Network Delay Time
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 a network of n nodes labeled from 0 to n-1, a list of travel times as directed edges [u, v, w], and a starting node k, find the time it takes for a signal to reach all nodes. Return -1 if any node cannot be reached.
Example 1
- Input
- times = [[2,1,1],[2,3,1],[3,4,1]], n = 4, k = 2
- Output
- 2
- Why
- Signal reaches node 1 at time 1, node 3 at time 2, and node 4 at time 2. Total time = 2.
Example 2
- Input
- times = [[1,2,1]], n = 2, k = 1
- Output
- 1
- Why
- Signal reaches node 2 from node 1 in time 1.
Constraints
- 1 <= k <= n <= 100
- 1 <= times.length <= 6000
- 1 <= w <= 100
How to think about it
Updated 2026-09-09The time for a signal to blanket the network is determined by the node that receives it last. This is the maximum over all single-source shortest path distances from starting node k. Because edge travel times are positive, Dijkstra's algorithm computes the shortest arrival time to every node, and the maximum finite distance is the answer.
Approaches, worst first
Bellman-Ford relaxation
time O(V * E) · space O(V)
Run n - 1 full edge relaxation passes from source k. While simple to implement, repeatedly scans all 6000 edges regardless of whether their source distances changed.
Dijkstra with min-heapWrite this one
time O(E log V) · space O(V + E)
Initialize all distances to infinity and `dist[k] = 0`. Use a priority queue to always pop the node with minimum arrival time and relax its outgoing edges. If all n nodes are reachable, return the maximum recorded distance; otherwise return -1.
Where people lose marks · 3
- Nodes are labeled 1-indexed (1 to n) or 0-indexed depending on the exact test suite; here `1 <= k <= n` indicates 1-based indexing, so size arrays as n + 1.
- If any node has a final distance of infinity after Dijkstra finishes, it is unreachable; return -1 immediately.
- Popping a stale entry from the priority queue without checking `time > dist[u]` wastes computation.
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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