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Average-case complexity
About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.
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TL;DR: In this paper, the authors studied the average case complexity of a linear multivariate problem (LMP) defined on functions of d variables and proved that tractability of an LMP in Λ std is equivalent to tractability in ǫ all, although the proof is no(constructive).
Abstract: We study the average case complexity of a linear multivariate problem (LMP) defined on functions of d variables. We consider two classes of information. The first Λ std consists of function values and the second Λ all of all continuous linear functionals. Tractability of LMP means that the average case complexity is O((1/e) p ) with p independent of d. We prove that tractability of an LMP in Λ std is equivalent to tractability in Λ all , although the proof is no(constructive. We provide a simple condition to check tractability in Λ all . We also address the optimal design problem for an LMP by using a relation to the worst case setting. We find the order of the average case complexity and optimal sample points for multivariate function approximation
26 citations
TL;DR: Two new cross-entropy (CE)-based sign-selection algorithms are developed, which adaptively adjusts the probability of "elite" samples, and stops whenever a PAR threshold is reached, and achieves up to 95 % complexity savings over the first.
Abstract: Sign-selection uses a set of subcarrier signs to reduce the peak-to-average power ratio (PAR) of orthogonal-frequency-division multiplexing (OFDM). However, the computational complexity (worst-case) is exponential in N, the number of subcarriers. Suboptimal sign-selection algorithms, achieving different tradeoffs between the PAR reduction and complexity, have thus been developed. For example, the derandomization method achieves high PAR reduction of O(log N) with relatively high complexity of O(N2). On the other hand, selective mapping (SLM) and partial transmit sequences (PTS) sacrifice the achievable PAR reduction for lower complexity. In this paper, we develop two new cross-entropy (CE)-based sign-selection algorithms. Our algorithms simultaneously updates the probabilities of the signs of all subcarriers. The first algorithm obtains a PAR lower than the above methods with a complexity level of O(N2). However, if the number of iterations is fixed, this algorithm obtains the same PAR reduction as derandomization, but with O(N log N) complexity. Practical PAR reduction algorithms require that the extra cost of PAR reduction must be small. Therefore, we propose the second algorithm, which adaptively adjusts the probability of "elite" samples, and stops whenever a PAR threshold is reached. Our second algorithm achieves up to 95 % complexity savings over the first (with only a 0.4-dB PAR reduction loss). The simulations confirm the complexity advantages of the proposed algorithms compared to SLM and derandomization.
26 citations
20 Sep 2011
TL;DR: The bit-complexity of the state-of-the art O(Δ) coloring algorithm is reduced without changing its time and message complexity to derive lower bounds on the time complexity for distributed algorithms as shown for the MIS and the coloring problems.
Abstract: We present tradeoffs between time complexity t, bit complexity b, and message complexity m. Two communication parties can exchange Θ(mlog(tb/m2)+b) bits of information for m < √bt and Θ(b) for m ≥ √bt. This allows to derive lower bounds on the time complexity for distributed algorithms as we demonstrate for the MIS and the coloring problems. We reduce the bit-complexity of the state-of-the art O(Δ) coloring algorithm without changing its time and message complexity. We also give techniques for several problems that require a time increase of tc (for an arbitrary constant c) to cut both bit and message complexity by Ω(log t). This improves on the traditional time-coding technique which does not allow to cut message complexity.
26 citations
TL;DR: It is proved that there is no efficient algorithm computing all minimal solutions, and the problem of complexity of computing minimal solutions is examined.
Abstract: Minimal solutions play a crucial role in describing all solutions of relational equations. For this reason, the problem of computing minimal solutions has for long been examined. The literature contains several algorithms for computing minimal solutions. Recently, contributions regarding computational complexity of the problem itself appeared. The complexity aspect is clearly of fundamental importance. However, the existing results contain serious flaws. In this paper, we inspect the existing contributions, clarify the flaws, examine the problem of complexity of computing minimal solutions, prove that there is no efficient algorithm computing all minimal solutions, and discuss further ramifications of our observations.
26 citations
Posted Content•
TL;DR: The main goal of this article is to explain the significance of the Knutson and Tao Saturation Theorem result in the context of Geometric Complexity Theory.
Abstract: We point out that the remarkable Knutson and Tao Saturation Theorem and polynomial time algorithms for LP have together an important and immediate consequence in Geometric Complexity Theory. The problem of deciding positivity of Littlewood-Richardson coefficients for GLn(C) belongs to P. Furthermore, the algorithm is strongly polynomial.
The main goal of this article is to explain the significance of this result in the context of Geometric Complexity Theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable Kac-Moody algebras.
26 citations