How do you solve Cycle Length Queries in a Tree on LeetCode?

Cycle Length Queries in a Tree (LeetCode #2509) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(1) space complexity. Key insight: Understanding the properties of binary trees helps in efficiently finding the cycle length.

What is the brute force solution for Cycle Length Queries in a Tree?

The brute force approach for Cycle Length Queries in a Tree is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves constructing the tree and finding the cycle length by traversing the nodes. This is straightforward but inefficient, especially for larger trees. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Cycle Length Queries in a Tree?

Cycle Length Queries in a Tree uses the following concepts: Array, Tree, Binary Tree. The recommended patterns to study are: Lowest Common Ancestor (LCA), Binary Tree Properties.

What are the interview tips for Cycle Length Queries in a Tree?

Always consider the properties of the data structure you're working with. Practice finding LCA in trees, as it's a common problem in interviews. Think about how to reduce the problem size — can you avoid full tree traversal?

What are common mistakes when solving Cycle Length Queries in a Tree?

Not recognizing that the tree structure allows for efficient depth calculations. Overcomplicating the cycle length calculation by trying to construct the tree.

#2509

Cycle Length Queries in a Tree

Hard

ArrayTreeBinary TreeLowest Common Ancestor (LCA)Binary Tree Properties

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(1)

💡

Intuition

Time O(n)Space O(1)

The optimal solution leverages the properties of binary trees and the concept of the Lowest Common Ancestor (LCA) to efficiently calculate the cycle length without constructing the tree explicitly.

⚙️

Algorithm

3 steps

1Step 1: For each query, find the depth of nodes a and b.
2Step 2: Find the Lowest Common Ancestor (LCA) of nodes a and b.
3Step 3: Calculate the cycle length using the formula: cycle_length = depth(a) + depth(b) - 2 * depth(LCA(a, b)) + 2.

solution.py21 lines

1class Solution:
2    def cycleLengthQueries(self, n, queries):
3        def depth(x):
4            return (x).bit_length() - 1
5        def lca(a, b):
6            while a != b:
7                if a > b:
8                    a //= 2
9                else:
10                    b //= 2
11            return a
12        results = []
13        for a, b in queries:
14            d_a = depth(a)
15            d_b = depth(b)
16            lca_node = lca(a, b)
17            d_lca = depth(lca_node)
18            cycle_length = d_a + d_b - 2 * d_lca + 2
19            results.append(cycle_length)
20        return results
21

ℹ

Complexity note: The optimal solution runs in O(n) time for each query due to the logarithmic depth calculations and the LCA finding process, making it efficient for larger trees.

1Understanding the properties of binary trees helps in efficiently finding the cycle length.
2The Lowest Common Ancestor (LCA) is crucial for calculating distances in trees.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.