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Recently, I need to construct an unique index from two integer. The best solution I found is the Pairing function.

Pairing function is an one to one and onto function that map two integers to a single integer. The definition as follows:

pair<-function(x,y){ 0.5*(x+y)*(x+y+1) + x } unpair<-function(z){ w= floor( (sqrt(8*z+1) - 1)/2 ) t = w*(w+1)/2 cbind(z-t,w-z+t) } foreach (i = 0:4,.combine=rbind) %do% { x<-0:i y<-i:0 key<-pair(x,y) unpair_key <- unpair(key) cbind(x,y,key=key,unpair_key=unpair_key) } x y key x y [1,] 0 0 0 0 0 [2,] 0 1 1 0 1 [3,] 1 0 2 1 0 [4,] 0 2 3 0 2 [5,] 1 1 4 1 1 [6,] 2 0 5 2 0 [7,] 0 3 6 0 3 [8,] 1 2 7 1 2 [9,] 2 1 8 2 1 [10,] 3 0 9 3 0 [11,] 0 4 10 0 4 [12,] 1 3 11 1 3 [13,] 2 2 12 2 2 [14,] 3 1 13 3 1 [15,] 4 0 14 4 0

If ordering of `x`

and `y`

is not important, we can swap `x`

and `y`

if `x>y`

. However, the Pairing function is not one to one and we can not back out `x`

and `y`

with `z`

pair<-cmpfun(function(x,y,ordering_matter=TRUE){ if (ordering_matter){ return(0.5*(x+y)*(x+y+1) + x) } else{ swap <- x>y return(0.5*(x+y)*(x+y+1) + (x* !swap) + (y*swap )) } }) foreach (i = 0:4,.combine=rbind) %do% { x<-0:i y<-i:0 key<-pair(x,y,ordering_matter=FALSE) unpair_key <- unpair(key) cbind(x,y,key=key,unpair_key=unpair_key) } x y key x y [1,] 0 0 0 0 0 [2,] 0 1 1 0 1 [3,] 1 0 1 0 1 [4,] 0 2 3 0 2 [5,] 1 1 4 1 1 [6,] 2 0 3 0 2 [7,] 0 3 6 0 3 [8,] 1 2 7 1 2 [9,] 2 1 7 1 2 [10,] 3 0 6 0 3 [11,] 0 4 10 0 4 [12,] 1 3 11 1 3 [13,] 2 2 12 2 2 [14,] 3 1 11 1 3 [15,] 4 0 10 0 4 >

If we have more than two integers, we can apply the Pairing function in a nested manner.

nestedPair<-function(x){ ncol_x = ncol(x) if(ncol_x==1){ return(x) } else if(ncol_x ==2) { return(pair(x[,1],x[,2])) } else if ( ncol_x > 2){ return(pair( x[,1] ,nestedPair(x[,2:ncol_x]) ) ) } } nestedUnpair<-function(x,order){ if(order==1){ return(unpair(x)) } else if(order >1) { out <- unpair(x) return(cbind(out[,1],nestedUnpair(out[,2],order-1))) } } x<-expand.grid(0:2,0:2,0:2) key <- nestedPair(x) unpair_key <- nestedUnpair(key,2) cbind(x=x,key=key,unpair_key=unpair_key) x.Var1 x.Var2 x.Var3 key unpair_key.1 unpair_key.2 unpair_key.3 1 0 0 0 0 0 0 0 2 1 0 0 2 1 0 0 3 2 0 0 5 2 0 0 4 0 1 0 3 0 1 0 5 1 1 0 7 1 1 0 6 2 1 0 12 2 1 0 7 0 2 0 15 0 2 0 8 1 2 0 22 1 2 0 9 2 2 0 30 2 2 0 10 0 0 1 1 0 0 1 11 1 0 1 4 1 0 1 12 2 0 1 8 2 0 1 13 0 1 1 10 0 1 1 14 1 1 1 16 1 1 1 15 2 1 1 23 2 1 1 16 0 2 1 36 0 2 1 17 1 2 1 46 1 2 1 18 2 2 1 57 2 2 1 19 0 0 2 6 0 0 2 20 1 0 2 11 1 0 2 21 2 0 2 17 2 0 2 22 0 1 2 28 0 1 2 23 1 1 2 37 1 1 2 24 2 1 2 47 2 1 2 25 0 2 2 78 0 2 2 26 1 2 2 92 1 2 2 27 2 2 2 107 2 2 2

To

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