Minimal Spanning Tree Algorithm

Not the travelling salesman problem. Algorithms include Prim's, Kruskals', I have implemented the greedy algorithm which is as follows: (it is a recursive algorithm) $T_0$ is a point.

• Suppose we have $T_{i}$, add an edge to it by finding the minimal edge between a point in the tree and a point that is not, join these to make $T_{i+1}$.

  Algorithm:
  

  1. Set $tree(i)\; <-\; (-n)$, for $i =1:n-1$
  2. Do $n-1$ times:
     • $dmin= min\{ dist(i, \vert tree(i)\vert)\; for\; i <n\; with\; tree(i)<0\}$
     • $i_{min}=$ index at which the minimum is attained.
     • Add edge : $tree(i_{min}) <- (-tree(i_{min}))$
     • Update list of nearest vertices: for $i =1:n-1$ do:
       • if $tree(i)<0$ and $dist(i,i_{min})< dist(i,-tree(i))$
         $tree(i)\; <-\; -i_{min}$

  The output to create will be a vector of numbers that correspond to where the $i$th point should be connected to.

  Here are some examples with Splus:

  mstree(vincr[1:10,  ])
  > mstree(vincr[1:10,  ])
  $x:
   [1]  4.20959759 -3.70445704  4.54166603  7.22903347  4.89603710  2.28360796
   [7]  0.00000000 -2.00589728  1.03156424  0.04926162
  $y:
   [1] -7.0463328 -6.4673190  0.4050449 -0.3888339 -2.2247181  0.0000000
   [7]  0.0000000 -3.2911484 -0.8079203 -1.5704904
  $mst:
  [1] 10  8  6  3  3  7 10 10 10
  
  mst:   vector of length nrow(x)-1 describing  the  edges  in  the
         minimal spanning tree.  The ith value in this vector is an
         observation number, indicating that this  observation  and
         the  ith observation should be linked in the minimal span-
         ning tree.
  
  $order:
        [,1] [,2] 
   [1,]    4    7
   [2,]    3   10
   [3,]    5    6
   [4,]    6    9
   [5,]    7    8
   [6,]   10    3
   [7,]    1    1
   [8,]    9    4
   [9,]    8    5
  [10,]    2    2
  order:    matrix, of size nrow(x) by 2, giving two types of  ord-
         ering:  The  first  column  presents the standard ordering
         from one extreme of the minimal spanning tree to the  oth-
         er.  This  starts on one end of a diameter and numbers the
         points in a certain order so that points close together in
         Euclidean  space  tend  to  be close in the sequence.  The
         second column presents the radial ordering, based on  dis-
         tance from the center of the minimal spanning tree.  These
         can be used to detect clustering.   See  below  for  graph
         theory definitions.
  

  plot(x, y)   # plot original data
  mst <- mstree(cbind(x, y), plane=F)   # minimal spanning tree
            # show tree on plot
  segments(x[seq(mst)], y[seq(mst)], x[mst], y[mst])
   
   i <- rbind(iris[,,1], iris[,,2], iris[,,3])
         tree <- mstree(i)    # multivariate planing
         plot(tree, type="n")  # plot data in plane
   text(tree, label=c(rep(1, 50), rep(2, 50), rep(3, 50))) # identify points
  
                  # get the absolute value stress e
         distp <- dist(i)
         dist2 <- dist(cbind(tree$x, tree$y))
         e <- sum(abs(distp - dist2))/sum(distp)
   0.1464094