What is the time complexity of Design Graph With Shortest Path Calculator?

The optimal solution has a time complexity of O(E * log(V)).

What is the space complexity of Design Graph With Shortest Path Calculator?

The space complexity is O(V + E).

What data structures are used to solve Design Graph With Shortest Path Calculator?

This problem uses Graph, Design, Heap (Priority Queue), Shortest Path. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Design Graph With Shortest Path Calculator?

Insight 1: Dijkstra's algorithm is the most suitable for finding the shortest path in a weighted graph efficiently, especially when we might add edges later. Insight 2: Using a priority queue (min-heap) allows us to efficiently select the node with the smallest distance from the source during the Dijkstra's algorithm, which optimizes the search process.

Which companies ask LeetCode 2642. Design Graph With Shortest Path Calculator in interviews?

This problem is frequently asked at Nike, Samsung.

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Design Graph With Shortest Path Calculator - LeetCode 2642 Solution

Design Graph With Shortest Path Calculator - Complete Solution Guide

Design Graph With Shortest Path Calculator is LeetCode problem 2642, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

There is a directed weighted graph that consists of n nodes numbered from 0 to n - 1 . The edges of the graph are initially represented by the given array edges where edges[i] = [from i , to i , edgeCost i ] meaning that there is an edge from from i to to i with the cost edgeCost i . Implement the Graph class: Graph(int n, int[][] edges) initializes the object with n nodes and the given edges. addEdge(int[] edge) adds an edge to the list of edges where edge = [from, to, edgeCost] . It is guarant

Detailed Explanation

The problem requires designing a `Graph` class that supports finding the shortest path between two nodes in a directed, weighted graph. The graph is initialized with a given number of nodes and a set of edges. The class must support adding new edges to the graph and querying the shortest path between any two nodes. If no path exists between two nodes, the `shortestPath` method should return -1.

Solution Approach

The solution utilizes Dijkstra's algorithm to compute the shortest path between two nodes in the graph. The graph is represented using an adjacency list, where each node maintains a list of its neighbors and the cost to reach them. A priority queue is used to store nodes based on their distance from the starting node, ensuring that the node with the smallest distance is processed first. The `shortestPath` method iteratively expands the search from the starting node until the destination node is reached or all reachable nodes have been explored. When a new edge is added using `addEdge`, it's simply appended to the adjacency list of the source node.

Step-by-Step Algorithm

Step 1: Initialize the `Graph` with the given number of nodes and edges. Create an adjacency list to represent the graph. Each element of the adjacency list represents a node, and each node contains a list of its neighbors (nodes it has a direct edge to) and the cost of traversing that edge.
Step 2: Implement the `addEdge` method to add a new edge to the graph. This involves adding the destination node and its corresponding cost to the adjacency list of the source node.
Step 3: Implement the `shortestPath` method. This method uses Dijkstra's algorithm.
Step 4: Create a `distances` array to store the shortest distance from the starting node (`node1`) to each node in the graph. Initialize all distances to infinity, except for the starting node, which is initialized to 0.
Step 5: Create a priority queue `pq` to store nodes to visit. Add the starting node with a distance of 0 to the priority queue.
Step 6: While the priority queue is not empty, extract the node `u` with the smallest distance from the priority queue. If the distance to `u` is greater than the current shortest distance to `u` (stored in `distances[u]`), then skip this node (this handles cases where the priority queue contains outdated distance values).
Step 7: If `u` is the destination node (`node2`), return the shortest distance to `u` (which is stored in `distances[u]`).
Step 8: Iterate through all the neighbors `v` of `u`. For each neighbor `v`, calculate the distance from the starting node to `v` through `u`. If this distance is smaller than the current shortest distance to `v` (stored in `distances[v]`), update `distances[v]` and add `v` to the priority queue with the updated distance.
Step 9: If the destination node is not reached after processing all reachable nodes, return -1, indicating that there is no path from the starting node to the destination node.

Key Insights

Insight 1: Dijkstra's algorithm is the most suitable for finding the shortest path in a weighted graph efficiently, especially when we might add edges later.
Insight 2: Using a priority queue (min-heap) allows us to efficiently select the node with the smallest distance from the source during the Dijkstra's algorithm, which optimizes the search process.
Insight 3: Representing the graph using an adjacency list is beneficial for both adding edges and finding neighbors of a node, especially when the graph might be sparse (not all nodes are connected to each other).

