International Journal of Foundations of Computer Science

Compressed sensing is a sparse sampling theory. Compared with the Nyquist-Shannon sampling theory, in compressed sensing one could reconstruct a sparse signal from a few linear and non-adaptive measurements. How to construct a good sensing matrix which captures the full information of a sparse signal is an important problem in compressed sensing. In this paper, we present a new deterministic construction using a linear or nonlinear code, which is a generalization of DeVore’s construction and Li et al.’s construction. By choosing some appropriate linear codes or nonlinear codes, we will construct some good binary sensing matrices which are superior to DeVore’s ones and Li et al.’s ones.

We consider the problem of scattering n robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering.

We first prove that n log n random bits are necessary to scatter n robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem.

Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in o(log log n) rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal (O(n log n) random bits are used) and time optimal (O(log log n) rounds are used). This improves the time complexity of previous results in the same setting by a log n factor.

Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes the time complexity gap with and without strong multiplicity detection (that is, O(1) time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach a constant value as desired otherwise).

We classify the growth of a k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the base-k logarithm of the joint spectral radius of any set of a special class of matrices determined by the k-kernel.

In this paper we present the first data structure for partially persistent B-trees with constant worst-case update time, where deletions are handled in a symmetrical to insertions manner. Our structure matches the query times of optimal partially persistent B-trees that do not support constant update time thus, we have managed to reduce the worst-case update time to a constant, without a penalty in the query times. The new data structure is produced by mixing two other data structures, (a) the partially persistent B-tree and (b) the balanced search tree with constant worst-case update time.

This paper studies some enumeration problems of binary sequences with period 2ⁿ based on the Games-Chan algorithm and a modified Stamp-Martin algorithm. We provide the exact number of critical error sequences of binary 2ⁿ-periodic sequences corresponding to the second critical point. The exact formula for the number of binary 2ⁿ-periodic sequences with exactly three critical points is also derived.

We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qⁿ over the finite field of size q. When d is constant, the algorithm needs 𝒪(n^d-1) quantum queries, which is optimal. Classically, Ω(n^d) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.