C / C++ / Java == How to operate from the minus to the max bond number in a large k dataset ?

#include <stdio.h>
#include <stdint.h>

int64_t count_numbers_in_range(int max_limit, int64_t range_start, int64_t range_end) {
int64_t total_count = 0;
while (range_start <= max_limit) {
total_count += (int64_t)(max_limit + 1) < range_end
? (int64_t)(max_limit + 1) – range_start
: range_end – range_start;
range_start *= 10;
range_end *= 10;
}
return total_count;
}

/**
* Computes the total number of numbers in the sequence up to max_limit.
*/
int64_t compute_sequence_size(int max_limit) {
int64_t total_size = 0;
// Count numbers starting with each digit (1 to 9)
for (int64_t digit = 1; digit <= 9; digit++) {
total_size += count_numbers_in_range(max_limit, digit, digit + 1);
}
return total_size;
}

int64_t find_kth_number_in_sequence(int max_limit, int kth_position) {
// Validate kth_position
int64_t total_size = compute_sequence_size(max_limit);
if (kth_position < 1 || kth_position > total_size) {
printf(“Error: kth_position %d is out of bounds. Sequence size is %lld.\n”,
kth_position, total_size);
return -1; // Indicate error
}

int64_t current_number = 1;
int remaining_positions = kth_position – 1;

while (remaining_positions > 0) {
int64_t count_in_current_branch = count_numbers_in_range(
max_limit, current_number, current_number + 1);
if (count_in_current_branch <= remaining_positions) {
current_number++;
remaining_positions -= count_in_current_branch;
} else {
current_number *= 10;
remaining_positions–;
}
}
return current_number;
}

int main() {
int max_limit = 13;
int kth_position = 2;

int64_t total_size = compute_sequence_size(max_limit);
printf(“Total numbers in sequence up to %d: %lld\n”, max_limit, total_size);

int64_t result = find_kth_number_in_sequence(max_limit, kth_position);
if (result != -1) {
printf(“The %d-th number in the sequence up to %d is: %lld\n”,
kth_position, max_limit, result);
}

return 0;
}

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Problem Context

The problem this script solves is a classic algorithmic challenge, often seen in competitive programming or coding interviews. It’s commonly known as the “K-th Smallest Number in Lexicographical Order” problem (e.g., LeetCode Problem 440). Here’s the setup:

• Input:
• n: An integer representing the upper bound of numbers to consider.
• k: The position (1-based index) of the number to find in the lexicographically sorted sequence.
• Sequence:
• Start with numbers 1 to 9.
• For each number, generate subsequent numbers by multiplying by 10 (e.g., 1 → 10 → 100 → 1000, 2 → 20 → 200, etc.), as long as the result is ≤ n.
• Sort these numbers lexicographically (as strings), not numerically. For example, 10 comes before 2 because “10” < “2” in lexicographical order.
• Output:
• The k-th number in this sorted sequence.

Example

• For n = 13, k = 2:
• The sequence of numbers ≤ 13 is: 1, 10, 100, 11, 12, 13, 2, 3, …, 9.
• Lexicographically sorted: 1, 10, 11, 12, 13, 2, 3, …, 9.
• The 2nd number (k = 2) is 10.

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Why Write This Script? (Purpose)

The script is designed to efficiently find the k-th number in this sequence without generating and sorting the entire list, which would be impractical for large n (e.g., n = 10^9). Here’s the motivation:

1. Efficiency:

• Generating all numbers up to n, sorting them lexicographically, and picking the k-th one has a time complexity of O(n log n) and space complexity of O(n), which is infeasible for large n.
• This script uses a tree-like traversal approach, counting numbers in each “branch” (e.g., numbers starting with 1, 2, …, 9) and narrowing down to the k-th number. The time complexity is O(log n * log n), which is much more efficient.

1. Algorithmic Challenge:

• This problem tests understanding of lexicographical ordering, tree traversal, and prefix-based counting.
• It requires creative thinking to avoid brute force and instead use a mathematical approach to count numbers in ranges.

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Use Case Scenarios

This script, while specific, has practical applications in several domains:

1. Competitive Programming and Coding Interviews:

• Problems like this are common in platforms like LeetCode, Codeforces, or HackerRank to test algorithmic skills.
• It evaluates a candidate’s ability to handle large datasets efficiently, think recursively, and optimize for time and space.

