Pattern this teaches:Union Find
Number of Provinces
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 1
- Core for
- 12 roles
The problem
An n x n matrix records which of n cities are directly linked: entry [i][j] is 1 when city i and city j are connected. Cities joined directly or through other cities form one province. Return how many provinces there are.
Example 1
- Input
- isConnected = [[1,1,0,0],[1,1,0,0],[0,0,1,1],[0,0,1,1]]
- Output
- 2
- Why
- Cities 0-1 form one province, cities 2-3 form another. No connections between the two groups.
Example 2
- Input
- isConnected = [[1,0,1,0],[0,1,0,1],[1,0,1,0],[0,1,0,1]]
- Output
- 2
- Why
- Cities 0 and 2 are connected (province 1). Cities 1 and 3 are connected (province 2).
Constraints
- n == isConnected.length
- 1 <= n <= 200
- isConnected[i][j] is 0 or 1
How to think about it
Updated 2026-09-29Provinces are connected components in disguise, encoded in an n x n adjacency matrix instead of an edge list. Finding the number of provinces means counting how many times an unvisited city must initiate an exhaustive search to sweep all of its transitively connected neighbors.
Approaches, worst first
DFS component sweep
time O(n^2) · space O(n)
Maintain a boolean array `visited` of length n. Loop i from 0 to n - 1: if city i is not visited, increment province count and recursively mark all j where `isConnected[i][j] === 1`.
Disjoint Set UnionWrite this one
time O(n^2 * α(n)) · space O(n)
Initialize DSU with count = n. Iterate over upper triangular indices 0 <= i < j < n: whenever `isConnected[i][j] === 1`, union i and j and decrement count if their sets were distinct.
Where people lose marks · 3
- Iterating over the adjacency matrix requires checking `isConnected[i][j] === 1 && i !== j` to avoid processing self-connections.
- Assuming symmetric matrix reading requires checking both directions; reading only upper triangle is sufficient for DSU but DFS requires full row checks.
- Confusing 0-indexed matrix dimensions with 1-based counts causes off-by-one loops that omit city 0 or trigger index out-of-bounds crashes.
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.
More Graph problems
Target Roles
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