Kruskal's Algorithm
Kruskal builds a minimum spanning tree by sorting edges by weight and taking the next cheapest edge that does not create a cycle.
Start with this mental model
Kruskal builds a minimum spanning tree by sorting edges by weight and taking the next cheapest edge that does not create a cycle.
How it works, without unnecessary jargon
Sort edges ascending.
Use Union-Find to test whether the two endpoints are already connected; if not, add the edge and union the components.
Read the algorithm as a sequence of decisions
Define directed/undirected and weighted/unweighted assumptions.
Choose the graph representation.
Track algorithm state and handle disconnected/cyclic cases explicitly.
Read a tiny input by hand and write down the state after every important operation. For Kruskal's Algorithm, the goal is to make the invariant visible rather than to memorize code.
O(E log E)
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
Good fit
- Use the algorithm only when its graph assumptions match the problem.
What usually goes wrong
- Forgetting disconnected components.
- Mixing weighted and unweighted shortest-path algorithms.
- Not protecting against revisits/cycles.
A professional answer shape
Connect Kruskal to Union-Find. The sorting step dominates, typically O(E log E).
What problem does Kruskal's Algorithm solve?
Kruskal builds a minimum spanning tree by sorting edges by weight and taking the next cheapest edge that does not create a cycle.
What should I mention about complexity?
Its typical cost here is O(E log E), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
What is the most common mistake?
Forgetting disconnected components.