How to Optimize the Runtime of a Recursive Subset Sum Algorithm

What strategies can I use to decrease the runtime of my recursive subset sum algorithm?

The subset sum problem is a classic computer science challenge that requires finding a subset of numbers from a list that sum up to a specified target number. Running this algorithm recursively can lead to inefficiencies due to overlapping subproblems. This answer explores strategies to optimize the runtime of the recursive approach.

def subset_sum(arr, target, memo={}):\n    # Check if the target is already computed\n    if target in memo: return memo[target]\n    # Base cases\n    if target == 0: return True\n    if target < 0 or len(arr) == 0: return False\n    # Include the last element or exclude it\n    result = subset_sum(arr[:-1], target - arr[-1], memo) or subset_sum(arr[:-1], target, memo)\n    memo[target] = result  # Store in memo\n    return result

Causes

• Exponential time complexity due to the recursive nature of the algorithm.
• Redundant calculations for overlapping subproblems and lack of memoization.
• Lack of pruning strategies that could eliminate unnecessary recursive calls.

Solutions

• Implement memoization to store previously computed results and avoid redundant calculations.
• Use dynamic programming for a more efficient solution that scales better.
• Incorporate pruning techniques that stop unnecessary recursive calls based on current sum comparisons.

Mistake: Not using memoization, leading to repeated calculations.

Solution: Implement memoization to store results of previous calls.

Mistake: Ignoring negative numbers that can directly affect the sum and lead to infinite recursion.

Solution: Handle negative numbers appropriately in the base case checks.

Mistake: Failing to prune recursive calls that obviously won't yield valid results.

Solution: Incorporate conditions to terminate unnecessary recursion early.