Complexity Analysis

Time Complexity: O(E * log(V))

Space Complexity: O(V + E)

Topics

This problem involves: Graph, Design, Heap (Priority Queue), Shortest Path.

Companies

Asked at: Nike, Samsung.

Design Graph With Shortest Path Calculator | LeetCode 2642

Hard

Graph Design Heap (Priority Queue)Shortest Path

Time: O(E * log(V))Space: O(V + E)

Solution Code

Available in:

import java.util.*;

class Graph {

    private int n;
    private List<List<Pair>> adj;

    public Graph(int n, int[][] edges) {
        this.n = n;
        this.adj = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            this.adj.add(new ArrayList<>());
        }
        for (int[] edge : edges) {
            int u = edge[0];
            int v = edge[1];
            int cost = edge[2];
            this.adj.get(u).add(new Pair(v, cost));
        }
    }

    public void addEdge(int[] edge) {
        int u = edge[0];
        int v = edge[1];
        int cost = edge[2];
        this.adj.get(u).add(new Pair(v, cost));
    }

    public int shortestPath(int node1, int node2) {
        int[] distances = new int[n];
        Arrays.fill(distances, Integer.MAX_VALUE);
        distances[node1] = 0;

        PriorityQueue<Pair> pq = new PriorityQueue<>(Comparator.comparingInt(p -> p.distance));
        pq.offer(new Pair(node1, 0));

        while (!pq.isEmpty()) {
            Pair curr = pq.poll();
            int dist = curr.distance;
            int u = curr.node;

            if (dist > distances[u]) {
                continue;
            }

            if (u == node2) {
                return dist;
            }

            for (Pair neighbor : adj.get(u)) {
                int v = neighbor.node;
                int weight = neighbor.distance;
                if (distances[u] + weight < distances[v]) {
                    int new_dist = distances[u] + weight;
                    distances[v] = new_dist;
                    pq.offer(new Pair(v, new_dist));
                }
            }
        }

        int result = distances[node2];
        return result == Integer.MAX_VALUE ? -1 : result;
    }

    private static class Pair {
        int node;
        int distance;

        public Pair(int node, int distance) {
            this.node = node;
            this.distance = distance;
        }
    }
}

/**
 * Your Graph object will be instantiated and called as such:
 * Graph obj = new Graph(n, edges);
 * obj.addEdge(edge);
 * int param_2 = obj.shortestPath(node1,node2);
 */

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Problem Info

Description

Problem 2642: Design Graph With Shortest Path Calculator - Detailed explanation and solution approaches.

Topics

Graph Design Heap (Priority Queue)Shortest Path

Companies

NikeSamsung

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:2642

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

9 steps

1
Step 1: Initialize the `Graph` with the given number of nodes and edges. Create an adjacency list to represent the graph. Each element of the adjacency list represents a node, and each node contains a list of its neighbors (nodes it has a direct edge to) and the cost of traversing that edge.
2
Step 2: Implement the `addEdge` method to add a new edge to the graph. This involves adding the destination node and its corresponding cost to the adjacency list of the source node.
3
Step 3: Implement the `shortestPath` method. This method uses Dijkstra's algorithm.
4
Step 4: Create a `distances` array to store the shortest distance from the starting node (`node1`) to each node in the graph. Initialize all distances to infinity, except for the starting node, which is initialized to 0.
5
Step 5: Create a priority queue `pq` to store nodes to visit. Add the starting node with a distance of 0 to the priority queue.
6
Step 6: While the priority queue is not empty, extract the node `u` with the smallest distance from the priority queue. If the distance to `u` is greater than the current shortest distance to `u` (stored in `distances[u]`), then skip this node (this handles cases where the priority queue contains outdated distance values).
7
Step 7: If `u` is the destination node (`node2`), return the shortest distance to `u` (which is stored in `distances[u]`).
8
Step 8: Iterate through all the neighbors `v` of `u`. For each neighbor `v`, calculate the distance from the starting node to `v` through `u`. If this distance is smaller than the current shortest distance to `v` (stored in `distances[v]`), update `distances[v]` and add `v` to the priority queue with the updated distance.
9
Step 9: If the destination node is not reached after processing all reachable nodes, return -1, indicating that there is no path from the starting node to the destination node.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(E * log(V))

💾

Space Complexity

O(V + E)

📝

Why?

