logical-or-of-two-binary-grids-represented-as-quad-trees

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Algorithms

A Binary Matrix is a matrix in which all the elements are either 0 or 1.

Given <code>quadTree1</code> and <code>quadTree2</code>. <code>quadTree1</code> represents a <code>n * n</code> binary matrix and <code>quadTree2</code> represents another <code>n * n</code> binary matrix.

Return a Quad-Tree representing the <code>n * n</code> binary matrix which is the result of logical bitwise OR of the two binary matrixes represented by <code>quadTree1</code> and <code>quadTree2</code>.

Notice that you can assign the value of a node to True or False when <code>isLeaf</code> is False, and both are accepted in the answer.

A Quad-Tree is a tree data structure in which each internal node has exactly four children. Besides, each node has two attributes:

<ul>
	<li><code>val</code>: True if the node represents a grid of 1&#39;s or False if the node represents a grid of 0&#39;s.</li>
	<li><code>isLeaf</code>: True if the node is leaf node on the tree or False if the node has the four children.</li>
</ul>

<pre>
class Node {
 public boolean val;
 public boolean isLeaf;
 public Node topLeft;
 public Node topRight;
 public Node bottomLeft;
 public Node bottomRight;
}</pre>

We can construct a Quad-Tree from a two-dimensional area using the following steps:

<ol>
	<li>If the current grid has the same value (i.e all <code>1&#39;s</code> or all <code>0&#39;s</code>) set <code>isLeaf</code> True and set <code>val</code> to the value of the grid and set the four children to Null and stop.</li>
	<li>If the current grid has different values, set <code>isLeaf</code> to False and set <code>val</code> to any value and divide the current grid into four sub-grids as shown in the photo.</li>
	<li>Recurse for each of the children with the proper sub-grid.</li>
</ol>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/new_top.png" style="width: 777px; height: 181px;" />
If you want to know more about the Quad-Tree, you can refer to the <a href="https://en.wikipedia.org/wiki/Quadtree">wiki</a>.

Quad-Tree format:

The input/output represents the serialized format of a Quad-Tree using level order traversal, where <code>null</code> 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 <code>[isLeaf, val]</code>.

If the value of <code>isLeaf</code> or <code>val</code> is True we represent it as 1 in the list <code>[isLeaf, val]</code> and if the value of <code>isLeaf</code> or <code>val</code> is False we represent it as 0.

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qt1.png" style="width: 550px; height: 196px;" /> <img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qt2.png" style="width: 550px; height: 278px;" />
<pre>
Input: quadTree1 = [[0,1],[1,1],[1,1],[1,0],[1,0]]
, quadTree2 = [[0,1],[1,1],[0,1],[1,1],[1,0],null,null,null,null,[1,0],[1,0],[1,1],[1,1]]
Output: [[0,0],[1,1],[1,1],[1,1],[1,0]]
Explanation: quadTree1 and quadTree2 are shown above. You can see the binary matrix which is represented by each Quad-Tree.
If we apply logical bitwise OR on the two binary matrices we get the binary matrix below which is represented by the result Quad-Tree.
Notice that the binary matrices shown are only for illustration, you don&#39;t have to construct the binary matrix to get the result tree.
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qtr.png" style="width: 777px; height: 222px;" />
</pre>

Example 2:

<pre>
Input: quadTree1 = [[1,0]], quadTree2 = [[1,0]]
Output: [[1,0]]
Explanation: Each tree represents a binary matrix of size 1*1. Each matrix contains only zero.
The resulting matrix is of size 1*1 with also zero.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>quadTree1</code> and <code>quadTree2</code> are both valid Quad-Trees each representing a <code>n * n</code> grid.</li>
	<li><code>n == 2x</code> where <code>0 &lt;= x &lt;= 9</code>.</li>
</ul>

intersect

Logical OR of Two Binary Grids Represented as Quad-Trees

Tree

Divide and Conquer

二进制矩阵中的所有元素不是 0 就是 1 。

给你两个四叉树，<code>quadTree1</code> 和 <code>quadTree2</code>。其中 <code>quadTree1</code> 表示一个 <code>n * n</code> 二进制矩阵，而 <code>quadTree2</code> 表示另一个 <code>n * n</code> 二进制矩阵。

