Construct Quad Tree
Problem Description
Given a n * n matrix grid of 0's and 1's only. We want to represent grid with a Quad-Tree.
Return the root of the Quad-Tree representing grid.
A Quad-Tree is a tree data structure in which each internal node has exactly four children. Besides, each node has two attributes:
val: True if the node represents a grid of 1's or False if the node represents a grid of 0's. Notice that you can assign thevalto True or False whenisLeafis False, and both are accepted in the answer.isLeaf: True if the node is a leaf node on the tree or False if the node has four children.
We can construct a Quad-Tree from a two-dimensional area using the following steps:
- If the current grid has the same value (i.e all
1'sor all0's) setisLeafTrue and setvalto the value of the grid and set the four children to Null and stop. - If the current grid has different values, set
isLeafto False and setvalto any value and divide the current grid into four sub-grids as shown in the photo. - Recurse for each of the children with the proper sub-grid.
If you want to know more about the Quad-Tree, you can refer to the wiki.
Quad-Tree format:
You don't need to read this section for solving the problem. This is only if you want to understand the output format here. The output represents the serialized format of a Quad-Tree using level order traversal, where null signifies a path terminator where no node exists below.
It is very similar to the serialization of the binary tree. The only difference is that the node is represented as a list [isLeaf, val].
If the value of isLeaf or val is True we represent it as 1 in the list [isLeaf, val] and if the value of isLeaf or val is False we represent it as 0.
Example 1:
Input: grid = [[0,1],[1,0]]
Output: [[0,1],[1,0],[1,1],[1,1],[1,0]]
Explanation: The explanation of this example is shown below:
Notice that 0 represents False and 1 represents True in the photo representing the Quad-Tree.
Example 2:
Input: grid = [[1,1,1,1,0,0,0,0],[1,1,1,1,0,0,0,0],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1],[1,1,1,1,0,0,0,0],[1,1,1,1,0,0,0,0],[1,1,1,1,0,0,0,0],[1,1,1,1,0,0,0,0]]
Output: [[0,1],[1,1],[0,1],[1,1],[1,0],null,null,null,null,[1,0],[1,0],[1,1],[1,1]]
Explanation: All values in the grid are not the same. We divide the grid into four sub-grids.
The topLeft, bottomLeft and bottomRight each has the same value.
The topRight have different values so we divide it into 4 sub-grids where each has the same value.
Explanation is shown in the photo below:
Constraints:
n == grid.length == grid[i].lengthn == 2xwhere0 <= x <= 6
Solve it on LeetCode
Approaches
1. Recursive Approach
Intuition:
The quad tree is a specialized tree used to partition a two-dimensional space by recursively subdividing it into four quadrants or regions. To construct a quad tree from a grid, we recursively divide the grid into four quadrants until each quadrant is a uniform section. If a section is not uniform (contains both 0s and 1s), we continue dividing it; otherwise, we create a leaf node.
Code:
- Time Complexity: O(N^2), where N is the length of the grid's side. In the worst case, we might check each cell to determine if a section is uniform.
- Space Complexity: O(log(N)), as the stack space used by the recursive calls at worst will be log of N due to the depth of the recursive tree.
2. Recursive Approach with Optimization
Intuition:
To optimize, instead of checking for uniform sections separately, integrate this directly into the recursive logic to avoid redundant checks. If during recursive splitting, all four quadrants return leaf nodes with the same value, we can merge them into a single leaf node.
Code:
- Time Complexity: O(N^2), similar to the basic approach, but avoids redundant checks.
- Space Complexity: O(log(N)), due to recursive stack similar to the basic approach.