Longest Increasing Subsequence

Question

I am learning dynamic programming and I have written down some code for longest increasing subsequence. I would like to know if there is any case or any area of improvement in the terms of optimization/programming style.

/**
 * LIS = Longest increasing subsequence.
 * Input = [10, 22, 9, 33, 21, 50, 41, 60, 80]
 * Output = [10, 22, 33, 50, 60, 80]
 * Created by gaurav on 1/7/15.
 */


function findSubsequence(arr){
    var allSubsequence = [],
        longestSubsequence = null,
        longestSubsequenceLength = -1;

    for(var i=0;i<arr.length;i++){          //i=1
        var subsequenceForCurrent = [arr[i]],
            current = arr[i],
            lastElementAdded = -1;
        for(var j=i;j<arr.length;j++){
            var subsequent = arr[j];
            if((subsequent > current) && (lastElementAdded<subsequent)){
                subsequenceForCurrent.push(subsequent);
                lastElementAdded = subsequent;
            }
        }
        allSubsequence.push(subsequenceForCurrent);
    }
    for(var i in allSubsequence){
        var subs = allSubsequence[i];
        if(subs.length>longestSubsequenceLength){
            longestSubsequenceLength = subs.length;
            longestSubsequence = subs;
        }
    }
    return longestSubsequence;
}


(function driver(){
    var sample = [87,88,91, 10, 22, 9,92, 94, 33, 21, 50, 41, 60, 80];
    console.log(findSubsequence(sample));
})();

Answer 1

This bit worries me:

lastElementAdded = -1;

It assumes that the minimum value in the array is zero. But really, an increasing sequence could start with -3456780 or something.

I'd use null or something non-numeric instead. Or simply add the first element automatically, and start the loop at index 1. You could use Number.NEGATIVE_INFINITY, but it has the same problem: A sequence could start at negative infinity.

Also, don't use for..in loops on arrays. A for..in loop enumerates an object's properties; it's not intended for iterating through an array's elements. Instead, use a regular for loop, or a forEach iterator.

Lastly, I'd change allSubsequence to allSubsequences (plural) simply because it more grammatically correct.

In terms of overall strategy, your current algorithm is doing a bit of unnecessary work. Given the example input, the first 3 subsequences it finds are:

[ 87, 88, 91, 92, 94 ]
[ 88, 91, 92, 94 ]
[ 91, 92, 94 ]

The last two aren't really interesting, since they're just subsequences of the first.

Now, this really isn't a problem for arrays as short as what you've got here. Still, it's a fun exercise, so I tried my hand at it. There's probably an even more elegant solution than what I'm proposing here, though. Algorithms aren't my strong suit, I'm afraid. But here's what I came up with:

1. Start a sequence with the first element of the input array
2. Iterate through the array
   • If a value is greater than the sequence's maximum, append it to sequence
   • If it's less and it's the first such value we've found, recurse with a subset of the input array, starting at the current index. Store the result.
3. Return whichever sequence - the current one, or the "fork" - is longer

Kinda hard to explain, actually. Hope the code below will help illustrate:

function findLongestIncreasingSequence(array) {
  var sequence = [],
      fork = null;

  // Always add the first value to the sequence
  sequence.push(array[0]);

  // Reduce the array with. Since no initial accumulator is given,
  // the first value in the array is used
  array.reduce(function (previous, current, index) {

    // If the current value is larger than the last, add it to the
    // sequence and return (i.e. check the next value)
    if(current > previous) {
      sequence.push(current);
      return current;
    }

    // If, however, the value is smaller, and we haven't had a fork
    // before, make one now, starting at the current value's index
    if(!fork && current < previous) {
      fork = findLongestIncreasingSequence(array.slice(index));
    }

    // Return the previous value if the current one is less or equal
    return previous;
  });

  // Compare the current sequence's length to the fork's (if any) and
  // return whichever one is larger
  return fork && fork.length > sequence.length ? fork : sequence;
}

Given the example input, you get:

findLongestIncreasingSequence(sample); // => [ 87, 88, 91, 92, 94 ]

Anyway, what it does is more or less this, where √ means the value was added to a sequence, and F means it "forked" and recursed

Initial input: 87  88  91  10  22  9  92  94  33  21  50  41  60  80
====================================================================
1st call:       √   √   √   F   .  .   √   √   .   .   .   .   .   .
2nd call:                   √   √  F   √   √   .   .   .   .   .   .
3rd call:                          √   √   √   F   .   .   .   .   .
4th call:                                      √   F   √   .   √   √
5th call:                                          √   √   F   √   √
6th call:                                                  √   √   √

Unfortunately, it's not tail-recursive, so call stack depth could be an issue.

And there's probably more optimizations that can be made. For instance, if the current sequence is already 5 items long, there's no reason to fork if the array only has 4 items left. And as I said, there's probably an even cleverer solution overall.

The alternative to the above would be to simply map out the indices at which a value is lower than the one preceding it. Then do the same thing (slice the array at those indices, find increasing sequence) one at a time instead of recursively, and compare sequence lengths at the end. Same result. But recursion is more fun :P

Answer 2

You might easily do this job by a very simple tail call optimized recursive function or by Array.prototype.reduce() functor.

function lis(a, r = [a[0]]){
  if(!a.length) return r;
  a.splice(0,1);
  r[r.length-1] < a[0] && r.push(a[0]);
  return lis(a,r);
}  
var arr = [-7, -10, 6, 22, 9, 33, 21, 50, 41, 60, 80 ];
console.log(lis(arr));

Or like

var arr = [-7, -10, 6, 22, 9, 33, 21, 50, 41, 60, 80 ],
    lis = arr.reduce((p,c,i) => i ? p[p.length-1] < c ? p.concat(c)
                                                      : p
                                  : [c] ,[]);
console.log(lis);