The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Chiouyng Lee - One of the best experts on this subject based on the ideXlab platform.
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embracing systolic super systolization of large scale circulant matrix vector multiplication on fpga with subquadratic Space Complexity
Field Programmable Gate Arrays, 2019Co-Authors: Jiafeng Xie, Chiouyng LeeAbstract:The recent advance in artificial intelligence (AI) technology has led to a new round of systolic structure innovation. Many AI accelerators have employed systolic structure to realize the core large-scale matrix-vector multiplication for high-performance processing, which has a Complexity of $o(n^2)$ for matrix size of $n\times n$ (difficult to be implemented on the field-programmable gate array (FPGA) platform). To overcome this drawback, in this paper, we propose a super systolization strategy to implement the core circulant matrix-vector multiplication into a systolic structure with subquadratic Space Complexity. The proposed effort is carried out through two stages of coherent interdependent efforts: (i) a novel matrix-vector multiplication algorithm based on Toeplitz matrix-vector product (TMVP) approach is proposed to obtain subquadratic Space Complexity; (ii) a series of optimization techniques are introduced to map the proposed algorithm into desired systolic structure. Finally, detailed Complexity analysis and comparison have been conducted to prove the efficiency of the proposed strategy. The proposed strategy is highly efficient and can be extended in many neural network based hardware implementation platforms.
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gaussian normal basis multiplier over gf 2 m using hybrid subquadratic and quadratic tmvp approach for elliptic curve cryptography
Iet Circuits Devices & Systems, 2017Co-Authors: Che Wun Chiou, Jimmin Lin, Yuhsien Sun, Chengmin Lee, Taipao Chuang, Chiouyng LeeAbstract:In recent years, subquadratric-and-quadratric Toeplitz matrix–vector product (TMVP) computations are widely used for the implementation of binary field multiplication in elliptic curve cryptography. Pure subquadratric TMVP structure involves significantly less Space Complexity and long computational delay, while quadratric TMVP structure involves larger Space Complexity and less computation delay. To optimise the tradeoff between time and Space complexities, this study presents a novel hybrid multiplier for Gaussian normal basis (GNB) in GF(2 m ) which combines subquadratic and quadratic structures. From the theoretical analysis, it is shown that the proposed hybrid multiplier can save ∼18% Space Complexity and 12% time Complexity than the existing GNB multiplier with pure TMVP decomposition.
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area efficient subquadratic Space Complexity digit serial multiplier for type ii optimal normal basis of gf 2 m using symmetric tmvp and block recombination techniques
IEEE Transactions on Circuits and Systems I-regular Papers, 2015Co-Authors: Chiouyng Lee, P K MeherAbstract:The type-II optimal normal basis (ONB) is popularly used to represent $GF(2^{m})$ for elliptic curve cryptosystems. It is shown in the literature that multiplication in binary fields, including those represented by type-II ONB, shifted polynomial basis, and dual basis, can be transformed into non-symmetric Toeplitz matrix-vector product (TMVP) formulation. In this paper, we show that type-II ONB multiplication can be realized by two symmetric TMVPs (STMVP). Moreover, we have proposed a novel folded TMVP block recombination (TMVPBR) for the computation of STMVP. Based on the proposed folded TMVPBR approach, we have proposed a new digit-serial structure for type-II ONB multiplication, while traditional parallel ONB multipliers are based on non-symmetric TMVPBR approach to achieve subquadratic Space Complexity architecture. The proposed digit-serial structure also involves subquadratic Space Complexity. By the theoretical analysis as well as from synthesis result, however, we find that the proposed architecture has significantly less area and less area-delay product compared to the existing digit-serial type-II ONB multipliers.
