# [R] Making a ranking algorithm more efficient

Peter Wolf s-plus at wiwi.uni-bielefeld.de
Wed Jun 2 11:09:13 CEST 2004

```let's start by defining x and y matching the properties of the points in
the picture
<<*>>=
x<-c(1,2,4,5,9,3,7,6); y<-c(10,8,5,3,2,9,4,7)
plot(x,y,pch=letters[1:8])

@
along each dimension we have to compare the coordinates of the points:
<<*>>=
outer(x,x,"<")

@
we get a logical matrix:
output-start
Wed Jun  2 10:36:52 2004
[,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]
[1,] FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[2,] FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[3,] FALSE FALSE FALSE  TRUE  TRUE FALSE  TRUE  TRUE
[4,] FALSE FALSE FALSE FALSE  TRUE FALSE  TRUE  TRUE
[5,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[6,] FALSE FALSE  TRUE  TRUE  TRUE FALSE  TRUE  TRUE
[7,] FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE
[8,] FALSE FALSE FALSE FALSE  TRUE FALSE  TRUE FALSE
output-end

results of dimension x and of dimension y have to be aggregated by "&"
<<*>>=
outer(x,x,"<")&outer(y,y,"<")

@
and we get:
output-start
Wed Jun  2 10:44:56 2004
[,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]
[1,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[2,] FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
[3,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE
[4,] FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE
[5,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[6,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[7,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[8,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
output-end

TRUE in position (i,j) indicates that x- and y-coordinates of point i
are smaller
than those of point j. The sum of row i shows the number of
points that dominate point i. The sum of col j shows the number of points
that are dominated by point j.

@
so we can compute the the number of dominated points by applying
function sum on the cols:
<<*>>=
apply(outer(x,x,"<")&outer(y,y,"<"),2,sum)

@
at last some polishing
<<*>>=
result<-apply(outer(x,x,"<")&outer(y,y,"<"),2,sum)+1
names(result)<-letters[1:8]
result

@
output-start
Wed Jun  2 10:59:03 2004
a b c d e f g h
1 1 1 1 1 2 2 3
output-end

in the case of more than 2 dimensions you have to add further
outer-operations

Peter Wolf

Waichler, Scott R wrote:

>I would like to make a ranking operation more efficient if possible.
>The goal is to rank a set of points representing objective
>function values such that points which are "dominated" by no
>others have rank 1, those which are dominated by one other point
>have rank 2, etc.  In the example with two dimensions below, objective
>functions 1 and 2 are to be minimized.  Points a-e are non-dominated,
>rank 1 points.  Point f is rank 2 because point b is superior, and
>point g is rank 2 because point d is superior.  Point h is rank 3
>because points c and d are both superior.
>
>       | a
>       |    f
>       |  b
>       |       h          (figure requires monospaced, plain-text
>display)
>Obj 1  |     c
>       |         g
>       |      d
>       |            e
>       |____________________
>
>              Obj 2
>
>
>I need to compute the ranks of the rows of a matrix, where each row
>represents a point in objective space and the columns
>contain the objective function values to be minimized.
>Rank is defined as the number of points with objective function
>values less than or equal to the current point (including the current
>point).  I've written the following function with two loops:
>
>  PARETO.RANK <- function(obj.array) {
>    obj.array <- unique(obj.array)
>
>    ind.row <- 1:nrow(obj.array)
>    ind.col <- 1:ncol(obj.array)
>
>    rank.vec <- rep(NA, length(ind.row)) # initialize
>
>    # Loop thru rows (points in objective function space)
>    for(i in ind.row) {
>      set <- ind.row
>      for (j in ind.col) {  # Loop thru objective functions
>        set <- set[ obj.array[set,j] <= obj.array[i,j] ]
>      }
>      rank.vec[i] <- length(set)
>    }
>    return(rank.vec)
>  } # end PARETO.RANK3()
>
>Can anyone think of a way to do this more efficiently, for example by
>vectorizing further?
>
>Thanks,
>Scott Waichler
>scott.waichler at pnl.gov
>
>______________________________________________
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