The time complexity is dominated by the Dijkstra's algorithm. Each node is added and extracted from the priority queue at most once, taking O(log V) time for each operation. We iterate through the neighbors of each node, which takes O(E) time in total. The space complexity is determined by the size of the adjacency list and the distances array.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Bellman-Ford Algorithm

Bellman-Ford can handle negative edge weights, unlike Dijkstra's. However, it's less efficient for graphs without negative edges.

Time: O(V * E)Space: O(V)

Trade-offs: Use Bellman-Ford if the graph might contain negative edge weights. Dijkstra's is more efficient if the graph only has non-negative weights.

Floyd-Warshall Algorithm

Floyd-Warshall computes the shortest path between all pairs of nodes in the graph.

Time: O(V^3)Space: O(V^2)

Trade-offs: Use Floyd-Warshall if you need to find the shortest path between all pairs of nodes. Dijkstra's is more efficient if you only need the shortest path between a single pair of nodes.

✨

Key Insights

1Insight 1: Dijkstra's algorithm is the most suitable for finding the shortest path in a weighted graph efficiently, especially when we might add edges later.
2Insight 2: Using a priority queue (min-heap) allows us to efficiently select the node with the smallest distance from the source during the Dijkstra's algorithm, which optimizes the search process.
3Insight 3: Representing the graph using an adjacency list is beneficial for both adding edges and finding neighbors of a node, especially when the graph might be sparse (not all nodes are connected to each other).

📚

Key Terms Explained

For Students

Graph

Nodes connected by edges. Like a social network or city map!

Edge Cases

Edge case 1: No path exists between `node1` and `node2`. The algorithm correctly returns -1 because the `distances[node2]` remains at its initial value (infinity) after the Dijkstra's algorithm is complete.
Edge case 2: The graph is disconnected. The Dijkstra's algorithm will only explore the connected component containing `node1`, and if `node2` is not in that component, the algorithm will correctly return -1.
Edge case 3: Adding an edge creates a shorter path. The algorithm correctly updates the distances array and priority queue to reflect the new shortest path.
Edge case 4: The weights are large, potentially causing integer overflow. The C solution uses (long) cast during comparison to prevent overflow. Other solutions can also use similar mitigations.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not handling the case where there is no path between the two nodes. The algorithm should return -1 in this case.

✗Common mistake 2: Forgetting to use a priority queue to efficiently select the node with the smallest distance. Using a simple queue or iterating through all nodes to find the minimum distance will result in a much slower algorithm.

✗Common mistake 3: Not checking `if dist > distances[u]` inside the `while` loop. This condition is crucial for preventing unnecessary processing of nodes with outdated distance values in the priority queue and ensures the algorithm's efficiency. Without this check, time complexity will increase significantly.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Navigation systems use shortest path algorithms to find the best route between two locations, considering factors such as distance, traffic, and road closures.

🏦Real-world scenario 2: Network routing protocols use shortest path algorithms to determine the most efficient path for data packets to travel between two computers in a network.

🎮Real-world scenario 3: Social networking sites can use shortest path algorithms to find the degree of separation between two users or recommend connections based on common friends.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Before coding, decide between BFS (shortest path, level-order) and DFS (exhaustive search, connected components). Draw the graph on paper first.

🎯 Interview Tip

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ Watch Out

Forgetting to mark nodes as visited, causing infinite loops. Also watch for disconnected components.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 1, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 13

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 6

❓

Frequently Asked Questions

How do I represent a graph in code?

Use adjacency list (HashMap or array of lists) for sparse graphs - it's space-efficient O(V+E). Use adjacency matrix for dense graphs or when you need O(1) edge lookup. For weighted graphs, store (neighbor, weight) pairs in adjacency lists.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Design Graph With Shortest Path Calculator"?

For this hard-level Graph problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether design techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?