1. Search and Indexing Systems:

• Lexicographical ordering is used in search engines, databases, or file systems where items (e.g., filenames, words) are sorted as strings.
• For example, if a system needs to find the k-th file in a lexicographically sorted list of filenames (e.g., “1.txt”, “10.txt”, “2.txt”), this algorithm can efficiently locate it without sorting all files.

1. Number Sequence Generation:

• In applications like generating dictionary-ordered sequences (e.g., for generating IDs, codes, or labels), this algorithm can help find the k-th entry efficiently.
• Example: A system generating product codes (1, 10, 100, 2, 20, …) might need the 1000th code without generating all prior codes.

1. Educational Tools:

• This script can be part of an educational tool teaching lexicographical ordering, prefix trees (tries), or algorithmic optimization.
• It demonstrates how to solve problems involving large datasets using counting and traversal instead of brute force.

1. Database Query Optimization:

• In databases with lexicographically ordered indexes (e.g., B-trees or tries), this algorithm can help efficiently locate the k-th entry in a range query without scanning all entries.

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Real-World Analogy

Imagine a library with books labeled using a numbering system: 1, 10, 100, 11, 12, 2, 20, …, 9, 90, etc., sorted lexicographically. The library has a limit on book numbers (e.g., ≤ n), and you need to find the 1000th book in this order. Instead of listing all books and sorting them, this script efficiently “navigates” the numbering system like a tree:

• Start at 1, count how many books start with 1 (1, 10, 100, 11, …).
• If the count is less than 1000, move to 2; otherwise, go deeper (e.g., to 10, then 100).
• Repeat until the 1000th book is found.

This approach saves time and memory, making it practical for large-scale systems.

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Why This Approach?

The script uses a prefix-based counting method to traverse the “tree” of numbers:

• Tree Structure: Numbers form a tree where each node (e.g., 1) has children (10, 100, 1000, …).
• Counting: For each prefix (e.g., 1), it counts how many numbers in that branch are ≤ n.
• Traversal: If the k-th number is in the current branch, go deeper (multiply by 10); otherwise, move to the next sibling (increment the prefix).

This method avoids generating the entire sequence, making it efficient for large n and k.

K-th number in a sequence

The code implements a solution to find the k-th number in a sequence where numbers are generated by multiplying a starting number by 10 in each step.

Code Explanation

countNumbersInRange(int n, long prefix1, long prefix2)

• Purpose: Counts how many numbers in the sequence fall within the range [prefix1, prefix2].
• Logic:
• Initialize steps = 0 to track the count of numbers.
• While prefix1 is less than or equal to n:
  • Add to steps the minimum of (long(n) + 1, prefix2) - prefix1. This calculates how many numbers exist between prefix1 and prefix2, capped by n + 1.
  • Multiply both prefix1 and prefix2 by 10 to move to the next “level” of numbers (e.g., from 1 to 10, 10 to 100, etc.).
• Return the total steps.

findKthNumber(int n, int k)

• Purpose: Finds the k-th smallest number in the sequence of numbers up to n, where numbers are generated by starting with 1 to 9 and repeatedly multiplying by 10 (e.g., 1, 10, 100, …).
• Logic:
• Start with curr = 1 (the first number in the sequence) and k-- (adjust k to 0-based counting).
• While k > 0:
  • Use countNumbersInRange to count how many numbers start with the current prefix curr (e.g., for curr = 1, count numbers like 1, 10, 100, etc., up to n).
  • If count <= k, the k-th number is not in this branch:
  • Subtract count from k.
  • Move to the next number at the same level (curr++), e.g., from 1 to 2.
  • Else, the k-th number is in this branch:
  • Multiply curr by 10 to go deeper (e.g., from 1 to 10).
  • Decrement k by 1.
• Return curr as the k-th number.

Example Walkthrough

Let’s trace the execution for n = 13, k = 2.

• The sequence of numbers up to n = 13 is: 1, 10, 100, 11, 12, 13, 2, 3, …, 9. These are generated by starting with 1 to 9 and multiplying by 10 as long as the result is ≤ n.
• Sorted order: 1, 10, 11, 12, 13, 2, 3, …, 9.
• We need the 2nd number (k = 2).

Step-by-Step Execution

1. Initialization:

• curr = 1, k = 2, then k-- → k = 1.

