 4056 of lecture notes in computer science, pages 280294. Stochastic collocation on unstructured multivariate meshes.
For several years their fourier algorithm 29 remained the only sublineartime fourier algorithm 3one can show that these methods can recover fourier approximations, y 2c n, for f. Selected publications computational mathematics science. These combinatorial constructions where then combined with improved variants of determinis compressed sensing techniques due to cormode et al. Our second result improves the algorithm further by utilizing a. The sparse fourier transform sft is a kind of discrete fourier transform dft for handling big data signals. A multiscale sublinear time fourier algorithm for noisy.
Our main focus is on sublineartime algorithms for combinatorial problems. Combinatorial sublineartime fourier algorithms springerlink. Indeed, in this case, since the list is close to monotone, it may not be too harmful to pass it. Some applications of coding theory in computational. Improved approximation guarantees for sublineartime fourier algorithms.
Combinatorial algorithms for compressed sensing, technical report dimacs tr. In this paper we build on the deterministic compressed sensing results of cormode and muthukrishnan cm 26, 6, 7 in order to develop the first known deterministic sublinear time sparse fourier. Sparse fourier transform in any constant dimension with nearlyoptimal sample complexity in sublinear time. The goal is to compute some property of a graph while only looking at a very small fraction of the nodes in the graph. Important topics within sublinear algorithms include data stream algorithms sublinear space, property testing sublinear time, and communication complexity sublinear communication but this list isnt. Combinatorial compressive sampling with applications by mark a. Randomized sublineartime algorithms which have a small. Artur czumaj christian sohler abstract in this paper we survey recent advances in the area of sublineartime algorithms.
Combinatorial sublineartime fourier algorithms, foundations of computational mathematics, vol. Nearly optimal sparse fourier transform haitham hassanieh mit piotr indyk. Randomized sublineartime monte carlo algorithms, which have a small. Sampleoptimal averagecase sparse fourier transform in. Gopher fast fourier transform gfft browse files at. We can represent a as a linear combination of basis. In this paper we develop the first known deterministic sublinear time sparse fourier transform algorithm which is guaranteed to produce accurate results. Combinatorial and graph algorithms lecture sublinear time graph algorithms lecturer. Combinatorial sublineartime fourier algorithms norbert wiener. In this paper, we survey some of the recent advances in this area. Sublinear time algorithms represent a new paradigm in computing, where an algorithm. Improved approximation guarantees for sublineartime fourier. We also show that truly sublineartime randomized algorithms are possible for the approximate decision problem.
In this paper we develop the first known deterministic sublinear time sparse fourier transform algorithm. The meeting is devoted to algorithms that are extremely efficient, in that the amount of resources they use is sublinear in the input size. Improved approximation guarantees for sublineartime. Adaptive sublinear time fourier algorithms advances. In 4, 5, a deterministic combinatorial fourier algorithm for estimating the best kterm fourier representation for a given frequency sparse signal, relying heavily on the chinese remainder theorem and combinatorial concepts, was introduced. Foundations of computational mathematics 10 3, 303338, 2010. We present a new deterministic algorithm for the sparse fourier transform problem, in which we seek to identify k. However, they employ random sampling techniques and thus fail to output good approximate answers with nonzero probability. Sparse fourier transform in any constant dimension with. Pdf signal approximation via the gopher fast fourier transform. May 20, 2004 abstract errorcorrecting codes and related combinatorial constructs play an important role in several recent and old results in computational complexity theory. Methods the fourier algorithms developed in 32 were obtained by utilizing modified combinatorial constructions related to oup testing matrices 24.
We study the problem of estimating the best k term fourier representation for a given frequency sparse signal i. Whats the frequency, kenneth sublinear fourier sampling. For the rounding algorithm, we see that there is no dependence on k until k 64. Sublinear time algorithms 1097 the property testing algorithm is allowed to output pass or fail. These combinatorial constructions where then combined with improved variants of deterministic compressed sensing techniques due to cormode et al. Read improved sparse fourier approximation results. In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm. Simple and practical algorithm for sparse fourier transform mit. Sublinear time is a daunting goal since it allows one to read only a miniscule fraction of the input. Citeseerx combinatorial sublineartime fourier algorithms. Shift finding in sublinear time columbia university. The fast fourier transform fft is a fundamental numerical algorithm whose importance in a wide variety of applications cannot be overstated.
N significant fourier coefficients from a signal of bandwidth n. Previous deterministic algorithms exhibit quadratic runtime scaling, while our algorithm scales linearly with k. Combinatorial compressive sampling with applications. Pdf improved bounds for a deterministic sublineartime. This paper improves on the bestknown runtime and measurement bounds for a recently proposed deterministic sublineartime sparse fourier transform algorithm hereafter called dsft. Sublinear time algorithms school of mathematical sciences. The binning has to be done in sublinear time, and thus these algorithms bin the fourier coef.
Martin strauss in the sparse recovery problem, we have a signal x 2rn that is sparse. But more recently, we have seen some major progress in sublineartime algorithms in the classical model of randomized and approximation algorithms. More details about such techniques can be found in a recent survey 10. Substringpattern matching in sublinear time using a. There are problems for which deterministic exact sublinear time algorithms are known. Nearly optimal deterministic algorithm for sparse walsh. Randomized sublinear time algorithms which have a small controllable probability of failure for each processed signal exist for solving this problem 24, 25.
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