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Lexicographical Order of Numbers: Step-by-Step Explanation

This guide explains how to solve the problem of generating numbers from 1 to n in lexicographical order using different programming languages. The solution involves a Depth-First Search (DFS) strategy to explore numbers by appending digits in sequence.

C++ Code Explanation

  1. Initialization:

    • Create a vector<int> to store the numbers in lexicographical order.
    • Iterate from numbers 1 to 9 (these act as root nodes for DFS exploration).

  2. DFS Implementation:

    • For each starting number, perform a DFS to explore all possible numbers by appending digits (0-9).
    • Each time a number is generated, add it to the result list.
    • Stop the DFS when the current number exceeds the limit n.

  3. Base Case:

    • If the current number being generated is greater than n, terminate the recursion for that branch.

Java Code Explanation

  1. Initialization:

    • Create a List<Integer> to store the lexicographically ordered numbers.
    • Iterate from 1 to 9, using each number as a starting point for DFS.

  2. DFS Implementation:

    • For each number, add it to the result list and then explore numbers by appending digits (0-9).
    • Recursively call DFS on the new number generated by appending digits.

  3. Base Case:

    • When a number exceeds n, terminate the recursion for that branch to avoid unnecessary exploration.

JavaScript Code Explanation

  1. Initialization:

    • Initialize an empty array result to store the final lexicographical sequence.
    • Iterate from 1 to 9, treating each number as the start of a DFS traversal.

  2. DFS Implementation:

    • For each starting number, recursively append digits (0-9) to generate new numbers.
    • Each generated number is added to the result array.

  3. Base Case:

    • Stop the recursion if the current number exceeds n.

Python Code Explanation

  1. Initialization:

    • Use a list result to hold the numbers in lexicographical order.
    • Loop through the numbers 1 to 9 and perform DFS starting with each.

  2. DFS Implementation:

    • Add the current number to the result list.
    • Recursively generate the next number by appending digits (0-9) to the current number.

  3. Base Case:

    • Stop recursion when the number being generated exceeds n.

Go Code Explanation

  1. Initialization:

    • Create an empty slice result to store numbers in lexicographical order.
    • Iterate over numbers 1 to 9 as the starting points for DFS exploration.

  2. DFS Implementation:

    • Perform DFS by appending digits (0-9) to the current number.
    • Append the valid numbers to the result slice.

  3. Base Case:

    • If the current number exceeds n, stop further recursion for that branch.

This solution uses Depth-First Search (DFS) in all languages to explore numbers in lexicographical order. The main idea is to treat each digit as a root and recursively generate new numbers by appending digits from 0 to 9 until the number exceeds n.