Add Profitable Schemes dynamic programming solutions in R

[Siddhram]

This PR introduces a complete and well-documented implementation of the Profitable Schemes problem (LeetCode 879) in R.

Overview
The implementation provides solutions for counting profitable schemes using dynamic programming. Both top-down memoized and bottom-up DP approaches are included, handling multi-dimensional states and modular arithmetic.

Features
Top-down DP with memoization:

• State: (n_remain, start_i, gain) with gain capped at 100
• Modulo 1e9+7
  Bottom-up DP:
• 2D DP over people and capped profit
• Modulo 1e9+7
  Example demonstrations included:
• Simple case
• Larger arrays
• Edge case with minProfit = 0
  Clear headers, complexity notes, and consistent style with other DP files.

Complexity
Time Complexity: O(n · minProfit · m)
Space Complexity:

• Top-down: O(n · minProfit · m)
• Bottom-up: O(n · minProfit)

Demonstration
Run the script to execute built-in examples:

Windows:

Rscript "R/dynamic_programming/profitable_schemes.r"

[siriak]

Same question as in a different PR, don't change formatting

[Copilot]

Pull Request Overview

This PR adds a comprehensive implementation of the Profitable Schemes dynamic programming problem (LeetCode 879) to the R algorithms repository. The implementation includes both top-down memoized and bottom-up DP approaches for counting profitable schemes using multi-dimensional states and modular arithmetic.

Key Changes

• Implements two dynamic programming solutions with memoization and bottom-up approaches
• Adds comprehensive documentation including complexity analysis and algorithm explanation
• Includes multiple test examples demonstrating various scenarios and edge cases

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 3 comments.

File	Description
dynamic_programming/profitable_schemes.r	New algorithm implementation with dual DP approaches and examples
DIRECTORY.md	Updated to include the new Profitable Schemes entry in the Dynamic Programming section

[Copilot]

The loop logic doesn't properly implement subset enumeration. This attempts to take all items from start_i onwards rather than considering each item as a take/skip decision. The loop should either iterate through all items with take/skip choices or implement a different recursive pattern.

Suggested change

	# Try all choices from start_i to end (subset enumeration by advancing index)
	for (take in start_i:(m - 1L)) {
	newGain <- gain + profit[take + 1L]
	if (newGain > maxGain) newGain <- maxGain
	poss <- (poss + dp(n_remain - group[take + 1L], take + 1L, newGain)) %% MOD
	}
	# Subset enumeration: take/skip current crime at start_i
	# Skip current crime
	poss <- (poss + dp(n_remain, start_i + 1L, gain)) %% MOD
	# Take current crime
	newGain <- gain + profit[start_i + 1L]
	if (newGain > maxGain) newGain <- maxGain
	poss <- (poss + dp(n_remain - group[start_i + 1L], start_i + 1L, newGain)) %% MOD

[Copilot]

The loop iterates from n to 0, but R sequences like n:0 when n=0 will create c(0, -1, -2, ..., 0) which is incorrect. Use seq(n, 0, by=-1) or add a condition to handle the n=0 case properly.

Suggested change

	for (people in n:0) {
	for (people in seq(n, 0, by = -1)) {

[Copilot]

When minProfit > maxGain (100), this creates an invalid range that will cause an error. The code should handle cases where minProfit exceeds the capped gain limit by returning 0.

[siriak]

We do not add Leetcode problems