1803

Author: lzl124631x

leetcode/1803. Count Pairs With XOR in a Range/README.md

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+# [1803. Count Pairs With XOR in a Range (Hard)](https://leetcode.com/problems/count-pairs-with-xor-in-a-range/)
+
+<p>Given a <strong>(0-indexed)</strong> integer array <code>nums</code> and two integers <code>low</code> and <code>high</code>, return <em>the number of <strong>nice pairs</strong></em>.</p>
+
+<p>A <strong>nice pair</strong> is a pair <code>(i, j)</code> where <code>0 &lt;= i &lt; j &lt; nums.length</code> and <code>low &lt;= (nums[i] XOR nums[j]) &lt;= high</code>.</p>
+
+<p>&nbsp;</p>
+<p><strong>Example 1:</strong></p>
+
+<pre><strong>Input:</strong> nums = [1,4,2,7], low = 2, high = 6
+<strong>Output:</strong> 6
+<strong>Explanation:</strong> All nice pairs (i, j) are as follows:
+    - (0, 1): nums[0] XOR nums[1] = 5 
+    - (0, 2): nums[0] XOR nums[2] = 3
+    - (0, 3): nums[0] XOR nums[3] = 6
+    - (1, 2): nums[1] XOR nums[2] = 6
+    - (1, 3): nums[1] XOR nums[3] = 3
+    - (2, 3): nums[2] XOR nums[3] = 5
+</pre>
+
+<p><strong>Example 2:</strong></p>
+
+<pre><strong>Input:</strong> nums = [9,8,4,2,1], low = 5, high = 14
+<strong>Output:</strong> 8
+<strong>Explanation:</strong> All nice pairs (i, j) are as follows:
+​​​​​    - (0, 2): nums[0] XOR nums[2] = 13
+&nbsp;   - (0, 3): nums[0] XOR nums[3] = 11
+&nbsp;   - (0, 4): nums[0] XOR nums[4] = 8
+&nbsp;   - (1, 2): nums[1] XOR nums[2] = 12
+&nbsp;   - (1, 3): nums[1] XOR nums[3] = 10
+&nbsp;   - (1, 4): nums[1] XOR nums[4] = 9
+&nbsp;   - (2, 3): nums[2] XOR nums[3] = 6
+&nbsp;   - (2, 4): nums[2] XOR nums[4] = 5</pre>
+
+<p>&nbsp;</p>
+<p><strong>Constraints:</strong></p>
+
+<ul>
+	<li><code>1 &lt;= nums.length &lt;= 2 * 10<sup>4</sup></code></li>
+	<li><code>1 &lt;= nums[i] &lt;= 2 * 10<sup>4</sup></code></li>
+	<li><code>1 &lt;= low &lt;= high &lt;= 2 * 10<sup>4</sup></code></li>
+</ul>
+
+
+**Related Topics**:  
+[Trie](https://leetcode.com/tag/trie/)
+
+## Solution 1. Trie
+
+### Complexity Analysis
+
+Adding a number into trie takes `O(16) = O(1)` time.
+
+The `count` will at most visit a perfect binary-tree like trie whose height is `16`, so there are at most `O(2^16-1) = O(1)` nodes. Hence the space complexity is `O(1)`.
+
+Since we repeat the above operations `N` times, the overall time complexity is `O(N)`.
+
+```cpp
+// OJ: https://leetcode.com/problems/count-pairs-with-xor-in-a-range/
+// Author: github.com/lzl124631x
+// Time: O(N)
+// Space: O(1)
+struct TrieNode {
+    TrieNode *next[2] = {};
+    int cnt = 0;
+};
+class Solution {
+    void add(TrieNode *node, int n) {
+        for (int i = 15; i >= 0; --i) {
+            int b = n >> i & 1;
+            if (node->next[b] == NULL) node->next[b] = new TrieNode();
+            node = node->next[b];
+            node->cnt++;
+        }
+    }
+    int count(TrieNode *node, int i, int n, int rl, int rh, int low, int high) {
+        if (rl >= low && rh <= high) return node->cnt;
+        if (rh < low || rl > high) return 0;
+        int b = n >> i & 1, r = b ^ 1, mask = ~(1 << i);
+        return (node->next[0] ? count(node->next[0], i - 1, n, rl & mask | (b << i), rh & mask | (b << i), low, high) : 0)
+            + (node->next[1] ? count(node->next[1], i - 1, n, rl & mask | (r << i), rh & mask | (r << i), low, high) : 0);
+    }
+public:
+    int countPairs(vector<int>& A, int low, int high) {
+        TrieNode root;
+        int ans = 0;
+        for (int n : A) {
+            ans += count(&root, 15, n, 0, (1 << 16) - 1, low, high);
+            add(&root, n);
+        }
+        return ans;
+    }
+};
+```