Abstract
The packing problem of unit equilateral triangles not only has the theoretical significance but also offers broad prospects in material processing and network resource optimization. Because this problem is nondeterministic polynomial (NP) hard and has the feature of continuity, it is necessary to limit the placements of unit equilateral triangles before optimizing and obtaining approximate solution (e.g., the unit equilateral triangles are not allowed to be rotated). This paper adopts a new quasi-human strategy to study the packing problem of unit equilateral triangles. Some new concepts are put forward such as side-clinging action, and an approximation algorithm for solving the addressed problem is designed. Time complexity analysis and the calculation results indicate that the proposed method is a polynomial time algorithm, which provides the possibility to solve the packing problem of arbitrary triangles.
Highlights
The solution of NP hard problem has both popularity and intractability, which is of great value in philosophy of science and real life
When a unit equilateral triangle is tangent to another unit equilateral triangle, zero, one, or two angle regions may be formed, which is shown in Figures 3, 4, and 5(a)–5(c), respectively
Let N > 0; and there are four triangles constituting the square container in P; R is the set of angle regions formed by four triangles constituting the square container
Summary
The solution of NP hard problem has both popularity and intractability, which is of great value in philosophy of science and real life. Packing problem of unit equilateral triangles is a special case of the two-dimension packing problem. Researches on the packing problem of unit equilateral triangles are theoretically significant to look for an efficient approximate algorithm for a NP hard issue, especially for a general packing problem. Based on the population control (PERM) strategy and corneroccupying approach, a new hybrid algorithm is proposed to solve the problem of packing equal or unequal circles into a larger circle container in [11]. The primary purpose of this paper is to study the packing problem of unit equilateral triangles according to the characteristic of the unit equilateral triangles and in the base of analysis of the general triangles packing problems. (iii) A new algorithm for solving the packing problem of unit equilateral triangles is presented based on the proposed quasi-human strategy
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