Subset sum problem

Here is my algorithm. It runs in O(2^(n/2)) and solves SubsetSum(1000, list-of-1000-ones) in 20 milliseconds. See the comments at the end of IVlad’s post.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Diagnostics;

namespace SubsetSum
{
    class Program
    {
        static void Main(string[] args)
        {
            var ns = new List<int>();
            for (int i = 0; i < 1000; i++) ns.Add(1);
            var s1 = Stopwatch.StartNew();
            bool result = SubsetSum(ns, 1000);
            s1.Stop();
            Console.WriteLine(result);
            Console.WriteLine(s1.Elapsed);
            Console.Read();
        }

        static bool SubsetSum(ist<int> nums, int targetL)
        {
            var left = new List<int> { 0 };
            var right = new List<int> { 0 };
            foreach (var n in nums)
            {
                if (left.Count < right.Count) left = Insert(n, left);
                else right = Insert(n, right);
            }
            int lefti = 0, righti = right.Count - 1;
            while (lefti < left.Count && righti >= 0)
            {
                int s = left[lefti] + right[righti];
                if (s < target) lefti++;
                else if (s > target) righti--;
                else return true;
            }
            return false;
        }

        static List<int> Insert(int num, List<int> nums)
        {
            var result = new List<int>();
            int lefti = 0, left = nums[0]+num;
            for (var righti = 0; righti < nums.Count; righti++)
            {

                int right = nums[righti];
                while (left < right)
                {
                    result.Add(left);
                    left = nums[++lefti] + num;
                }
                if (right != left) result.Add(right);
            }
            while (lefti < nums.Count) result.Add(nums[lefti++] + num);
            return result;
        }
    }
}

And here is an improved version that prunes the sets:

static bool SubsetSum(List<int> nums, int target)
{
    var remainingSum = nums.Sum();
    var left = new List<int> { 0 };
    var right = new List<int> { 0 };
    foreach (var n in nums)
    {
        if (left.Count == 0 || right.Count == 0) return false;
        remainingSum -= n;
        if (left.Count < right.Count) left = Insert(n, left, target - remainingSum - right.Last(), target);
        else right = Insert(n, right, target - remainingSum - left.Last(), target);
    }
    int lefti = 0, righti = right.Count - 1;
    while (lefti < left.Count && righti >= 0)
    {
        int s = left[lefti] + right[righti];
        if (s < target) lefti++;
        else if (s > target) righti--;
        else return true;
    }
    return false;
}

static List<int> Insert(int num, List<int> nums, int min, int max)
{
    var result = new List<int>();
    int lefti = 0, left = nums[0]+num;
    for (var righti = 0; righti < nums.Count; righti++)
    {

        int right = nums[righti];
        while (left < right)
        {
            if (min <= left && left <= max) result.Add(left);
            left = nums[++lefti] + num;
        }
        if (right != left && min <= right && right <= max) result.Add(right);
    }
    while (lefti < nums.Count)
    {
        left = nums[lefti++] + num;
        if (min <= left && left <= max) result.Add(left);
    } 
    return result;
}

This last one solved the problem with 100000 ones in about 5 milliseconds (but this is a best case of the algorithm, with real world data it will be slower).

For your use this algorithm is probably fast enough (and I don’t see any obvious improvements). If you enter 10,000 products with a random price between 0 and 20 and your goal is to sum to 500 this is solved in 0.04 seconds on my laptop.

Edit: I just read on Wikipedia that the best known algorithm is O(2^(n/2)*n). This one is O(2^(n/2)). Do I get the Turing Award?