On an infinite number line (x-axis), we drop given squares in the order they are given.
The i-th square dropped (positions[i] = (left, side_length)) is a square with the left-most point being positions[i][0] and sidelength positions[i][1].
The square is dropped with the bottom edge parallel to the number line, and from a higher height than all currently landed squares.  We wait for each square to stick before dropping the next.
The squares are infinitely sticky on their bottom edge, and will remain fixed to any positive length surface they touch (either the number line or another square).  Squares dropped adjacent to each other will not stick together prematurely.


Return a list ans of heights.  Each height ans[i] represents the current highest height of any square we have dropped, after dropping squares represented by positions[0], positions[1], ..., positions[i].


Example 1:

Input: [[1, 2], [2, 3], [6, 1]]
Output: [2, 5, 5]
Explanation:

After the first drop of positions[0] = [1, 2]:
_aa
_aa
-------
The maximum height of any square is 2.

After the second drop of positions[1] = [2, 3]:
__aaa
__aaa
__aaa
_aa__
_aa__
--------------
The maximum height of any square is 5.  
The larger square stays on top of the smaller square despite where its center
of gravity is, because squares are infinitely sticky on their bottom edge.

After the third drop of positions[1] = [6, 1]:
__aaa
__aaa
__aaa
_aa
_aa___a
--------------
The maximum height of any square is still 5.

Thus, we return an answer of [2, 5, 5].





Example 2:

Input: [[100, 100], [200, 100]]
Output: [100, 100]
Explanation: Adjacent squares don't get stuck prematurely - only their bottom edge can stick to surfaces.



Note:
1 .
1 .
1 .

Difficulty: interview

Starter code:

class Solution:
    def fallingSquares(self, positions: List[List[int]]) -> List[int]:
        

Solutions:
class Solution:
     def fallingSquares(self, positions):
         """
         :type positions: List[List[int]]
         :rtype: List[int]
         """
         height = [0]
         pos = [0]
         res = []
         max_h = 0
         for left, side in positions:
             i = bisect.bisect_right(pos, left)
             j = bisect.bisect_left(pos, left + side)
             high = max(height[i - 1:j] or [0]) + side
             pos[i:j] = [left, left + side]
             height[i:j] = [high, height[j - 1]]
             max_h = max(max_h, high)
             res.append(max_h)
         return res        

class Solution:
     def fallingSquares(self, positions):
         """
         :type positions: List[List[int]]
         :rtype: List[int]
         """
         to_cor = {}
         for x, l in positions:
             to_cor[x] = 1
             to_cor[x+l] = 1
         for i, v in enumerate(sorted(to_cor.keys())):
             to_cor[v] = i
         over = Overlapper(len(to_cor))
         ans = []
         mx = 0
         for x, l in positions:
             tmx = over.getMax(to_cor[x], to_cor[x+l])+l
             mx = max(mx, tmx)
             ans.append(mx)
             over.Update(to_cor[x], to_cor[x+l], tmx)
         return ans
         
         
 class Overlapper:
     def __init__(self, n):
         self.line = [0]*n
     
     def getMax(self, a, b):
         return max(self.line[a:b])
     
     def Update(self, a, b, mx):
         self.line[a:b] = [mx]*(b-a)

class Solution:
     def fallingSquares(self, positions):
         """
         :type positions: List[List[int]]
         :rtype: List[int]
         
         COORDINATE COMPRESSION IS THE WAY TO GO - CHECKOUT OFFICAL LEETCODE ANSWER.  THIS WORKS TOO.
         """
 #         if positions == []: return []
 #         def overlap(int1, int2):
 #             return not (int1[0] > int2[1] or int1[1] < int2[0])
 
 # #       sorted disjoint interval list
 #         ints = [positions[0][0], positions[0][0]+positions[0][1]-1] # forgot to make this x+y-1 - wrote y
 #         intheights = {(positions[0][0], positions[0][0]+positions[0][1]-1): positions[0][1]} # same as above
 #         maxheights = [positions[0][1]]
         
 #         for i in range(1, len(positions)):
 #             # print(positions[i], ints, intheights)
 #             start2 = positions[i][0]
 #             end2 = positions[i][0] + positions[i][1] - 1
             
 #             leftpos = bisect.bisect_left(ints, start2)
 #             rightpos = bisect.bisect_right(ints, end2)
 #             newints = ints[:(leftpos // 2 * 2)]
 #             if leftpos % 2 == 1:
 #                 newints.append(ints[leftpos-1])
 #                 newints.append(start2-1)
 #                 intheights[(ints[leftpos-1], start2-1)] = intheights[(ints[leftpos-1], ints[leftpos])]
             
 #             left = leftpos // 2 * 2
 #             right = (rightpos + 1) // 2 * 2
 #             maxSoFar = 0
 #             lastheight = 0
 #             while left < right:
 #                 maxSoFar = max(maxSoFar, intheights[(ints[left], ints[left+1])])
 #                 lastheight = intheights[(ints[left], ints[left+1])]
 #                 del intheights[(ints[left], ints[left+1])]
 #                 left += 2
             
 #             newints.append(start2)
 #             newints.append(end2)
 #             intheights[(start2, end2)] = maxSoFar + (end2-start2+1)
 #             maxheight = max(maxheights[-1], intheights[(start2, end2)])
 #             maxheights.append(maxheight)
 
 #             if rightpos % 2 == 1:
 #                 newints.append(end2+1)
 #                 newints.append(ints[rightpos])
 #                 intheights[(end2+1, ints[rightpos])] = lastheight
 #             newints.extend(ints[right:])  # forgot this line
 #             ints = newints
             
             
 #         return maxheights
         ints = []
         for left, size in positions:
             ints.append(left)
             ints.append(left+size-1)
         
         index = {num:idx for idx, num in enumerate(sorted(ints))}
         heights = [0]*2*len(positions)
         
         def query(left, right):
             return max(heights[index[left]: index[right]+1])
         
         def updateHeights(left, right, newheight):
             for i in range(index[left], index[right]+1):
 #                 no need for max here.  You are updating unless the height is negative
                 heights[i] = newheight
         
         maxheights = []
         for left, size in positions:
             right = left+size -1
             newheight = query(left, right) + size
             updateHeights(left, right, newheight)
             if not maxheights or maxheights[-1] <= newheight:
                 maxheights.append(newheight)
             else:
                 maxheights.append(maxheights[-1])
         
         return maxheights