Swarm-Based Spreading Points

Abstract

In this paper we propose a Swarm-based Spreading Points algorithm (SSP) for improving the solutions for packing problems. The SSP repositions the initial set of points and evolves it to improve the minimum distance between points. During the evolving process, for each point, a feasible direction of movement is computed according to its nearest neighbors so that the shortest pairwise distance between the point and other points can be increased along this direction (if any). Our experiments showed that the SSP algorithm can improve certain best-known solutions for some problems previously reported in the literature.

Appendix

Given a predefined calculation precision, SSP is convergent. A proof of the convergence is given below.

Let c be an n-dimensional vector and the arrangement of its elements, from its first element denoted by c [1] to the last c[n], corresponds to the arrangement of d _i for all n points ordered by ascend, i.e., c[i] ≤ c[i + 1] for any 1 ≤ i ≤ n-1. Given a specified calculation precision and the number of points n, the set of c, denoted by C, is bounded (finite). The inequality c ₁ > c ₂ holds if and only if there exists an integer i (1 ≤ i≤n) such that c ₁[j] ≥ c ₂[j] ∀j < i and c ₁[i] > c ₂[i] where c ₁, c ₂∈C.

Let c _t and c _t+1∈C be arrangements of shortest pairwise distances for n points at iterations t and t + 1 respectively. The elements in c _t and c _t+1 are ordered by ascend.

Convergence.

Let t be the current iteration and t + 1 be the iteration after movement of a point p _i (suppose the position index of the point p _i is also i in c _t ). When the point p _i moves along the direction r _i (t), for any point p _k in the nearest neighbors n _i (t) of p _i , the d _k does not decrease (increases if the point p _i also belongs to n _k (t)); for other points p _j (p _j is not in n _i (t)), let d _ij be the distance from the point p _i to the point p _j , and then $d_{ij} \ge d_{i}^{ + }$ according to the definition of $d_{i}^{ + }$. After the point p _i moves a step $a\left( {d_{i}^{ + } - d_{i} } \right)$ from iteration t to t + 1, the distance $d_{ij} \left( {t + 1} \right) \, \ge d_{i}^{ + } \left( t \right) - a\left( {d_{i}^{ + } \left( t \right) - d_{i} \left( t \right)} \right) > d_{i} \left( t \right)$ since a is less than 1. So for p _j , if d _j (t) ≤ d _i (t) at iteration t, d _j (t + 1) does not change at iteration t + 1 since d _ij (t + 1) > d _i (t) and the point p _i does not become a neighbor of p _j at iteration t + 1. If d _j (t) > d _i (t) at iteration t, d _j (t + 1) is still larger than d _i (t) at iteration t + 1 even though the movement of the point p _i affects d _j (i.e., the point p _i becomes a nearest neighbor of the point p _j at iteration t + 1). In summary, (1) for the point p _u , the position of which in c _t is before that of the point p _i (d _u (t) ≤ d _i (t)), the d _u (t + 1) does not decrease at iteration t + 1; (2) for the point p _v , the position of which in c _t is after that of the point p _i (d _v (t) > d _i (t)), the d _v (t + 1) is still larger than d _i (t); (3) for the point p _i , its d _i (t + 1) increases. Hence c _t+1 increases according to the above definition when there exists any movable point. Because C is bounded (finite), c _t will converge with t → ∞.