Greedy Property Research Articles

Extensions and New Characterizations of Some Greedy-Type Bases

Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.

Periodic representations in Salem bases

We prove that all algebraic bases β allow an eventually periodic representation of the elements of ℚ(β) with a finite alphabet of digits $${\cal A}$$ . Moreover, the classification of bases allowing that those representations have the socalled weak greedy property is given. The decision problem whether a given pair (β, $${\cal A}$$ ) allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.

Residual-Decaying-Based Informed Dynamic Scheduling for Belief-Propagation Decoding of LDPC Codes

Belief-propagation (BP) algorithm and its variants are well-established methods for iterative decoding of LDPC codes. Among them, residual belief-propagation (RBP), which is the most primitive and representative informed dynamic scheduling (IDS) strategy, can significantly accelerate the convergence speed. However, RBP decoding suffers from a poor convergence error-rate performance due to its greedy property, which is one of the challenging issues in the design of IDS strategies. To tackle this problem, a novel IDS scheme, namely residual-decaying-based residual belief-propagation (RD-RBP) algorithm, is presented in this paper. In RD-RBP, a decaying mechanism is introduced to manipulate the residuals of those check-to-variable messages, preventing the decoding resources from being unreasonably occupied by a small group of edges in the Tanner graph. The greediness is therefore alleviated and better performance of convergence error-rate is achieved. Besides, a two-stage scheduling scheme combining prior-art variable-node and variable-to-check-edge RBP (V-VCRBP) with RD-RBP, named V-VC-RD-RBP, is proposed for achieving both fast convergence speed and a low convergence error-rate. The simulation results validate the advantages of the proposed schemes.

A distributed routing for wireless sensor networks with mobile sink based on the greedy embedding

Mobile sinks can be a solution to solve the problem that energy consumption of sensor nodes is not balanced in Wireless Sensor Networks (WSN). Caused by the sink mobility, the paths between the sensor nodes and the sink change with time. It is necessary to find a protocol that can find efficient routings between the mobile sink and sensor nodes but do not consume too many network resources. In this paper, we propose an algorithm called virtual-node greedy embedding (VGE) to embed the topology of a WSN into hyperbolic plane without the physical geographic information. This algorithm gives each node a virtual coordinate that guarantees the greedy forwarding of the packets from sensor nodes to the sink when there is no failed node in WSN, even if some obstacles exist in the network. VGE keeps the greedy property when new nodes join the network after the initial embedding and does not change the node coordinates assigned previously. Based on the characteristic of embedding process, modified gravity-pressures (MGP) algorithm is proposed to find a routing path quickly when there are some failed nodes in WSN. Furthermore, particular solutions of the parameters in VGE are presented with concise style, and the simulations show that our algorithm has an efficient performance.

Robust Independence Systems

An independence system $\mathcal{F}$ is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number $\alpha> 0$, a set $X\in\mathcal{F}$ is said to be $\alpha$-robust if for any $k$, it includes an $\alpha$-approximation of the maximum $k$-independent set, where a set $Y$ in $\mathcal{F}$ is called $k$-independent if the size $|Y|$ is at most $k$. In this paper, we show that every independence system has a $1/\sqrt{\mu(\mathcal{F})}$-robust independent set, where $\mu(\mathcal{F})$ denotes the exchangeability of $\mathcal{F}$. Our result contains a classical result for matroids and the ones of Hassin and Rubinstein [SIAM J. Discrete Math., 15 (2002), pp. 530--537] for matchings and Fujita, Kobayashi, and Makino [SIAM J. Discrete Math., 27 (2013), pp. 1234--1256] for matroid $2$-intersections, and provides better bounds for the robustness for many independence systems such as $b$-matchings, hypergraph matchings, matroid $p$-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.

A switching algorithm for design of optimal increasing binary filters over large windows

All known approaches for the design of increasing translation-invariant binary window filters involve combinatoric searches. This paper proposes a new switching algorithm having the advantage that the search is over a smaller set than other algorithms. Beginning with an estimate from image realizations of the optimal generic (nonincreasing) window function, the algorithm switches (exchanges) a set of observation vectors (templates) between the optimal function's kernel and the kernel's complement. There are many such “switching sets” that provide a kernel defining an increasing filter. The optimal increasing filter is the one corresponding to the switching set that produces the minimal increase in error over the optimal generic filter. The core of the search problem is the inversion set of the optimal filter. The inversion set is composed of all vectors in the kernel lying beneath a nonkernel vector in the lattice of observation vectors and all nonkernel vectors lying above a kernel vector. The new algorithm, which is based on an error-related greedy property, recursively eliminates the inversion set until the optimal increasing filter is obtained. For purposes of computational efficiency, the actual implementation may be based on a relaxation of the original construction, so that the result may be based on a relaxation of the original construction, so that the result may be suboptimal. For the various models tested, the relaxed algorithm has proven to be optimal or very close to optimal. Besides its good estimation precision, the new algorithm has three noteworthy properties: first, it is applicable to relatively large windows; second, it operates directly on the input data via estimates of the determining conditional probabilities; and third, the degree of relaxation serves as an input parameter to the algorithm, so that computation time can be bounded for large windows and the algorithm can run to full optimality for small windows.