1. First Iteration (k = 1):

• Call countNumbersInRange(13, 1, 2):
  • prefix1 = 1, prefix2 = 2, steps = 0.
  • First loop: prefix1 = 1 ≤ 13:
  • steps += min((long(13) + 1, 2) - 1) = min(14, 2) - 1 = 2 - 1 = 1.
  • prefix1 = 1 * 10 = 10, prefix2 = 2 * 10 = 20.
  • Second loop: prefix1 = 10 ≤ 13:
  • steps += min((long(13) + 1, 20) - 10) = min(14, 20) - 10 = 14 - 10 = 4.
  • prefix1 = 10 * 10 = 100, prefix2 = 20 * 10 = 200.
  • Third loop: prefix1 = 100 > 13, break.
  • Return steps = 1 + 4 = 5.
• count = 5, k = 1:
  • Since count = 5 > k = 1, the k-th number is in this branch (starting with 1).
  • curr = 1 * 10 = 10, k-- → k = 0.

1. Second Iteration (k = 0):

• Since k = 0, exit the loop.
• Return curr = 10.

Result

• The 2nd number in the sequence is 10.
• The sorted sequence up to 13 is: 1, 10, 11, 12, 13, 2, 3, …, 9.
• Position 1: 1, Position 2: 10. So, the answer is correct.

Final Answer

For n = 13, k = 2, the k-th number is 10.

Let’s analyze the system and determine the relevance of private and public in this context.

• POST /api/v1/short-urls: To create a short URL from a long URL.
• GET /api/v1/short-urls/{alias}: To retrieve the original long URL using the short URL alias.

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Understanding the URL Shortening System

Components and Flow

1. Clients:

• Represented by two users with laptops, interacting with the system via HTTP requests.

1. API Gateway:

• Acts as an entry point for client requests.
• Handles routing of requests to the appropriate backend service (TinyURL Server).
• Supports two endpoints:
  • POST /api/v1/short-urls: A client sends a long URL to create a short URL.
  • GET /api/v1/short-urls/{alias}: A client requests the original long URL using the short URL alias (e.g., alias3).

1. TinyURL Server:

• Processes the requests:
  • For POST, it generates a short URL (alias) and writes the mapping (short URL → long URL) to the database.
  • For GET, it retrieves the long URL from the database using the alias.

1. Database:

• Stores the mapping between short URLs (aliases) and long URLs.
• The server performs:
  • write short URL: Stores the short URL and its corresponding long URL.
  • get long URL: Retrieves the long URL based on the short URL alias.

Flow Examples

• Shortening a URL:
• Client sends: POST /api/v1/short-urls with a long URL (e.g., “https://example.com/very/long/url”).
• API Gateway routes the request to the TinyURL Server.
• TinyURL Server generates a short alias (e.g., “alias3”) and writes the mapping to the database.
• Response: The short URL (e.g., “https://tinyurl.com/alias3”) is returned to the client.
• Retrieving a Long URL:
• Client sends: GET /api/v1/short-urls/alias3.
• API Gateway routes the request to the TinyURL Server.
• TinyURL Server queries the database to get the long URL.
• Response: The long URL (e.g., “https://example.com/very/long/url”) is returned to the client.

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Are private and public Useful Here?

The concepts of private and public typically apply to access control in programming languages like C++, Java, or Python, where they determine the visibility of class members (methods, variables).

class TinyURLServer {
public:
// Public method to handle POST request for creating a short URL
std::string createShortURL(const std::string& long_url) {
std::string alias = generateAlias();
writeToDatabase(alias, long_url);
return alias;
}

// Public method to handle GET request for retrieving the long URL
std::string getLongURL(const std::string& alias) {
return readFromDatabase(alias);
}

private:
// Private method to generate a unique alias
std::string generateAlias() {
// Logic to generate a unique alias (e.g., “alias3”)
return “alias3”;
}

// Private method to write to the database
void writeToDatabase(const std::string& alias, const std::string& long_url) {
// Database write logic
}

// Private method to read from the database
std::string readFromDatabase(const std::string& alias) {
// Database read logic
return “https://example.com/very/long/url”;
}
};

In the context of this URL shortening system, we need to consider whether private and public access modifiers are relevant and useful, both at the code level and at the system architecture level.

