Pre-Computation Techniques & Hashing in Competitive Programming

Introduction

Competitive programming often involves solving complex problems within tight time constraints. One common challenge is the repeated computation of factorials, especially when dealing with large inputs. In this article, we'll explore the basics of pre-computation techniques and hashing, specifically focusing on how to efficiently compute factorials modulo a large prime number. This discussion is based on the Competitive Programming Course.

Problem Statement

Imagine you're given multiple test cases, and in each test case, you need to compute the factorial of a number n. The results must be computed modulo 10^9+7. This problem is a classic example where pre-computation can save a significant amount of time.

Understanding Factorials

A factorial of a non-negative integer n (denoted as n!) is the product of all positive integers less than or equal to n. Mathematically, it can be expressed as:

Modulo Operation

In competitive programming, results often need to be computed modulo a large prime number to avoid overflow and ensure the results fit within standard data types. A common modulus used is 10^9+7. Thus, for our factorial computation, we'll be using:

n! %(10^9+7)

Importance of Pre-Computation

Without pre-computation, calculating the factorial for each test case individually would be inefficient, especially for large values of n. Instead, we can pre-compute the factorials for all possible values up to the maximum n in advance. This allows us to answer each test case in constant time O(1).

Steps for Pre-Computation

1. Initialize an Array: Create an array named 'factorial' where 'factorial[i]' will store 'i!' modulo (10^9+7).
2. Base Case: Set 'factorial [0]' to 1 because 0! = 1.
3. Iterative Computation: For each 'i' from 1 to the maximum 'n', compute 'factorial[i]' as ('factorial[i-1]' times 'i') modulo (10^9+7).

Example

Consider the following pre-computation:

• factorial [0] = 1
• factorial [1] = (factorial [0] × 1) % (10^9 + 7) = 1
• factorial [2] = (factorial [1] × 2) % (10^9 + 7) = 2
• factorial [3] = (factorial [2] × 3) % (10^9 + 7) = 6
• And so on...

Efficiently Handling Multiple Test Cases

Once the pre-computation is done, handling multiple test cases becomes straightforward:

• For each test case, simply return factorial[n].
• This approach ensures that each query is processed in O (1) time.

Conclusion

Pre-computation techniques are invaluable in competitive programming. By calculating factorials modulo (10^9 + 7), we can efficiently manage numerous test cases with substantial inputs. This approach not only enhances performance but also streamlines the solving of intricate problems. Continue to delve into and practice these methods to improve your problem-solving abilities in competitive programming. Meanwhile, enjoy coding!

Additional Resources

For more in-depth reading and practice, consider exploring these resources:

Competitive Programming Course

Introduction to Hashing Techniques

Advanced Pre-Computation Methods