How do you solve Incremental Even-Weighted Cycle Queries on LeetCode?

Incremental Even-Weighted Cycle Queries (LeetCode #3887) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n α(n)) time complexity and O(n) space complexity. Key insight: Cycles must maintain even weight sums.

What is the time complexity of Incremental Even-Weighted Cycle Queries?

The optimal solution for Incremental Even-Weighted Cycle Queries runs in O(n α(n)) time and uses O(n) space. Union-Find operations are nearly constant time, leading to efficient edge processing.

What is the brute force solution for Incremental Even-Weighted Cycle Queries?

The brute force approach for Incremental Even-Weighted Cycle Queries is Brute Force, with O(n²) time complexity and O(1) space complexity. Check every edge and see if adding it creates an odd-weight cycle. This method is straightforward but inefficient. The optimal Optimal Solution approach improves this to O(n α(n)).

What algorithm or data structure is used to solve Incremental Even-Weighted Cycle Queries?

Incremental Even-Weighted Cycle Queries uses the following concepts: Union-Find, Graph Theory. The recommended patterns to study are: Union-Find, Graph Traversal.

What are the interview tips for Incremental Even-Weighted Cycle Queries?

Explain your thought process clearly. Discuss edge cases and constraints. Practice similar graph problems.

What are common mistakes when solving Incremental Even-Weighted Cycle Queries?

Not tracking parity correctly. Ignoring cycle checks before union.

#3887

Incremental Even-Weighted Cycle Queries

Hard

Union-FindGraph TheoryUnion-FindGraph Traversal

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

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

💡

Intuition

Time O(n α(n))Space O(n)

Use Union-Find to manage connected components and track parity. Each node can represent a bit, ensuring cycles maintain even weights.

⚙️

Algorithm

3 steps

1Step 1: Initialize a Union-Find structure to manage components and their parities.
2Step 2: For each edge, check if it can be added without creating an odd-weight cycle using the Union-Find structure.
3Step 3: If it can be added, union the nodes and update their parities accordingly.

solution.py33 lines

1class UnionFind:
2    def __init__(self, n):
3        self.parent = list(range(n))
4        self.rank = [0] * n
5        self.parity = [0] * n
6    def find(self, x):
7        if self.parent[x] != x:
8            orig_parent = self.parent[x]
9            self.parent[x] = self.find(self.parent[x])
10            self.parity[x] ^= self.parity[orig_parent]
11        return self.parent[x]
12    def union(self, x, y, weight):
13        rootX = self.find(x)
14        rootY = self.find(y)
15        if rootX != rootY:
16            if self.rank[rootX] > self.rank[rootY]:
17                self.parent[rootY] = rootX
18                self.parity[rootY] = self.parity[x] ^ self.parity[y] ^ weight
19            else:
20                self.parent[rootX] = rootY
21                self.parity[rootX] = self.parity[x] ^ self.parity[y] ^ weight
22                if self.rank[rootX] == self.rank[rootY]:
23                    self.rank[rootY] += 1
24    def add_edges_optimal(n, edges):
25        uf = UnionFind(n)
26        count = 0
27        for u, v, w in edges:
28            if uf.find(u) != uf.find(v):
29                uf.union(u, v, w)
30                count += 1
31            elif (uf.parity[u] ^ uf.parity[v] ^ w) != 0:
32                continue
33        return count

ℹ

Complexity note: Union-Find operations are nearly constant time, leading to efficient edge processing.

1Cycles must maintain even weight sums.
2Union-Find helps manage components and parity efficiently.

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