Introduction

Floyd's Triangle is a fascinating geometric pattern named after the renowned computer scientist Robert W.Floyd. It is a right-angled triangular sequence of natural numbers, where each row represents the consecutive natural numbers arranged in a triangular pattern.

Understanding

Let's start by understanding the logic behind generating Floyd's Triangle. The pattern is created by filling the rows of a triangle with consecutive natural numbers. The first row contains one number, the second row contains two numbers, and so on. The nth row contains n numbers.

1 
2 3 
4 5 6 
7 8 9 10 

As we can observe, each row contains consecutive natural numbers, and the pattern forms a right-angled triangle.

Coding

#include <iostream>

void generateFloydsTriangle(int rows) {
    int number = 1;

    // Outer loop for rows
    for (int i = 1; i <= rows; ++i) {
        
        // Inner loop for columns
        for (int j = 1; j <= i; ++j) {
            std::cout << number << " ";
            ++number;
        }

        // Move to the next line after each row
        std::cout << std::endl;
    }
}

int main() {
    int rows;

    // Input the number of rows
    std::cout << "Enter the number of rows for Floyd's Triangle: ";
    std::cin >> rows;

    // Generate and display Floyd's Triangle
    generateFloydsTriangle(rows);

    return 0;
}

Complexity Analysis

Time Complexity: The time complexity of this implementation is O(n^2), where n is the number of rows. This is because we are using nested loops, and the number of iterations is proportional to the sum of the first n natural numbers.

Space Complexity: The space complexity is relatively straightforward and can be considered constant, O(1). This is because the amount fo additional memory used does not scale with the size of the input (number of rows). The primary use of memory is for the integer variables used in the loops (i, j, and number), which take up a constant amount of space regardless of the input size. There is no dynamic allocation or data structures that grow with the input size.