Find maximum of 'y' coordinates from a list of duplicate 'x' coordinates

Question

I have a multidimensional-variable list, suppose for example {{x1,y1},{x2,y2}..}. I have duplicate values for the 'x' coordinates and I need to find for the duplicate 'x' elements the corresponding minimum 'y'. A sample of this list is the following:

l1={{1, 1.43E-46}, {21, 2.79E-48}, {41, 3.22E-45}, {41,1.74E-46}, {81, 2.77E-46}, {121,9.97E-48}, {161, 1.24E-45}, {181,1.19E-45}}

I need to rewrite the list with only those elements from duplicate 'x' coordinates with a minimum value of 'y', for the former list this would be:

l1={{1, 1.43E-46}, {21, 2.79E-48}, {41,1.74E-46}, {81, 2.77E-46}, {121,9.97E-48}, {161, 1.24E-45}, {181,1.19E-45}}

Since from the duplicate coordinates 'x', the minimum 'y' is:

 {41,1.74E-46}

Answer 1

This works:

l1 = {{1, 1.43 10^-46}, {21, 2.79 10^-48}, {41, 3.22 10^-45}, {41, 
    1.74 10^-46}, {81, 2.77 10^-46}, {121, 9.97 10^-48}, {161, 
    1.24 10^-45}, {181, 1.19 10^-45}};

Map[First[SortBy[#, Last]] &, SplitBy[l1, First]]

(*
==> {{1, 1.43*10^-46}, {21, 2.79*10^-48}, {41, 1.74*10^-46}, {81,
   2.77*10^-46}, {121, 9.97*10^-48}, {161, 1.24*10^-45}, {181, 
  1.19*10^-45}}
*)

The first step is to use SplitBy with the first element as the criterion. Then the size of the last element is tested in SortBy.

Answer 2

l1 = {{1, 1.43 10^-46}, {21, 2.79 10^-48}, {41, 3.22 10^-45}, {41,1.74 10^-46}, 
      {81, 2.77 10^-46}, {121, 9.97 10^-48}, {161, 1.24 10^-45}, {181, 1.19 10^-45}};

I version 10, you can also use MinimalBy:

    Join@@MinimalBy[Last]/@GatherBy[l1, First]
    (* {{1, 1.43*10^-46}, {21, 2.79*10^-48}, {41, 1.74*10^-46}, {81, 2.77*10^-46}, 
        {121, 9.97*10^-48}, {161, 1.24*10^-45}, {181, 1.19*10^-45}} *)

or Merge:

List@@@Normal@Merge[#->#2&@@@l1, Min]
(* {{1, 1.43*10^-46}, {21, 2.79*10^-48}, {41, 1.74*10^-46}, {81, 2.77*10^-46}, 
    {121, 9.97*10^-48}, {161, 1.24*10^-45}, {181, 1.19*10^-45}} *)