Pattern this teaches:Union Find
Redundant Connection
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 2
- Core for
- 12 roles
The problem
Given an undirected graph that started as a tree with n nodes but has one extra edge added, find and return the edge that creates a cycle. The edges are given as pairs [u, v].
Example 1
- Input
- edges = [[1,2],[2,3],[3,4],[4,5],[5,1]]
- Output
- [5,1]
- Why
- Edges form a cycle 1-2-3-4-5-1. The edge [5,1] closes the cycle and is the redundant one.
Example 2
- Input
- edges = [[1,2],[1,3],[1,4],[2,5],[3,5]]
- Output
- [3,5]
- Why
- Edges form a cycle 1-2-5-3-1. The edge [3,5] closes the cycle and is the redundant one.
Constraints
- 1 <= edges.length <= 1000
- 1 <= edges[i][0], edges[i][1] <= edges.length
How to think about it
Updated 2026-09-29A tree with n nodes starts with zero edges and gains connectivity one edge at a time without ever closing a loop. As you stream through edges from left to right, test if both endpoints already share the same component. The first edge connecting two nodes that are already in the same set closes a cycle and must be returned.
Approaches, worst first
DFS reachability per edge
time O(n^2) · space O(n)
Maintain an adjacency list. For every edge [u, v], run a DFS from u to check if v is already reachable. If so, return [u, v]; otherwise add the undirected edge to the adjacency list.
Disjoint Set Union with path compressionWrite this one
time O(n * α(n)) · space O(n)
Initialize DSU with n elements. For each edge [u, v], find the root representatives of u and v. If `find(u) === find(v)`, the edge creates a cycle; otherwise merge their sets via `union`. Directly handles stream order and hits inverse Ackermann time per operation.
Where people lose marks · 3
- Node labels are 1-indexed up to n, so allocating parent arrays of size n instead of n + 1 causes out-of-bounds indexing.
- The problem requires returning the LAST edge in the input array that creates a cycle; testing from left to right naturally returns the duplicate edge upon arrival.
- Forgetting path compression in `find()` degrades DSU operations to O(n) worst case on degenerate linear trees.
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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