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subquadratic Space Complexity digit serial multipliers over gf 2 m using generalized a b way karatsuba algorithm
IEEE Transactions on Circuits and Systems I-regular Papers, 2015Co-Authors: Chiouyng Lee, P K MeherAbstract:Karatsuba algorithm (KA) is popularly used for high-precision multiplication by divide-and-conquer approach. Recently, subquadratic digit-serial multiplier based on $(a,2)$ -way KA decomposition is proposed in [1]. In this paper, we extend a $(a,2)$ -way KA to derive a generalized $(a,b)$ -way KA decomposition with $a\neq b$ . We have shown that $(a,2)$ -way KA and mult-way KA are special cases of the proposed $(a,b)$ -way KA decomposition. Based on the proposed KA decomposition, we have established two types of subquadratic digit-serial multipliers, namely, the KA-based multiplier and the recombined KA-based multiplier. From theoretical analysis, as well as, from synthesis results we have shown that the proposed KA-based multipliers have significantly less delay and less area-delay product (ADP) compared to the existing naive digit-serial multipliers.
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subquadratic Space Complexity digit serial multiplier over binary extension fields using toom cook algorithm
International Symposium on Integrated Circuits, 2014Co-Authors: Chiouyng Lee, P K Meher, Wenyo LeeAbstract:In this paper, we present a new (4,2)-way Toom-Cook algorithm using finite field interpolation. The proposed algorithm uses multi-evaluation scheme to construct a digit-serial multiplier over GF(2m) which involves subquadratic Space-Complexity. From theoretical analysis, it is found that the proposed architecture has O(mlog4 5) Space Complexity and O(mlog4 2) latency, which is significantly less than traditional digit-serial multipliers.
David P Woodruff - One of the best experts on this subject based on the ideXlab platform.
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adaptive matrix vector product
Symposium on Discrete Algorithms, 2017Co-Authors: Santosh Vempala, David P WoodruffAbstract:We consider the following streaming problem: given a hardwired m × n matrix A together with a poly(mn)-bit hardwired string of advice that may depend on A, for any x with coordinates x1, . . . xn presented in order, output the coordinates of A · x in order. Our focus is on using as little memory as possible while computing A · x; we do not count the size of the output tape on which the coordinates of A · x are written; for some matrices A such as the identity matrix, a constant number of words of Space is achievable. Such an algorithm has to adapt its memory contents to the changing structure of A and exploit it on the fly. We give a nearly tight characterization, for any number of passes over the coordinates of x, of the Space Complexity of such a streaming algorithm. Our characterization is constructive, in that we provide an efficient algorithm matching our lower bound on the Space Complexity. The essential parameters, streaming rank and multi-pass streaming rank of A, might be of independent interest, and we show they can be computed efficiently. We give several applications of our results to computing Johnson-Lindenstrauss transforms. Finally, we note that we can characterize the optimal Space Complexity when the coordinates of A · x can be written in any order.
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Streaming Space Complexity of Nearly All Functions of One Variable on Frequency Vectors
arXiv: Data Structures and Algorithms, 2016Co-Authors: Vladimir Braverman, David P Woodruff, Stephen R. Chestnut, Lin F. YangAbstract:A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or poly-logarithmic, Space. Given a function $g$, we study the Space Complexity of approximating $\sum_{i=1}^n g(|f_i|)$, where $f\in\mathbb{Z}^n$ is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when $g$ is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class of functions $g$, there is a Space-efficient approximation algorithm for the sum if and only if $g$ satisfies the condition. The functions $g$ that we are able to characterize include all convex, concave, monotonic, polynomial, and trigonometric functions, among many others, and is the first such characterization for non-monotonic functions. Thus, for nearly all functions of one variable, we answer the open question from the celebrated paper of Alon, Matias and Szegedy (1996).
