Network Delay Time Leetcode Problem 743 [Python Solution]

In this blog post, we will delve into the Network Delay Time problem, which is categorized as an advanced graph problem on LeetCode.

The goal of this problem is to find the minimum time it takes for a signal to travel from a given source node to all other nodes in a network.

We will discuss the problem overview, constraints, a brute-force approach, and an efficient Python solution.

So, let's dive right in!

Problem Overview

You are given a network of n nodes, labeled from 1 to n.

Additionally, you are provided with a list of travel times as directed edges in the format times[i] = (u, v, w), where u is the source node, v is the target node, and w is the time it takes for a signal to travel from the source to the target.

The task is to send a signal from a given node k and return the minimum time it takes for all n nodes to receive the signal.

If it is impossible for all nodes to receive the signal, you should return -1. Let's illustrate this with an example:

Example 1:

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

Output:
2

In this example, we start at node 2 and send a signal.

The signal takes 1 unit of time to reach node 1 and 1 unit of time to reach node 3. Then, it takes another unit of time to reach node 4. After 2 units of time, all nodes have received the signal.

Understanding the Constraints

Before we proceed with the solutions, let's understand the constraints mentioned in the problem:

1 <= k <= n <= 100: The number of nodes (n) can range from 1 to 100, and the source node (k) is within this range.
1 <= times.length <= 6000: The number of edges is at least 1 but can go up to 6000.
times[i].length == 3: Each edge is represented as a triple (u, v, w) with source, target, and weight.
1 <= u, v <= n: The source and target nodes are within the range of 1 to n.
u != v: The source node is not the same as the target node.
0 <= w <= 100: The weight (time) of each edge is non-negative and can be up to 100.
All pairs (u, v) are unique: There are no multiple edges between the same source and target nodes.

Now that we have a clear understanding of the problem and its constraints, let's explore the solution approaches.

Network Delay Time Python Code Solution

Brute-force Approach

We'll start by discussing a brute-force approach to solving the Network Delay Time problem.

While this approach may not be the most efficient, it helps us understand the problem before diving into more optimized solutions.

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def networkDelayTime(self, times: List[List[int]], n: int, k: int) -&gt; int:
    # Initialize a dictionary to store the graph&#39;s edges (adjacency list).

edges = collections.defaultdict(list)

    # Populate the adjacency list with the edges.

for u, v, w in times:
        edges[u].append((v, w))

    # Initialize a min-heap to track nodes to visit, starting with (0, k).

minHeap = [(0, k)]

    # Initialize a set to keep track of visited nodes.

visit = set()

    # Initialize a variable to track the time taken.

t = 0

    # Perform the Dijkstra&#39;s algorithm using a min-heap.

while minHeap:
        # Pop the node with the minimum weight.

w1, n1 = heapq.heappop(minHeap)

        # If the node has already been visited, continue.

if n1 in visit:
            continue

        # Mark the node as visited and update the time.

visit.add(n1)
        t = w1

        # Explore the neighbors of the current node.

for n2, w2 in edges[n1]:
            if n2 not in visit:
                # Add the neighbor to the min-heap with updated weight.

heapq.heappush(minHeap, (w1 + w2, n2))

    # If we visited all nodes, return the time; otherwise, return -1. return t if len(visit) == n else -1

The above code implements Dijkstra's algorithm using a min-heap (priority queue) to find the minimum time it takes for the signal to reach all nodes.

It efficiently explores the network and keeps track of visited nodes and their distances from the source node.

The time complexity of this approach is O(E * log(V)), where E is the number of edges (times) and V is the number of nodes (n).

Now, let's move on to a more efficient approach.

Efficient Approach with Dijkstra's Algorithm

The efficient approach to solving the Network Delay Time problem involves using Dijkstra's algorithm.

This algorithm is specifically designed for finding the shortest path in a graph with weighted edges.

We'll provide a high-level overview of the efficient approach without going into the implementation details, as we have already discussed the code.

Dijkstra's algorithm operates as follows:

Initialize a priority queue (min-heap) with the source node and an initial distance of 0.
While the priority queue is not empty, repeat the following steps:
a.

Pop the node with the minimum distance from the priority queue.
b.

If this node has already been visited, skip it.
c.

Mark the node as visited and update the time (distance).
d.

Explore all neighboring nodes and calculate the total distance from the source.
e.

Add unvisited neighbors to the priority queue with their updated distances.
3. After the algorithm completes, check if all nodes have been visited.

If yes, return the time; otherwise, return -1. Dijkstra's algorithm guarantees the shortest path because it explores nodes in increasing order of their distances from the source.

By using a priority queue, it efficiently selects the next node to visit.

Time and Space Complexity

Time Complexity

The efficient approach, which uses Dijkstra's algorithm with a priority queue, has a time complexity of O(E * log(V)), where E represents the number of edges (times) and V represents the number of nodes (n).

This is a highly efficient algorithm for solving the problem, even for large graphs.

Space Complexity

The space complexity of the solution is O(E + V), where E represents the number of edges (times) and V represents the number of nodes (n).

This space is used for storing the graph's edges in an adjacency list, the min-heap, and the set to keep track of visited nodes.

The space usage is reasonable and won't cause memory issues for typical inputs.

Reasoning Behind Our Approach

Dijkstra's algorithm is a

well-known and widely used graph algorithm for finding the shortest path.

In this problem, we adapted Dijkstra's algorithm to calculate the minimum time it takes for a signal to travel from a source node to all other nodes in a network.

The key insights behind our approach are as follows:

We create an adjacency list to represent the graph, making it easy to access neighbors for each node.
We use a priority queue (min-heap) to efficiently select the node with the shortest distance from the source node.

This ensures that we explore nodes in increasing order of their distances.

We keep track of visited nodes to avoid revisiting them, ensuring that we find the minimum time efficiently.

By using Dijkstra's algorithm, we guarantee the correctness of our solution and achieve a time complexity of O(E * log(V)), making it an optimal choice for this problem.

Conclusion

In this blog post, we tackled the Network Delay Time problem, a challenging graph problem on LeetCode.

We discussed the problem's overview, constraints, and provided both a brute-force and an efficient Python solution.

The efficient solution leverages Dijkstra's algorithm with a priority queue to find the minimum time it takes for a signal to traverse a network.

We hope this explanation and code have been helpful in understanding how to approach this problem.

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For more details and to practice the problem, you can visit the Network Delay Time on LeetCode.

Thank you for reading, and happy coding!