Count Unreachable Pairs of Nodes in Undirected Graph
A medium Graph problem included in Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 1
- Core for
- 11 roles
- Platform
- LeetCode
The problem
Given n nodes labeled from 0 to n-1 and a list of undirected edges, return the number of pairs of different nodes that are unreachable from each other.
Example 1
- Input
- n = 3, edges = [[0,1],[0,2]]
- Output
- 0
- Why
- All nodes are reachable from each other, so no unreachable pairs.
Example 2
- Input
- n = 7, edges = [[0,2],[0,5],[2,4],[1,6],[5,4]]
- Output
- 14
- Why
- Components are {0,2,4,5}, {1,6}, {3}. Unreachable pairs: 4*2+4*1+2*1=14.
Constraints
- 1 <= n <= 10^5
- 0 <= edges.length <= 2 * 10^5
How to think about it
Updated 2026-09-09Total pairs of distinct nodes is `n * (n - 1) / 2`. Within any connected component of size s, all `s * (s - 1) / 2` pairs are reachable from each other. Subtracting the reachable pairs of every component from the total pairs yields the number of unreachable pairs in a single arithmetic pass.
Approaches, worst first
Connected component sizes with combinatorial subtraction
time O(V + E) · space O(V + E)
Find sizes of all connected components using DFS or BFS. Total possible pairs is `n * (n - 1) / 2`. Subtract `size * (size - 1) / 2` for each component to find unreachable pairs. BigInt or 64-bit integers prevent overflow for n up to 10^5.
Running suffix sum multiplicationWrite this one
time O(V + E) · space O(V + E)
Gather component sizes. Iterate through each component of size s, multiplying s by `remainingNodes - s` and adding to total, then decrementing remainingNodes by s.
Where people lose marks · 3
- Integer overflow: with n = 10^5, the total number of pairs can exceed 5 * 10^9, overflowing standard 32-bit signed integers.
- Isolated nodes with no edges are components of size 1 and must be included in the calculation.
- When all nodes form a single connected component, the answer is 0 unreachable pairs.
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.
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