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on the exact Space Complexity of sketching and streaming small norms
Symposium on Discrete Algorithms, 2010Co-Authors: Daniel M Kane, Jelani Nelson, David P WoodruffAbstract:We settle the 1-pass Space Complexity of (1 ± e)-approximating the Lp norm, for real p with 1 ≤ p ≤ 2, of a length-n vector updated in a length-m stream with updates to its coordinates. We assume the updates are integers in the range [--M, M]. In particular, we show the Space required is Θ(e-2 log(mM) + log log(ns)) bits. Our result also holds for 0
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the data stream Space Complexity of cascaded norms
Foundations of Computer Science, 2009Co-Authors: T S Jayram, David P WoodruffAbstract:We consider the problem of estimating cascaded aggregates over a matrix presented as a sequence of updates in a data stream. A cascaded aggregate P ±Q is defined by evaluating aggregate Q repeatedly over each row of the matrix, and then evaluating aggregate P over the resulting vector of values. This problem was introduced by Cormode andMuthukrishnan, PODS, 2005 [CM]. We analyze the Space Complexity of estimating cascaded norms on an n £d matrix to within a small relative error. Let Lp denote the p-th norm, where p is a non-negative integer. We abbreviate the cascaded normLk ±Lp by Lk,p . (1) For any constant k ¸ p ¸ 2, we obtain a 1-pass e O(n1i2/kd1i2/p )-Space algorithm for estimating Lk,p . This is optimal up to polylogarithmic factors and resolves an open question of [CM] regarding the Space Complexity of L4,2. We also obtain 1-pass Space-optimal algorithms for estimating L1,k and Lk,1. (2)We prove a Space lower bound of(n1i1/k ) on estimating Lk,0 and Lk,1, resolving an open question due to Indyk, IITK Data StreamsWorkshop (Problem 8), 2006. We also resolve two more questions of [CM] concerning Lk,2 estimation and block heavy hitter problems. Ganguly, Bansal and Dube (FAW, 2008) claimed an e O(1)-Space algorithm for estimating Lk,p for any k,p 2 [0,2]. Our lower bounds show this claimis incorrect.
P K Meher - One of the best experts on this subject based on the ideXlab platform.
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area efficient subquadratic Space Complexity digit serial multiplier for type ii optimal normal basis of gf 2 m using symmetric tmvp and block recombination techniques
IEEE Transactions on Circuits and Systems I-regular Papers, 2015Co-Authors: Chiouyng Lee, P K MeherAbstract:The type-II optimal normal basis (ONB) is popularly used to represent $GF(2^{m})$ for elliptic curve cryptosystems. It is shown in the literature that multiplication in binary fields, including those represented by type-II ONB, shifted polynomial basis, and dual basis, can be transformed into non-symmetric Toeplitz matrix-vector product (TMVP) formulation. In this paper, we show that type-II ONB multiplication can be realized by two symmetric TMVPs (STMVP). Moreover, we have proposed a novel folded TMVP block recombination (TMVPBR) for the computation of STMVP. Based on the proposed folded TMVPBR approach, we have proposed a new digit-serial structure for type-II ONB multiplication, while traditional parallel ONB multipliers are based on non-symmetric TMVPBR approach to achieve subquadratic Space Complexity architecture. The proposed digit-serial structure also involves subquadratic Space Complexity. By the theoretical analysis as well as from synthesis result, however, we find that the proposed architecture has significantly less area and less area-delay product compared to the existing digit-serial type-II ONB multipliers.
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subquadratic Space Complexity digit serial multipliers over gf 2 m using generalized a b way karatsuba algorithm
IEEE Transactions on Circuits and Systems I-regular Papers, 2015Co-Authors: Chiouyng Lee, P K MeherAbstract:Karatsuba algorithm (KA) is popularly used for high-precision multiplication by divide-and-conquer approach. Recently, subquadratic digit-serial multiplier based on $(a,2)$ -way KA decomposition is proposed in [1]. In this paper, we extend a $(a,2)$ -way KA to derive a generalized $(a,b)$ -way KA decomposition with $a\neq b$ . We have shown that $(a,2)$ -way KA and mult-way KA are special cases of the proposed $(a,b)$ -way KA decomposition. Based on the proposed KA decomposition, we have established two types of subquadratic digit-serial multipliers, namely, the KA-based multiplier and the recombined KA-based multiplier. From theoretical analysis, as well as, from synthesis results we have shown that the proposed KA-based multipliers have significantly less delay and less area-delay product (ADP) compared to the existing naive digit-serial multipliers.
