A non iterative method of separation of points by planes in n dimensions and its application

We demonstrate an algorithm that can separate any number of points in n-dimensional space from one another by planes. Given a set G of Nf points, along with their coordinates, in n-dimensional X-Space, the algorithm partitions all the Nf points by using planes such that no two points are left un-separated by some plane. The algorithm proceeds by transferring points from G to another set S in n-dimensional X-space, to the same coordinate position in S as it was in G. However initially S has very few points and very few planes, but the few planes are so chosen that all the points in S are already separated from each other by the few planes contained in it. Each point is then chosen at random from G and then transferred to S one by one, matters are so arranged that if necessary new planes are drawn in S so that the points in S will always be separated. In order to do this each point from G is labeled as a member of S if it is separated from others in S, otherwise it is labeled as a “special” point which is not yet separated. The collection proceeds (at random) as new points are added to S from G and each new point is either separate and becomes a member of S or labeled as “special”. As soon as n “special” points are collected they are all separated by a single extra plane, which is then added to the collection. The algorithm is made possible because a new concept called Orientation Vector is used. This vector is a Hamming vector and is associated with each point and has all the information necessary to ascertain if two points are separate or not. The algorithm proceeds till S contains all the planes which separate all the points. The method is non iterative, it will always halt successfully and the algorithm strictly follows Shannon’s principle of making optimal use of information as it advances stage by stage. It has the property of restart, if new points are needed to be separated the algorithm can continue from where it left off. At some later stage if the dimension of the data (n to n+r) is increased the algorithm can still continue from where it left off, and tackle the new data points which are of a higher dimension, after some minor modifications. 1 ar X iv :1 50 9. 08 74 2v 5 [ cs .C G ] 2 3 O ct 2 01 5 A proof is provided with a worked example. The computational Complexity is of O(n.Nlog(N)) +O(nlog(N)), where N is the given number of points and $n$ is the dimension of space. In summary this paper describes a non-iterative algorithm of separating a given set of points in n-dimension by planes. Its application to data retrieval problems in very large medical data bases is also given. Possible future applications are