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    LeetCode link: 494. Target Sum, difficulty: Medium.

    You are given an integer array nums and an integer target.

    You want to build an expression out of nums by adding one of the symbols + and - before each integer in nums and then concatenate all the integers.

    • For example, if nums = [2, 1], you can add a + before 2 and a - before 1 and concatenate them to build the expression +2-1.

    Return the number of different expressions that you can build, which evaluates to target.

    Example 1:

    Input: nums = [1,1,1,1,1], target = 3

    Output: 5

    Explanation:

    -1 + 1 + 1 + 1 + 1 = 3
    +1 - 1 + 1 + 1 + 1 = 3
    +1 + 1 - 1 + 1 + 1 = 3
    +1 + 1 + 1 - 1 + 1 = 3
    +1 + 1 + 1 + 1 - 1 = 3

    Example 2:

    Input: nums = [1], target = 1

    Output: 1

    Constraints:

    • 1 <= nums.length <= 20
    • 0 <= nums[i] <= 1000
    • 0 <= sum(nums[i]) <= 1000
    • -1000 <= target <= 1000

    Intuition
    PreviousNext

    This problem is quite difficult if you have not solved similar problems before. So before you start working on this question, it is recommended that you first work on another relatively simple question 416. Partition Equal Subset Sum that is similar to this one.

    Pattern of "Dynamic Programming"

    "Dynamic Programming" requires the use of the dp array to store the results. The value of dp[i][j] can be converted from its previous (or multiple) values ​​through a formula. Therefore, the value of dp[i][j] is derived step by step, and it is related to the previous dp record value.

    "Dynamic programming" is divided into five steps

    1. Determine the meaning of each value of the array dp.
    2. Initialize the value of the array dp.
    3. Fill in the dp grid data in order according to an example.
    4. Based on the dp grid data, derive the recursive formula.
    5. Write a program and print the dp array. If it is not as expected, adjust it.

    Detailed description of these five steps

    1. Determine the meaning of each value of the array dp.
      • First determine whether dp is a one-dimensional array or a two-dimensional array. A one-dimensional rolling array means that the values ​​of the array are overwritten at each iteration. Most of the time, using one-dimensional rolling array instead of two-dimensional array can simplify the code; but for some problems, such as operating "two swappable arrays", for the sake of ease of understanding, it is better to use two-dimensional array.
      • Try to use the meaning of the return value required by the problem as the meaning of dp[i] (one-dimensional) or dp[i][j] (two-dimensional). It works about 60% of the time. If it doesn't work, try other meanings.
      • Try to save more information in the design. Repeated information only needs to be saved once in a dp[i].
      • Use simplified meanings. If the problem can be solved with boolean value, don't use numeric value.
    2. Initialize the value of the array dp. The value of dp involves two levels:
      1. The length of dp. Usually: condition array length plus 1 or condition array length.
      2. The value of dp[i] or dp[i][j]. dp[0] or dp[0][0] sometimes requires special treatment.
    3. Fill in the dp grid data in order according to an example.
      • The "recursive formula" is the core of the "dynamic programming" algorithm. But the "recursive formula" is obscure. If you want to get it, you need to make a table and use data to inspire yourself.
      • If the original example is not good enough, you need to redesign one yourself.
      • According to the example, fill in the dp grid data "in order", which is very important because it determines the traversal order of the code.
      • Most of the time, from left to right, from top to bottom. But sometimes it is necessary to traverse from right to left, from bottom to top, from the middle to the right (or left), such as the "palindrome" problems. Sometimes, it is necessary to traverse a line twice, first forward and then backward.
      • When the order is determined correctly, the starting point is determined. Starting from the starting point, fill in the dp grid data "in order". This order is also the order in which the program processes.
      • In this process, you will get inspiration to write a "recursive formula". If you can already derive the formula, you do not need to complete the grid.
    4. Based on the dp grid data, derive the recursive formula.
      • There are three special positions to pay attention to: dp[i - 1][j - 1], dp[i - 1][j] and dp[i][j - 1], the current dp[i][j] often depends on them.
      • When operating "two swappable arrays", due to symmetry, we may need to use dp[i - 1][j] and dp[i][j - 1] at the same time.
    5. Write a program and print the dp array. If it is not as expected, adjust it.
      • Focus on analyzing those values that are not as expected.

    After reading the above, do you feel that "dynamic programming" is not that difficult? Try to solve this problem. 🤗

    Step-by-Step Solution

    1. Determine the meaning of the dp[j]
      • dp[j] represents that by using the first i nums, the number of different expressions that you can build, which evaluates to j.
      • dp[j] is an integer.

