How to find the longest monotone subsequence of a sequence?

Question

I have a list, say {5,4,10,16,2,1}. I want the longest monotone subsequence. In this case the longest monotone sequence is {5,4,2,1}.

Answer 1

The following function will find the longest ascending or the longest descending monotonic sequence in a list of numbers. When the longest ascending sequence is of the same length as the longest descending sequence, it returns both.

longestMonotonicSeq[data : {_?NumberQ ..}] :=
  Module[{up, down},
    up = LongestOrderedSequence[data, #1 >= #2 &];
    down = LongestOrderedSequence[data, #2 >= #1 &];
    Switch[Sign[Length[up] - Length[down]],
      1, up,
     -1, down,
      0, {up, down}]]

Tests

longestMonotonicSeq[{5, 4, 10, 16, 2, 1}]

{5, 4, 2, 1}

longestMonotonicSeq[{10, 11, 3, 25/2, 2, 1, 20, 42.}]

{10, 11, 25/2, 20, 42.}

longestMonotonicSeq[{10, 3, 16, 2, 1, 17, 20}]

{{3, 16, 17, 20}, {10, 3, 2, 1}}