Author: root

Published on: October 1, 2023

Author: Coding Expert

Problem Description

A salesman is visiting various cities to sell products. The salesman knows the prices of products that can be sold in each city, as well as the travel costs between cities. The salesman needs to set a goal and find the most efficient route to achieve that goal. In other words, the salesman must visit each city only once and find a route that allows him to return home while maximizing his profit.

Input

The input consists of the number of cities n, an array of sale prices prices, and a 2D array of travel costs costs. The prices are represented by prices[i] for the price of the product in the i-th city, and the travel costs are represented by costs[i][j] for the cost of traveling from city i to city j.

Output

The function should return the maximum profit that the salesman can achieve by returning home slowly.

Constraints

• 2 ≤ n ≤ 10
• 0 ≤ prices[i] ≤ 1000
• 0 ≤ costs[i][j] ≤ 1000

Problem-Solving Approach

This problem is similar to the ‘Traveling Salesman Problem’ and can be solved using backtracking or dynamic programming. Essentially, the salesman needs to try all possible combinations to optimize the route that visits all cities and returns home.

Step 1: Understanding the Problem

Since the salesman must visit all cities, he needs to explore all paths between cities while considering the item sale profits and travel costs for each path. The goal is to calculate profits and costs to select the optimal route.

Step 2: Designing the Algorithm

To solve this problem, follow these steps:

• Assume each city as the starting city and visit each city while exploring all possible paths.
• Calculate the sale profits and travel costs for each path to update the optimal profit.
• After visiting all cities, calculate the cost to return to the original city as well.

Step 3: Implementation

Now let’s move on to the implementation phase. We will write a function in JavaScript to find the maximum profit of the salesman.

function maxProfit(n, prices, costs) {
    let maxProfit = -Infinity;

    function backtrack(currentCity, visited, currentProfit) {
        // If all cities are visited, return home.
        if (visited.length === n) {
            const returnCost = costs[currentCity][0]; // Cost to return to the starting city
            const totalProfit = currentProfit - returnCost; // Calculate total profit
            maxProfit = Math.max(maxProfit, totalProfit);
            return;
        }

        // Visit each city.
        for (let nextCity = 0; nextCity < n; nextCity++) {
            if (!visited.includes(nextCity)) {
                visited.push(nextCity); // Record city visit
                const nextProfit = currentProfit + prices[nextCity]; // Calculate profit for next city
                const travelCost = costs[currentCity][nextCity]; // Travel cost
                backtrack(nextCity, visited, nextProfit - travelCost); // Recursive call
                visited.pop(); // Remove visit record (backtracking)
            }
        }
    }

    // Start from the 0th city
    backtrack(0, [0], prices[0]);

    return maxProfit;
}

// Example usage
const n = 4;
const prices = [100, 70, 90, 40];
const costs = [
    [0, 10, 15, 20],
    [10, 0, 35, 25],
    [15, 35, 0, 30],
    [20, 25, 30, 0]
];

console.log(maxProfit(n, prices, costs));
        

Step 4: Code Explanation

The maxProfit function defined above performs the following tasks:

• currentCity: Tracks the current city.
• visited: Tracks the cities visited so far.
• currentProfit: Tracks the cumulative profit so far.

We recursively explore each city. After visiting all cities, we calculate the cost to return home to update the total profit.

Example Test

When running the code, the maxProfit function will return the maximum profit. It is advisable to experiment with various input values to observe the performance of the algorithm.

Conclusion

In this lesson, we explored the Traveling Salesman Problem. It is important to understand the theoretical background and implementation methods that frequently appear in coding tests. By exploring various routes and quantitatively calculating the optimal profit, we learned how to utilize the powerful features of JavaScript.

In the next session, we will cover another algorithm problem. If you have any questions or feedback, please leave a comment!

In this lecture, we will cover one of the frequently asked questions in coding tests, which is the “Calculating the Area of a Polygon” problem. We will conduct in-depth learning on how to implement an algorithm to calculate the area of a polygon using JavaScript.

Problem Description

Write a function to calculate the area of a polygon given the coordinates of its vertices. The vertices of the polygon are sorted either in a clockwise or counterclockwise direction, and the vertex coordinates are represented as integers in a two-dimensional coordinate system.

Input

• The number of vertices of the polygon n (3 ≤ n ≤ 1000)
• The coordinates of n vertices (x1, y1), (x2, y2), ..., (xn, yn)

Output

The area of the polygon should be printed rounded to two decimal places.

Solution Process

There are several methods to calculate the area of a polygon. Here, we will use the most common “Shoelace Formula (or Polygon Area Formula)”. This formula allows us to easily calculate the area of a polygon.

