# Symmetric list with maximum in center

How to sort a list, e.g.

RandomInteger[{0, 100}, 50]


that the Maximum is in the center and all other elements are placed symetrically in descending order around the center?

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One way:

SeedRandom[1];

#[[1 ;; ;; 2]] ~Join~ Reverse[#[[2 ;; ;; 2]]] &@
Sort[RandomInteger[{0, 100}, 50]]
(*
{0, 3, 4, 10, 15, 18, 24, 26, 29, 30, 38, 43, 44, 47, 54, 65, 68, 68, \
70, 74, 76, 83, 86, 93, 100, 100, 97, 90, 86, 80, 75, 73, 69, 68, 67, \
56, 48, 45, 44, 43, 33, 30, 28, 25, 23, 17, 14, 6, 4, 1}
*)


Edit: Or, following @Mr.Wizard's suggestion,

#[[1 ;; ;; 2]] ~Join~ #[[Floor[Length[#], 2] ;; 1 ;; -2]] &@
Sort[RandomInteger[{0, 100}, 50]]

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Both shorter and faster: #[[;; ;; 2]] ~Join~ #[[-2 ;; 1 ;; -2]] & – Mr.Wizard Jul 20 '14 at 15:16
@Mr.Wizard I guess I never thought about spanning backwards -- Thanks! – Michael E2 Jul 20 '14 at 15:18
Yeah, it's a nice trick for reversing, but I think I got the spec wrong the first time, now corrected. Check it, will you? – Mr.Wizard Jul 20 '14 at 15:19
@Mr.Wizard Actually, as I started to edit, I think we have to worry whether the length of the list is odd or even. – Michael E2 Jul 20 '14 at 15:19
@MichaelE2 yes, having to worry about the length being odd or even makes things a lot uglier – acl Jul 20 '14 at 15:26

For practice we could write the same thing as in Michael E2's answer with Range and Extract:

symSort[list_] := Module[
{l = Length@list},
Extract[Sort@list, List /@ Join @@ Range[{1, Floor[l, 2]}, {l, 1}, {2, -2}]]
]


Example:

SeedRandom[1];
symSort[RandomInteger[{0, 100}, 50]]

(* {0,3,4,10,15,18,24,26,29,30,38,43,44,47,54,65,68,68,
70,74,76,83,86,93,100,100,97,90,86,80,75,73,69,68,67,56,48,45,
44,43,33,30,28,25,23,17,14,6,4,1} *)


For lists with an even number of items we could also do

symSort[list_] := Join[First@#, Reverse@Last@#] &@Transpose@Partition[Sort@list, 2]

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Although this isn't quite as fast as Michael's revised method using negative stepping for reversal I find the following satisfyingly short and clean:

Join[#, Reverse @ #][[;; ;; 2]] &


Both methods are still very fast so I'd personally use this one. :-)

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It takes a little thought to see that it is always correct, but it's quite clever and clean, as you say. I believe it uses twice as much memory, but that's probably not much of a practical limitation. +1 – Michael E2 Jul 21 '14 at 13:24