BestBox V2

BestBox

Create a data structure for each box

* Data about the box itself (x,y,z) (name?) 
* bestDepth (initialize to 1)
* bestBox (initialize to null)

Sort Boxes ascending by size (x * y * z) (NLogN)

Starting with the smallest box: 
    Compare backwards with all smaller boxes. 
        Create a temporary list of smaller boxes that will fit inside it.
        Pick one of the smaller boxes with the highest bestDepth. 
           This box is your bestBox. Your bestDepth is its bestDepth + 1.
               (Those values are guaranteed to be filled in correctly because we are proceeding from smaller boxes).
        If no boxes can fit inside leave bestDepth at 1 and bestBox at null

// You only calculate each box once in this method. 
// If you revisit a box you only have to use the already calculated values. N calculations, each having to look at most N other boxes' values (which are already calculated).
// So This is O(N^2). 

Find the box with the biggest bestDepth. This is your best box. To find the complete chain, iterate down through the bestBox value. (O(N) for each of these steps)

O(N^2) + 2(O(N)) = O(N^2)