Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree — LeetCode 1489 Python Solution

HardUnion FindGraphMinimum Spanning TreeSortingStrongly Connected Component

Problem: #1489
Pattern: Union-Find
Reading time: 7 min
Source: leetcode.com

The problem

Given a weighted undirected connected graph with n vertices numbered from 0 to n - 1, and an array edges where edges[i] = [ai, bi, weighti] represents a bidirectional and weighted edge between nodes ai and bi. A minimum spanning tree (MST) is a subset of the graph's edges that connects all vertices without cycles and with the minimum possible total edge weight.

Example

Input: n = 5, edges = [[0,1,1],[1,2,1],[2,3,2],[0,3,2],[0,4,3],[3,4,3],[1,4,6]]
Output: [[0,1],[2,3,4,5]]
Explanation: The figure above describes the graph.

Python solution

Python

class UnionFind:
    def __init__(self, n):
        self.p = list(range(n))
        self.n = n

    def union(self, a, b):
        if self.find(a) == self.find(b):
            return False
        self.p[self.find(a)] = self.find(b)
        self.n -= 1
        return True

    def find(self, x):
        if self.p[x] != x:
            self.p[x] = self.find(self.p[x])
        return self.p[x]


class Solution:
    def findCriticalAndPseudoCriticalEdges(
        self, n: int, edges: List[List[int]]
    ) -> List[List[int]]:
        for i, e in enumerate(edges):
            e.append(i)
        edges.sort(key=lambda x: x[2])
        uf = UnionFind(n)
        v = sum(w for f, t, w, _ in edges if uf.union(f, t))
        ans = [[], []]
        for f, t, w, i in edges:
            uf = UnionFind(n)
            k = sum(z for x, y, z, j in edges if j != i and uf.union(x, y))
            if uf.n > 1 or (uf.n == 1 and k > v):
                ans[0].append(i)
                continue

            uf = UnionFind(n)
            uf.union(f, t)
            k = w + sum(z for x, y, z, j in edges if j != i and uf.union(x, y))
            if k == v:
                ans[1].append(i)
        return ans

Complexity

Measure	Complexity
Time	O(V+E)
Space	O(V) auxiliary

Pattern: Union-Find

Merge groups and ask whether two things are connected, both in near-constant time. LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree is filed here because LeetCode tags it Union Find, which is the vocabulary this hub collects.

The union-find guide has the Python template for the pattern and the 83 LeetCode problems that use it.

Frequently asked questions

How hard is LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree?: LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree is rated Hard on LeetCode.
What is the time complexity of LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree?: The Python solution on this page runs in O(V+E).
What is the space complexity of LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree?: The Python solution on this page uses O(V) auxiliary space.
What topics does LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree cover?: LeetCode 1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree is tagged Union Find, Graph, Minimum Spanning Tree, Sorting and Strongly Connected Component on LeetCode.