LeetCode #1971 — EASY

Find if Path Exists in Graph

Build confidence with an intuition-first walkthrough focused on union-find fundamentals.

The Problem

Problem Statement

There is a bi-directional graph with n vertices, where each vertex is labeled from 0 to n - 1 (inclusive). The edges in the graph are represented as a 2D integer array edges, where each edges[i] = [u_i, v_i] denotes a bi-directional edge between vertex u_i and vertex v_i. Every vertex pair is connected by at most one edge, and no vertex has an edge to itself.

You want to determine if there is a valid path that exists from vertex source to vertex destination.

Given edges and the integers n, source, and destination, return true if there is a valid path from source to destination, or false otherwise.

Example 1:

Input: n = 3, edges = [[0,1],[1,2],[2,0]], source = 0, destination = 2
Output: true
Explanation: There are two paths from vertex 0 to vertex 2:
- 0 → 1 → 2
- 0 → 2

Example 2:

Input: n = 6, edges = [[0,1],[0,2],[3,5],[5,4],[4,3]], source = 0, destination = 5
Output: false
Explanation: There is no path from vertex 0 to vertex 5.

Constraints:

1 <= n <= 2 * 10⁵
0 <= edges.length <= 2 * 10⁵
edges[i].length == 2
0 <= u_i, v_i <= n - 1
u_i != v_i
0 <= source, destination <= n - 1
There are no duplicate edges.
There are no self edges.

Step 01

Brute Force Baseline

Problem summary: There is a bi-directional graph with n vertices, where each vertex is labeled from 0 to n - 1 (inclusive). The edges in the graph are represented as a 2D integer array edges, where each edges[i] = [ui, vi] denotes a bi-directional edge between vertex ui and vertex vi. Every vertex pair is connected by at most one edge, and no vertex has an edge to itself. You want to determine if there is a valid path that exists from vertex source to vertex destination. Given edges and the integers n, source, and destination, return true if there is a valid path from source to destination, or false otherwise.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Union-Find

Example 1

3
[[0,1],[1,2],[2,0]]
0
2

Example 2

6
[[0,1],[0,2],[3,5],[5,4],[4,3]]
0
5

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1971: Find if Path Exists in Graph
class Solution {
    private int destination;
    private boolean[] vis;
    private List<Integer>[] g;

    public boolean validPath(int n, int[][] edges, int source, int destination) {
        this.destination = destination;
        vis = new boolean[n];
        g = new List[n];
        Arrays.setAll(g, i -> new ArrayList<>());
        for (var e : edges) {
            int u = e[0], v = e[1];
            g[u].add(v);
            g[v].add(u);
        }
        return dfs(source);
    }

    private boolean dfs(int i) {
        if (i == destination) {
            return true;
        }
        vis[i] = true;
        for (var j : g[i]) {
            if (!vis[j] && dfs(j)) {
                return true;
            }
        }
        return false;
    }
}

// Accepted solution for LeetCode #1971: Find if Path Exists in Graph
func validPath(n int, edges [][]int, source int, destination int) bool {
	vis := make([]bool, n)
	g := make([][]int, n)
	for _, e := range edges {
		u, v := e[0], e[1]
		g[u] = append(g[u], v)
		g[v] = append(g[v], u)
	}
	var dfs func(int) bool
	dfs = func(i int) bool {
		if i == destination {
			return true
		}
		vis[i] = true
		for _, j := range g[i] {
			if !vis[j] && dfs(j) {
				return true
			}
		}
		return false
	}
	return dfs(source)
}

# Accepted solution for LeetCode #1971: Find if Path Exists in Graph
class Solution:
    def validPath(
        self, n: int, edges: List[List[int]], source: int, destination: int
    ) -> bool:
        def dfs(i: int) -> bool:
            if i == destination:
                return True
            if i in vis:
                return False
            return any(dfs(j) for j in g[i])

        g = [[] for _ in range(n)]
        for u, v in edges:
            g[u].append(v)
            g[v].append(u)
        vis = set()
        return dfs(source)

// Accepted solution for LeetCode #1971: Find if Path Exists in Graph
impl Solution {
    pub fn valid_path(n: i32, edges: Vec<Vec<i32>>, source: i32, destination: i32) -> bool {
        let n = n as usize;
        let source = source as usize;
        let destination = destination as usize;

        let mut g = vec![Vec::new(); n];
        let mut vis = vec![false; n];

        for e in edges {
            let u = e[0] as usize;
            let v = e[1] as usize;
            g[u].push(v);
            g[v].push(u);
        }

        fn dfs(g: &Vec<Vec<usize>>, vis: &mut Vec<bool>, i: usize, destination: usize) -> bool {
            if i == destination {
                return true;
            }
            vis[i] = true;
            for &j in &g[i] {
                if !vis[j] && dfs(g, vis, j, destination) {
                    return true;
                }
            }
            false
        }

        dfs(&g, &mut vis, source, destination)
    }
}

// Accepted solution for LeetCode #1971: Find if Path Exists in Graph
function validPath(n: number, edges: number[][], source: number, destination: number): boolean {
    const g: number[][] = Array.from({ length: n }, () => []);
    for (const [u, v] of edges) {
        g[u].push(v);
        g[v].push(u);
    }
    const vis = new Set<number>();
    const dfs = (i: number) => {
        if (i === destination) {
            return true;
        }
        if (vis.has(i)) {
            return false;
        }
        vis.add(i);
        return g[i].some(dfs);
    };
    return dfs(source);
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n + m)

Space

O(n + m)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(n) space

Track components with a list or adjacency matrix. Each union operation may need to update all n elements’ component labels, giving O(n) per union. For n union operations total: O(n²). Find is O(1) with direct lookup, but union dominates.

UNION-FIND

O(α(n)) time

O(n) space

With path compression and union by rank, each find/union operation takes O(α(n)) amortized time, where α is the inverse Ackermann function — effectively constant. Space is O(n) for the parent and rank arrays. For m operations on n elements: O(m × α(n)) total.

Shortcut: Union-Find with path compression + rank → O(α(n)) per operation ≈ O(1). Just say “nearly constant.”

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.