Shoelace Formula

For the given vertices (x1, y1), (x2, y2), ..., (xn, yn), the area A is calculated as follows:

A = (1/2) * | Σ (xi * yi+1 - yi * xi+1) | 

Here, i+1 is set to return to 1 when i reaches n using modular arithmetic. This formula considers the contributions of all edges of the polygon in calculating the area.

JavaScript Code Implementation

Let’s implement the above formula in code. Below is the code written in JavaScript.

function calculatePolygonArea(vertices) {
    let n = vertices.length;
    let area = 0;

    for (let i = 0; i < n; i++) {
        let x1 = vertices[i][0];
        let y1 = vertices[i][1];
        let x2 = vertices[(i + 1) % n][0];
        let y2 = vertices[(i + 1) % n][1];

        area += (x1 * y2) - (y1 * x2);
    }

    return Math.abs(area / 2).toFixed(2);
}

// Example
let vertices = [[0, 0], [4, 0], [4, 3], [0, 4]];
console.log(calculatePolygonArea(vertices)); // 12.00

Code Explanation

• The calculatePolygonArea function takes an array of vertex coordinates vertices as input.
• It calculates the number of polygon vertices n.
• It initializes the area area to 0, then calculates the area for all vertices.
• It adds the contribution to the area using the current vertex (xi, yi) and the next vertex (xi+1, yi+1).
• It connects the last vertex with the first vertex through modulus operation to complete the area calculation.
• It returns the calculated area rounded to two decimal places.

Test Cases

If you have checked the code, let’s add the following test cases.

let testVertices1 = [[0, 0], [0, 2], [2, 2], [2, 0]]; // Rectangle
let testVertices2 = [[0, 0], [4, 0], [4, 3], [0, 4]]; // Irregular Polygon

console.log(calculatePolygonArea(testVertices1)); // 4.00
console.log(calculatePolygonArea(testVertices2)); // 12.00

Conclusion

In this lecture, we explored the theory behind calculating the area of a polygon along with an example of its implementation in JavaScript. I believe that understanding the formula to calculate the area of a polygon and implementing it in actual code will help in coding tests.

The problem of calculating the area of a polygon is frequently asked in real coding tests, so be sure to firmly grasp the basic theories and problem-solving processes. In the next lecture, we will cover another algorithm problem, so I hope for your continued interest.

Problem Description

The Lowest Common Ancestor (LCA) is the problem of finding the most recent ancestor node of two nodes. This problem is very important in tree data structures and is a frequently asked topic in coding tests and interviews. In this course, we will cover how to find the LCA using JavaScript and examine the process of solving actual algorithm problems.

Problem Definition

Given a binary tree and two nodes A and B, write a function that finds and returns the lowest common ancestor of the two nodes.

Input Format

• The number of nodes is N, where 1 ≤ N ≤ 10^4.
• Each node has a unique integer ID.
• The IDs of the two nodes A and B are provided.

Output Format

• Returns the ID of the node representing the lowest common ancestor of the two nodes.

Examples

Example 1

        Input: 
        Tree Structure:
              1
            /   \
          2      3
         / \    / \
        4   5  6   7

        A = 4, B = 5
        Output: 2  // Lowest Common Ancestor: 2
    

Example 2

        Input: 
        Tree Structure:
              1
            /   \
          2      3
         / \    / \
        4   5  6   7

        A = 6, B = 7
        Output: 3  // Lowest Common Ancestor: 3
    

Problem Solving Process

1. Define Tree Node Structure

First, we need to define the node structure to represent a binary tree. Each node must have information about at least the parent and child nodes. In JavaScript, we can define the node structure as follows.

    class TreeNode {
        constructor(id) {
            this.id = id;  // Node's ID
            this.left = null;  // Left child node
            this.right = null;  // Right child node
        }
    }
    

2. Create the Tree

Let’s create the example tree from the problem. The code below shows how to construct the tree.

    const root = new TreeNode(1);
    root.left = new TreeNode(2);
    root.right = new TreeNode(3);
    root.left.left = new TreeNode(4);
    root.left.right = new TreeNode(5);
    root.right.left = new TreeNode(6);
    root.right.right = new TreeNode(7);
    

3. Find the Lowest Common Ancestor

Now we will implement a function to actually find the LCA. The common method for finding the lowest common ancestor in a binary tree is to traverse the tree recursively and return the node when both nodes A and B are found.

    function findLCA(root, A, B) {
        if (root === null) {
            return null;
        }

        // If the current node is A or B
        if (root.id === A || root.id === B) {
            return root;
        }

        const leftLCA = findLCA(root.left, A, B);
        const rightLCA = findLCA(root.right, A, B);

        // If LCA is found in both children, current node is LCA
        if (leftLCA && rightLCA) {
            return root;
        }
        
        return leftLCA !== null ? leftLCA : rightLCA;
    }
    

4. Test the Function

Now let’s test the LCA function we wrote. You can call the function with the following code to output the result.

    const A = 4;
    const B = 5;
    const lcaNode = findLCA(root, A, B);

    console.log(`Lowest Common Ancestor: ${lcaNode.id}`);  // Output: Lowest Common Ancestor: 2
    

Complexity Analysis

The time complexity of this algorithm is O(N), where N is the number of nodes. This is because we need to visit every node once. The space complexity is O(H), where H is the height of the tree. In the worst case, H can be close to N (when the tree is skewed).

