What is the optimal time complexity of solving Network Delay Time?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Network Delay Time?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Network Delay Time?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Network Delay Time?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Network Delay Time

Learn how to solve the 'Network Delay Time' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

You are given a network of n nodes, labeled from 1 to n. You are also given times, a list of travel times as directed edges times[i] = [ui, vi, wi], where ui is the source node, vi is the target node, and wi is the time it takes for a signal to travel from source to target.

We will send a signal from a given node k. Return the minimum time it takes for all the n nodes to receive the signal. If it is impossible for all the n nodes to receive the signal, return -1.

Write a function networkDelayTime(times: List[List[int]], n: int, k: int) -> int.

Constraints

•1 <= k <= n <= 100
•1 <= len(times) <= 6000
•times[i].length == 3
•1 <= ui, vi <= n
•ui != vi
•0 <= wi <= 100
•All the pairs (ui, vi) are unique

Examples

Example 1

Input

times = [[2,1,1],[2,3,1],[3,4,1]], n = 4, k = 2

Output

Explanation

The signal starts at node 2. It reaches 1 and 3 in 1 unit of time, and 4 in 2 units of time.

Example 2

Input

times = [[1,2,1]], n = 2, k = 1

Output

Explanation

Signal reaches node 2 from node 1 in 1 unit of time.

Example 3

Input

times = [[1,2,1]], n = 2, k = 2

Output

-1

Explanation

Signal starts at node 2, but there is no path from node 2 to node 1. So node 1 never receives it.

Need a Hint?

Represent graph node connections as an adjacency list/matrix, then use standard BFS or DFS graph traversal.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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