We propose efficient algorithms for Sorting k-Sets in Bins. The Sorting k-Sets in Bins problem can be described as follows. We are given numbered n bins with k balls in each bin. Balls in the i-th bin are numbered n-i+1. We can only swap balls between adjacent bins. Our task is to move all of the balls to the same numbered bins. For this problem, we give an efficient greedy algorithm with $rac{k+1}{4}n^2+O(k+n)$ swaps and provide a detailed analysis for k=3. In addition, we give a more efficient recursive algorithm using $rac{15}{16}n^2+O(n)$ swaps for k=3.

Menger's sponge (MS) is a kind of three-dimensional fractal structure. To analyze non-resonant electromagnetic properties of MS composed of isotropic paraelectric material, a novel, high-speed computation method employing simple recursion equations in terms of scattering amplitudes for two MS's with adjacent stage numbers, which are the parameters describing structural differences of MS's, is formulated. Within the scope of non-resonant electromagnetic phenomena, scattering patterns, forward and backward scattering amplitudes, and total cross sections of MS are investigated as a function of stage number and incident plane waves, and behaviors typical to fractal structures are extracted from the numerical results of the above equations. In addition, scattering properties at infinite stage number are discussed.

Recursive programs are often easier to read and write than iterative ones, but their execution frequently requires large numbers of procedure calls and stack operations. This causes problems in program optimization related to inline coding and the locality of data references. In addition to these problems, defining programs recursively sometimes leads to repetitive execution of similar computations, causing programs to have exponential time complexity. As a result, recursion removal methods, which transform a given recursive program to an iterative one without using the stack and increasing the amount of computation time, have been studied since the 1970s. In 1998, our group proposed a recursion removal method for a linear recursive program. In this paper, we extend the method to deal with non-linear recursive programs with one descent function (RPODs), which are programs of the form f(x) = if p(x) then b(x) else a(c(x),f(d(x)),f(d2(x)),...,f(dn(x))). First, we define the cumulative function for an RPOD. Next, based on the new cumulative function, several transformation techniques for RPODs are shown. These include a unified method of deriving logarithmic-order iterative programs or loop-free programs. Finally, the relationships between our method and various tupling strategies are discussed.