Kruskal's Algorithm (MST)
A medium Graph problem included in Love Babbar 450, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 2
- Core for
- 11 roles
- Platform
- GeeksforGeeks
The problem
Given a weighted undirected graph with n vertices and a list of edges [u, v, weight], find the minimum spanning tree using Kruskal's algorithm and return the total weight of the MST.
Example 1
- Input
- n = 4, edges = [[0,1,1],[0,2,4],[1,2,2],[1,3,6],[2,3,3]]
- Output
- 6
- Why
- Kruskal sorts edges by weight and picks (0,1)=1, (1,2)=2, (2,3)=3 without forming cycles. Total = 6.
Example 2
- Input
- n = 3, edges = [[0,1,5],[1,2,3],[0,2,1]]
- Output
- 4
- Why
- Sorted by weight: (0,2)=1, (1,2)=3, (0,1)=5. Pick first two for MST. Total = 4.
Constraints
- 1 <= n <= 100
- 0 <= edges.length <= n * (n - 1) / 2
- 1 <= edge weight <= 1000
How to think about it
Updated 2026-09-09Sort all edges globally by weight. Greedily accept edges from lightest to heaviest: if an edge connects two nodes that are not yet connected, it is guaranteed to belong to some minimum spanning forest. Use disjoint set union to check connectivity in near constant time and stop after picking n - 1 edges.
Approaches, worst first
Edge sort with DFS cycle checking
time O(E log E + E * V) · space O(V + E)
Sort all edges by weight in ascending order. Maintain an adjacency list for the growing tree. Before adding each candidate edge `(u, v)`, run a full DFS from u to verify whether v is already reachable. Running a complete graph traversal per candidate edge incurs O(E * V) time.
Global sort with DSU cycle preventionWrite this one
time O(E log E) · space O(V)
Sort all edges ascending by weight. Initialize DSU for n vertices. For each edge `(u, v, w)`, call `find(u)` and `find(v)`. If they differ, unite them, add w to MST weight sum, and increment edge count. Break early when n - 1 edges have been included.
Where people lose marks · 3
- Neglecting to break early once n - 1 edges are selected forces unnecessary DSU lookups on remaining edges.
- Sorting edges in descending order builds a maximum spanning tree instead of a minimum spanning tree.
- When the graph has fewer than n - 1 edges total, spanning the graph is impossible.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
Companies that have asked it
Tags taken from the problem's own GeeksforGeeks page — not a copied list.
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