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subquadratic Space Complexity digit serial multiplier over binary extension fields using toom cook algorithm
International Symposium on Integrated Circuits, 2014Co-Authors: Chiouyng Lee, P K Meher, Wenyo LeeAbstract:In this paper, we present a new (4,2)-way Toom-Cook algorithm using finite field interpolation. The proposed algorithm uses multi-evaluation scheme to construct a digit-serial multiplier over GF(2m) which involves subquadratic Space-Complexity. From theoretical analysis, it is found that the proposed architecture has O(mlog4 5) Space Complexity and O(mlog4 2) latency, which is significantly less than traditional digit-serial multipliers.
Stephane Rovedakis - One of the best experts on this subject based on the ideXlab platform.
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fast self stabilizing minimum spanning tree construction using compact nearest common ancestor labeling scheme
arXiv: Distributed Parallel and Cluster Computing, 2013Co-Authors: Lelia Blin, Shlomi Dolev, Maria Potopbutucaru, Stephane RovedakisAbstract:We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The Space Complexity of our solution is $O(\log^2n)$ bits and it converges in $O(n^2)$ rounds. Thus, this algorithm improves the convergence time of previously known self-stabilizing asynchronous MST algorithms by a multiplicative factor $\Theta(n)$, to the price of increasing the best known Space Complexity by a factor $O(\log n)$. The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with only $O(\log^2n)$ bits.
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fast self stabilizing minimum spanning tree construction using compact nearest common ancestor labeling scheme
International Symposium on Distributed Computing, 2010Co-Authors: Lelia Blin, Shlomi Dolev, Maria Potopbutucaru, Stephane RovedakisAbstract:We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The Space Complexity of our solution is O(log2 n) bits and it converges in O(n2) rounds. Thus, this algorithm improves the convergence time of all previously known self-stabilizing asynchronous MST algorithms by a multiplicative factor Θ(n), to the price of increasing the best known Space Complexity by a factor O(log n). The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with only O(log2 n) bits.
Changho Seo - One of the best experts on this subject based on the ideXlab platform.
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efficient multiplier based on hybrid approach for toeplitz matrix vector product
Information Processing Letters, 2018Co-Authors: Ku-young Chang, Sun Mi Park, Dowon Hong, Changho SeoAbstract:Abstract We propose a hybrid approach for a Toeplitz matrix–vector product (TMVP) of size k ⋅ 2 i 3 j , where k ≥ 1 and i , j ≥ 0 . It is possible to make trade-offs between time and Space complexities for a TMVP by choosing values k, i, and j properly. We show that the multiplier based on the proposed hybrid TMVP approach has lower Space as well as time complexities than other subquadratic Space Complexity multipliers for five fields recommended by NIST. Moreover, for those five fields, the Space complexities of the proposed multiplier are reduced by a minimum 59 % and a maximum 77 % compared with quadratic Space Complexity multiplier.
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new block recombination for subquadratic Space Complexity polynomial multiplication based on overlap free approach
IEEE Transactions on Computers, 2017Co-Authors: Sun Mi Park, Dowon Hong, Ku-young Chang, Changho SeoAbstract:In this paper, we present new parallel polynomial multiplication formulas which result in subquadratic Space Complexity. The schemes are based on a recently proposed block recombination of polynomial multiplication formula. The proposed two-way, three-way, and four-way split polynomial multiplication formulas achieve the smallest Space complexities. Moreover, by providing area-time tradeoff method, the proposed formulas enable one to choose a parallel formula for polynomial multiplication which is suited for a design environment.