1. At the Code Level (Implementation of TinyURL Server)

If the TinyURL Server is implemented in a language that supports access modifiers (e.g., C++, Java, Python), private and public are useful for encapsulating the server’s logic:

• The methods that handle the API endpoints should be public because they need to be accessible to the API Gateway or external callers.
• Here, createShortURL and getLongURL are public because they are the entry points for the API Gateway to interact with the TinyURL Server.
• Methods like generateAlias, writeToDatabase, and readFromDatabase are private because they are internal implementation details that shouldn’t be exposed to external callers. This encapsulation:
  • Prevents misuse (e.g., calling writeToDatabase directly with invalid data).
  • Allows the server to change its internal logic (e.g., how aliases are generated) without affecting external code.
• Usefulness:
• public ensures the API Gateway can call the necessary methods.
• private protects the internal logic, improving security and maintainability.

2. At the System Architecture Level (API Endpoints and Access Control)

In the context of the API design shown in the diagram, private and public can be interpreted as access control for the API endpoints and services:

• Public:
• The API endpoints (POST /api/v1/short-urls and GET /api/v1/short-urls/{alias}) are public in the sense that they are exposed to external clients (users of the TinyURL service).
• These endpoints need to be publicly accessible so that clients can shorten URLs and retrieve long URLs.
• Private:
• The interaction between the TinyURL Server and the Database (e.g., write short URL, get long URL) should be private in the sense that it’s internal to the system and not exposed to external clients.
• The API Gateway shouldn’t expose database operations directly to clients. For example:
  • A client should not be able to directly call a POST /api/v1/database/write endpoint to modify the database.
  • Instead, the TinyURL Server handles database operations internally, ensuring that:
  • Data validation occurs (e.g., checking if the long URL is valid).
  • Security measures are applied (e.g., preventing SQL injection).
  • The database schema can change without affecting clients.
• Usefulness:
• Public API Endpoints: Necessary for clients to use the service.
• Private Database Operations: Ensures security and encapsulation at the system level. Clients only interact with the API Gateway, and the TinyURL Server controls database access.

3. In the Context of the Previous Code (K-th Number Sequence)

The earlier discussion was about a C implementation of the k-th number in a lexicographical sequence. That code didn’t directly relate to the URL shortening system, but let’s connect the two:

• The URL shortening system might need to generate unique aliases (e.g., “alias1”, “alias2”, “alias3”, …).
• One way to generate aliases is to use a sequence of numbers (1, 2, 3, …) and map them to strings (e.g., 1 → “alias1”).
• The k-th number sequence script could be used to generate these numbers efficiently, especially if the aliases are based on a lexicographically ordered sequence.
• In that case:
• The find_kth_number_in_sequence function would be private within the TinyURL Server, as it’s an internal implementation detail for generating aliases.
• The POST /api/v1/short-urls endpoint would be public, as it’s the external interface for clients to create short URLs.

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Why private and public Are Useful in This System

1. Encapsulation:

• private methods (e.g., database operations, alias generation) hide the internal workings of the TinyURL Server, making the system easier to maintain and modify.
• Example: If the database changes from SQL to NoSQL, only the private database methods need to be updated; the public API endpoints remain unchanged.

1. Security:

• Keeping database operations private prevents clients from directly manipulating the database, reducing the risk of attacks (e.g., SQL injection, data corruption).
• public endpoints can have additional security measures (e.g., authentication, rate limiting) applied at the API Gateway.

1. Modularity:

• public endpoints define a clear contract for clients, while private methods allow the TinyURL Server to organize its logic internally.
• This separation makes it easier to scale the system (e.g., adding new features like URL expiration without changing the public API).

1. Reusability:

• private helper functions (e.g., generateAlias) can be reused within the TinyURL Server for other operations (e.g., generating aliases for different types of URLs).

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Conclusion

Yes, private and public are useful for this URL shortening system:

• At the Code Level: public methods expose the necessary functionality (e.g., handling POST and GET requests), while private methods encapsulate internal logic (e.g., database operations, alias generation).
• At the System Level: The API endpoints are public to allow client access, but database interactions are private to ensure security and encapsulation.
• Connection to the K-th Number Script: If the TinyURL Server uses a sequence-based alias generation (like the k-th number script), that logic would be private within the server, while the API endpoints remain public.

This design aligns with good software engineering practices, ensuring the system is secure, maintainable, and scalable.