Conclusion

In this course, we learned in detail about how to find the lowest common ancestor using JavaScript. This concept is a fundamental basis for solving various tree-related problems and will help in understanding tree traversal algorithms. Since it is frequently asked in coding tests and interviews, make sure to master it. In the next course, we will cover other tree-related problems, so stay tuned!

In this course, we will cover how to solve JavaScript coding test problems. The topic of the problem is to find the minimum value among prime and palindromic numbers. We will guide you through the problem-solving process step by step, along with the necessary algorithms.

Problem Definition

We have a given integer N. We need to find the minimum value among all numbers that are both prime and palindromic from 1 to N. If such a number does not exist, return -1.

Example Problems

• Input: N = 31
  Output: 3 (Among the primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, the palindromic number is 3)
• Input: N = 11
  Output: 11 (The prime number 11 is a palindromic number)
• Input: N = 1
  Output: -1 (1 is neither a prime nor a palindromic number)

Problem Solving Approach

To solve the problem, we will use the following methods.

1. Find prime numbers from 1 to N.
2. Filter the prime numbers to find those that are palindromic.
3. Find the minimum value among the palindromic numbers.

Step-by-Step Implementation

Step 1: Finding Prime Numbers

To find prime numbers, we can use a simple Sieve of Eratosthenes algorithm. This algorithm is an efficient way to find prime numbers with a time complexity of O(n log log n).

function findPrimes(n) {
        const isPrime = Array(n + 1).fill(true);
        isPrime[0] = isPrime[1] = false; // 0 and 1 are not prime numbers.
        
        for (let i = 2; i * i <= n; i++) {
            if (isPrime[i]) {
                for (let j = i * i; j <= n; j += i) {
                    isPrime[j] = false; // Exclude multiples of i from primes
                }
            }
        }
        
        const primes = [];
        for (let i = 2; i <= n; i++) {
            if (isPrime[i]) {
                primes.push(i);
            }
        }
        return primes;
    }

Step 2: Finding Palindromic Numbers

A palindromic number is an integer that reads the same forwards and backwards. For example, 121 and 1331 are palindromic numbers.

function isPalindrome(num) {
        const str = num.toString();
        return str === str.split('').reverse().join('');
    }

Step 3: Finding the Minimum Value

Now we gather the numbers that are both prime and palindromic and find the minimum value among them.

function findMinPalindromicPrime(n) {
        const primes = findPrimes(n);
        const palindromicPrimes = primes.filter(isPalindrome);
        
        return palindromicPrimes.length > 0 ? Math.min(...palindromicPrimes) : -1;
    }

Complete Code

Now let's look at the final code that combines all the steps.

function isPalindrome(num) {
        const str = num.toString();
        return str === str.split('').reverse().join('');
    }

    function findPrimes(n) {
        const isPrime = Array(n + 1).fill(true);
        isPrime[0] = isPrime[1] = false;

        for (let i = 2; i * i <= n; i++) {
            if (isPrime[i]) {
                for (let j = i * i; j <= n; j += i) {
                    isPrime[j] = false;
                }
            }
        }

        const primes = [];
        for (let i = 2; i <= n; i++) {
            if (isPrime[i]) {
                primes.push(i);
            }
        }
        return primes;
    }

    function findMinPalindromicPrime(n) {
        const primes = findPrimes(n);
        const palindromicPrimes = primes.filter(isPalindrome);
        
        return palindromicPrimes.length > 0 ? Math.min(...palindromicPrimes) : -1;
    }

    // Example usage
    console.log(findMinPalindromicPrime(31)); // Output: 3
    console.log(findMinPalindromicPrime(11)); // Output: 11
    console.log(findMinPalindromicPrime(1));  // Output: -1

Conclusion

In this course, we implemented an algorithm to solve the problem of finding the minimum value among prime and palindromic numbers. We explored the process of efficiently finding prime numbers using the Sieve of Eratosthenes, checking for palindromic numbers, and ultimately finding the minimum value that meets the criteria. To tackle such problems, both algorithmic thinking and programming skills are essential. I hope you continue to enhance your skills through various coding challenges.