How do I find the elements in a list that return the highest value for a function?

Question

Suppose we have a list of values and a function f. I want to find which of the elements maximizes the return value of the function in Mathematica. Let call the function ListMaxArg. On the following example,

ListMaxArg[Total, {{1, 2, 3}, {10}}]

it should return {10}. Is there any such function in Mathematica's library? If not, what is the simplest way to write it? I can for sure write the function using a loop, but I am looking for something more functional style.

Answer 1

While using Ordering as in Mr.Wizard's answer will most likely be the fastest non-compiled solution, it will not return all possible arguments that maximize the return value. Here's a simple way of writing a function that does this:

listMaxArg[f_, list_] := With[{max = f /@ list // Max}, Select[list, f@# == max &]]

Here's an example that compares the two solutions:

a = {{1, 2, 3}, {10}, {6, 4}};
listMaxArg[Total, a] 
(* Out[1]= {{10}, {6, 4}} *)

listMaxArgMrWiz[Total, a]
(*Out[2]= {{6, 4}} *)

Answer 2

If I understand the question this should be fastest:

listMaxArg[f_, L_List] := L ~Extract~ Ordering[f /@ L, -1]

listMaxArg[Total, {{1, 2, 3}, {10}}]

{10}

You could also find the top n values by using -n as the second argument to Ordering.
This could be included in the function, e.g. listMaxArg[f_, L_List, n_Integer] := . . .

---

This method should be fast for finding multiple maximum values:

listMaxArg[f_, L_List] := L ~Extract~ Position[#, Max@#] &[f /@ L]

listMaxArg[Total, {{1, 2, 3}, {10}, {6, 2}, {3, 7}}]

{{10}, {3, 7}}

---

I argue the superiority of the Extract-Position method (from Arnoud Buzing) over Select on the basis of timings. I will use Tr in the place of Total as it is faster on Packed Arrays, and therefore better shows the overhead of each method.

listMaxArgRM[f_, list_] :=
  With[{max = f /@ list // Max}, Select[list, f@# == max &]]

listMaxArgMrW[f_, L_List] :=
  L ~Extract~ Position[#, Max@#] &[f /@ L]

SeedRandom[1]
list = RandomInteger[7, #] & /@ RandomInteger[{1, 5}, 1000000];

r1 = listMaxArgRM[Tr, list]; // AbsoluteTiming
r2 = listMaxArgMrW[Tr, list]; // AbsoluteTiming
r1 === r2

{0.6770387, Null}

{0.2080119, Null}

True

As you can see listMaxArgMrW is over three times as efficient as listMaxArgRM, at least in version 7.