Removing pairs that meet a given criterion from a list

Let $(x_{min},x_{max})$ and $(y_{min},y_{max})$ represent lowerbound and upperbound acceptable values for some coordinate $c_i$.

Setting $x_{min} = y_{min} = 1$ & $x_{max} = y_{max} = 9$, how can I take a list of values, e.g.:

val = {{0, 10}, {2, 10}, {4, 4}, {5, 6}, {10, 4}};

and use DeleteCases to get rid of coordinates such as {0, 10}, {2,10}, & {10,4} (the examples in the above list) where either the $x$ or $y$ component of a pair is outside of the specified upper and lower bounds for acceptable values? Is DeleteCases an appropriate tool for large arrays?

Answer

You can achieve the same end with DeleteCases or Cases.

f signals a coordinate that is out of bounds.

g signals that a coordinate is in bounds.

f[n_] := n < 1 \[Or] n > 9
g[n_] := 1 <= n <= 9

Four ways to filter out the outlying points:

v[k] generates 10^k values between {0, 0} and {10, 10})

v[k_] := Table[{r, r}, {10^k}];

1. 
   

   Delete the points that have an out-of-bounds coordinate.

   
   

   DeleteCases[v[k], {x_, y_} /; f[x] \[Or] f[y]]
   



2. 
   

   Delete the points that are not in-bounds.

   
   

   DeleteCases[v[k], {x_, y_} /; \[Not] (g[x] \[And] g[y])]
   



3. 
   
   

   Keep points that are not out-of-bounds.

   
   

   Cases[v[k], {x_, y_} /; \[Not] (f[x] \[Or] f[y])]
   



4. 
   

   Keep points that are in-bounds.

   
   

   Cases[v[k], {x_, y_} /; (g[x] \[And] g[y])]
   

Timings

    r = Table[{AbsoluteTiming[DeleteCases[v[k], {x_, y_} /; f[x] \[Or] f[y]];],

          AbsoluteTiming[DeleteCases[v[k], {x_, y_} /; \[Not] (g[x] \[And]g[y])];], 
          AbsoluteTiming[Cases[v[k], {x_, y_} /; \[Not] (f[x] \[Or] f[y])];],
          AbsoluteTiming[Cases[v[k], {x_, y_} /; (g[x] \[And] g[y])];]}, {k,  3, 7}]
    Grid@Prepend[ r[[All, All, 1]], {"Delete f", "Delete not-g", "Cases not-f", "Cases g"}]

The results with DeleteCases and Cases (from 10^3 to 10^7 points) show that DeleteCases with function g gives the fastest analysis.