请你返回一个表示 <code>n * n</code> 二进制矩阵的四叉树，它是 <code>quadTree1</code> 和 <code>quadTree2</code> 所表示的两个二进制矩阵进行 按位逻辑或运算 的结果。

注意，当 <code>isLeaf</code> 为 False 时，你可以把 True 或者 False 赋值给节点，两种值都会被判题机制 接受 。

四叉树数据结构中，每个内部节点只有四个子节点。此外，每个节点都有两个属性：

<ul>
	<li><code>val</code>：储存叶子结点所代表的区域的值。1 对应 True，0 对应 False；</li>
	<li><code>isLeaf</code>: 当这个节点是一个叶子结点时为 True，如果它有 4 个子节点则为 False 。</li>
</ul>

<pre>
class Node {
 public boolean val;
&nbsp; &nbsp; public boolean isLeaf;
&nbsp; &nbsp; public Node topLeft;
&nbsp; &nbsp; public Node topRight;
&nbsp; &nbsp; public Node bottomLeft;
&nbsp; &nbsp; public Node bottomRight;
}</pre>

我们可以按以下步骤为二维区域构建四叉树：

<ol>
	<li>如果当前网格的值相同（即，全为 <code>0</code> 或者全为 <code>1</code>），将 <code>isLeaf</code> 设为 True ，将 <code>val</code> 设为网格相应的值，并将四个子节点都设为 Null 然后停止。</li>
	<li>如果当前网格的值不同，将 <code>isLeaf</code> 设为 False， 将 <code>val</code> 设为任意值，然后如下图所示，将当前网格划分为四个子网格。</li>
	<li>使用适当的子网格递归每个子节点。</li>
</ol>

<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/new_top.png" style="height: 181px; width: 777px;" />

如果你想了解更多关于四叉树的内容，可以参考 <a href="https://baike.baidu.com/item/%E5%9B%9B%E5%8F%89%E6%A0%91/8557650">百科</a>。

四叉树格式：

输出为使用层序遍历后四叉树的序列化形式，其中 <code>null</code> 表示路径终止符，其下面不存在节点。

它与二叉树的序列化非常相似。唯一的区别是节点以列表形式表示 <code>[isLeaf, val]</code> 。

如果 <code>isLeaf</code> 或者 <code>val</code> 的值为 True ，则表示它在列表&nbsp;<code>[isLeaf, val]</code> 中的值为 1 ；如果 <code>isLeaf</code> 或者 <code>val</code> 的值为 False ，则表示值为 0 。

&nbsp;

示例 1：

<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qt1.png" style="height: 196px; width: 550px;" /> <img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qt2.png" style="height: 278px; width: 550px;" />

<pre>
输入：quadTree1 = [[0,1],[1,1],[1,1],[1,0],[1,0]]
, quadTree2 = [[0,1],[1,1],[0,1],[1,1],[1,0],null,null,null,null,[1,0],[1,0],[1,1],[1,1]]
输出：[[0,0],[1,1],[1,1],[1,1],[1,0]]
解释：quadTree1 和 quadTree2 如上所示。由四叉树所表示的二进制矩阵也已经给出。
如果我们对这两个矩阵进行按位逻辑或运算，则可以得到下面的二进制矩阵，由一个作为结果的四叉树表示。
注意，我们展示的二进制矩阵仅仅是为了更好地说明题意，你无需构造二进制矩阵来获得结果四叉树。
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/11/qtr.png" style="height: 222px; width: 777px;" />
</pre>

示例 2：

<pre>
输入：quadTree1 = [[1,0]]
, quadTree2 = [[1,0]]
输出：[[1,0]]
解释：两个数所表示的矩阵大小都为 1*1，值全为 0 
结果矩阵大小为 1*1，值全为 0 。
</pre>

&nbsp;

提示：

<ul>
	<li><code>quadTree1</code> 和 <code>quadTree2</code> 都是符合题目要求的四叉树，每个都代表一个 <code>n * n</code> 的矩阵。</li>
	<li><code>n == 2x</code>&nbsp;，其中 <code>0 &lt;= x &lt;= 9</code>.</li>
</ul>