A lower bound on the computational complexity of the QR decomposition on a shared memory SIMD computer

Abstract In this paper, a method using bintree structure to express the states of the packing space of rectangular packing is proposed. Through the sequential decomposition of the packing space, the optimal packing scheme of various sized rectangular packing can be obtained by every time putting the optimal piece that satisfies specular conditions toward the current packing space and by locating it at the up-left corner of the current packing space. Different optimal packing schemes that satisfy different demands can be obtained by adjusting the values of the ordering factors KA and KB. A parallel algorithm based on SIMD-CREW shared-memory computer is designed through the analysis of the parallelism of the bintree expression. The whole packing process is clearly expressed by the bintree. The computational complexity of the algorithm is shown to be O(n2logn). Both the experimental results and the comparison with other sequential packing algorithms have proved that the parallel packing algorithm is efficient. What is more, it nearly doubles the problem solving speed.

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KNOWLEDGE STATES FOR THE CACHING PROBLEM IN SHARED MEMORY MULTIPROCESSOR SYSTEMS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006504 ◽

2009 ◽

Vol 20 (01) ◽

pp. 167-183 ◽

Cited By ~ 1

Author(s):

WOLFGANG BEIN ◽

LAWRENCE L. LARMORE ◽

RÜDIGER REISCHUK

Keyword(s):

Lower Bound ◽

Shared Memory ◽

Competitive Analysis ◽

Online Algorithm ◽

Data Access ◽

Global Memory ◽

Multiprocessor Systems ◽

Optimal Behavior ◽

Knowledge States ◽

Shared Memory Multiprocessor

Multiprocessor systems with a global shared memory provide logically uniform data access. To hide latencies when accessing global memory each processor makes use of a private cache. Several copies of a data item may exist concurrently in the system. To guarantee consistency when updating an item a processor must invalidate copies of the item in other private caches. To exclude the effect of classical paging faults, one assumes that each processor knows its own data access sequence, but does not know the sequence of future invalidations requested by other processors. Performance of a processor with this restriction can be measured against the optimal behavior of a theoretical omniscient processor, using competitive analysis. We present a [Formula: see text]-competitive randomized online algorithm for this problem for cache size of 2, and prove a matching lower bound on the competitiveness. The algorithm is derived with the help of a new concept we call knowledge states. Finally, we show a lower bound of [Formula: see text] on the competitiveness for larger cache sizes.

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On NACK-Based rDWS Algorithm for Network Coded Broadcast

Entropy ◽

10.3390/e21090905 ◽

2019 ◽

Vol 21 (9) ◽

pp. 905 ◽

Cited By ~ 1

Author(s):

Sovanjyoti Giri ◽

Rajarshi Roy

Keyword(s):

Computational Complexity ◽

Lower Bound ◽

Finite Field ◽

Probabilistic Analysis ◽

Queue Length ◽

Efficiency Analysis ◽

Entropy Analysis ◽

Erasure Channels ◽

Coding Strategy ◽

High Computational Complexity

The Drop when seen (DWS) technique, an online network coding strategy is capable of making a broadcast transmission over erasure channels more robust. This throughput optimal strategy reduces the expected sender queue length. One major issue with the DWS technique is the high computational complexity. In this paper, we present a randomized version of the DWS technique (rDWS), where the unique strength of the DWS, which is the sender’s ability to drop a packet even before its decoding at receivers, is not compromised. Computational complexity of the algorithms is reduced with rDWS, but the encoding is not throughput optimal here. So, we perform a throughput efficiency analysis of it. Exact probabilistic analysis of innovativeness of a coefficient is found to be difficult. Hence, we carry out two individual analyses, maximum entropy analysis, average understanding analysis, and obtain a lower bound on the innovativeness probability of a coefficient. Based on these findings, innovativeness probability of a coded combination is analyzed. We evaluate the performance of our proposed scheme in terms of dropping and decoding statistics through simulation. Our analysis, supported by plots, reveals some interesting facts about innovativeness and shows that rDWS technique achieves near-optimal performance for a finite field of sufficient size.

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A Computational Perspective on Network Coding

Mathematical Problems in Engineering ◽

10.1155/2010/436354 ◽

2010 ◽

Vol 2010 ◽

pp. 1-11

Author(s):

Qin Guo ◽

Mingxing Luo ◽

Lixiang Li ◽

Yixian Yang

Keyword(s):

Graph Theory ◽

Computational Complexity ◽

Lower Bound ◽

Network Coding ◽

Upper Bound ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Multicast Networks ◽

Computational Perspective ◽

Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

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DISASSEMBLING TWO-DIMENSIONAL COMPOSITE PARTS VIA TRANSLATIONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195993000051 ◽

1993 ◽

Vol 03 (01) ◽

pp. 71-84 ◽

Cited By ~ 6

Author(s):

DORON NUSSBAUM ◽

JÖRG-RÜDIGER SACK

Keyword(s):

Computational Complexity ◽

Lower Bound ◽

Two Dimensional ◽

Composite Part ◽

Disassembly Sequence

This paper deals with the computational complexity of disassembling 2-dimensional composite parts (comprised of M simple n-vertex polygons) via collision-free translations. The first result of this paper is an O(Mn+M log M) algorithm for computing a sequence of translations (performed in a common direction) to disassemble composite parts. The algorithm improves on the O(Mn log Mn) bound previously established for this problem and is easily seen to be optimal. The problem had been posed by Nurmi and by Toussaint. The second result of this paper is an Ω(Mn+M log M) lower bound proof for the problem of detecting whether a composite part can be disassembled, or contains interlocking subparts. Thus, detecting the existence of a disassembly sequence is as hard as computing one. As a consequence, the algorithm for computing a disassembly sequence is optimal also for the detecting problem.

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Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

Special Matrices ◽

10.1515/spma-2017-0015 ◽

2017 ◽

Vol 5 (1) ◽

pp. 202-224 ◽

Cited By ~ 1

Author(s):

Dimitrios Christou ◽

Marilena Mitrouli ◽

Dimitrios Triantafyllou

Keyword(s):

Computational Complexity ◽

Qr Decomposition ◽

Greatest Common Divisor ◽

Computational Simulations ◽

Matrix Representations ◽

Basis Matrix ◽

Matrix Methods ◽

Structured Matrix ◽

Bezout Matrix ◽

Column Pivoting

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

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Witnessing matrix identities and proof complexity

International Journal of Algebra and Computation ◽

10.1142/s021819671850011x ◽

2018 ◽

Vol 28 (02) ◽

pp. 217-256

Author(s):

Fu Li ◽

Iddo Tzameret

Keyword(s):

Computational Complexity ◽

Lower Bound ◽

Finite Basis ◽

Proof Complexity ◽

Polynomial Identities ◽

Open Problems ◽

Complete Proof ◽

Proof Systems ◽

Matrix Identity ◽

Matrix Identities

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

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