    2. Determine the dp array's initial value

      • Use an example. We didn't use the Example 1: Input: nums = [1,1,1,1,1], target = 3 because it is too special and is not a good example for deriving a formula.
      • I made up an example: nums = [1,2,1,2], target = 4.
      • First, determine the size of the knapsack.
        • The target value may be very small, such as 0, so it alone cannot determine the size of the knapsack.
        • The sum of nums should also be taken into account to fully cover all knapsack sizes.
        • target may be negative, but considering that + and - are added to num arbitrarily, the dp[j] should be symmetrical around 0. So the result of negative target dp[target] is equal to dp[abs(target)].
        • So the size of the knapsack can be max(sum(nums), target) + 1.

      • Second, determine what are the items. The items are the nums in this problem.

        So after initialization, the 'dp' array would be:
#   0 1 2 3 4 5 6
#   1 0 0 0 0 0 0 # dp
# 1
# 2 
# 1
# 2

      • dp[0] is set to 1, indicating that an empty knapsack can be achieved by not using any nums. In addition, it is used as the starting value, and the subsequent dp[j] will depend on it. If it is 0, all values of dp[j] will be 0.

      • dp[j] = 0 (j != 0), indicating that it is impossible to get j with no nums.

    3. According to an example, fill in the dp grid data "in order".

      1. Use the first num '1'.
#   0 1 2 3 4 5 6
#   1 0 0 0 0 0 0
# 1 0 1 0 0 0 0 0 # dp
# 2
# 1
# 2

      2. Use the second num '2'.
#   0 1 2 3 4 5 6
#   1 0 0 0 0 0 0
# 1 0 1 0 0 0 0 0
# 2 0 1 0 1 0 0 0
# 1
# 2

      3. Use the third num '1'.
#   0 1 2 3 4 5 6
#   1 0 0 0 0 0 0
# 1 0 1 0 0 0 0 0
# 2 0 1 0 1 0 0 0
# 1 2 0 2 0 1 0 0
# 2

      4. Use the fourth num '2'.
#   0 1 2 3 4 5 6
#   1 0 0 0 0 0 0
# 1 0 1 0 0 0 0 0
# 2 0 1 0 1 0 0 0
# 1 2 0 2 0 1 0 0
# 2 4 0 3 0 2 0 1 # dp

    4. According to the dp grid data, derive the "recursive formula".

      dp[j] = dp[abs(j - nums[i])] + dp[j + nums[i]]
      • If j < nums[i], dp[j - nums[i]] will raise array index out of range exception. So we use the dp[abs(j - num)] which is equal to it, because the dp[j] are symmetrical around 0, such as dp[-j] equals to dp[j] (-j is an imaginary index).
      • When j + nums[i] >= dp.length, dp[j + nums[i]] must be 0 to prevent interference.

    5. Write a program and print the dp array. If it is not as expected, adjust it.

    Complexity

    Time complexity

    O(n * sum)

    Space complexity

    O(n * sum)

    Solution in languages: C# Python C++ Java JavaScript Go Ruby

    C# # 

    public class Solution
{
    public int FindTargetSumWays(int[] nums, int target)
    {
        target = Math.Abs(target);

        var dp = new int[Math.Max(nums.Sum(), target) + 1];
        dp[0] = 1;

        foreach (var num in nums)
        {
            var dc = (int[])dp.Clone();

            for (var j = 0; j < dp.Length; j++)
            {
                dp[j] = dc[Math.Abs(j - num)] + (j + num < dp.Length ? dc[j + num] : 0);
            }
        }

        return dp[target];
    }
}

    Python # 

    class Solution:
    def findTargetSumWays(self, nums: List[int], target: int) -> int:
        target = abs(target)

        dp = [0] * (max(sum(nums), target) + 1)
        dp[0] = 1

        for num in nums:
            dc = dp.copy()

            for j in range(len(dp)):
                dp[j] = dc[abs(j - num)] + (dc[j + num] if j + num < len(dp) else 0)

        return dp[target]

    C++ # 

    class Solution {
public:
    int findTargetSumWays(vector<int>& nums, int target) {
        auto sum = reduce(nums.begin(), nums.end());
        target = abs(target);

        auto dp = vector<int>(max(sum, target) + 1);
        dp[0] = 1;

        for (auto num : nums) {
            auto dc = dp;

            for (auto j = 0; j < dp.size(); j++) {
                dp[j] = dc[abs(j - num)] + (j + num < dp.size() ? dc[j + num] : 0);
            }
        }

        return dp[target];
    }
};

    Java # 

    class Solution {
    public int findTargetSumWays(int[] nums, int target) {
        var sum = IntStream.of(nums).sum();
        target = Math.abs(target);

        var dp = new int[Math.max(sum, target) + 1];
        dp[0] = 1;

        for (var num : nums) {
            var dc = dp.clone();

            for (var j = 0; j < dp.length; j++) {
                dp[j] = dc[Math.abs(j - num)] + (j + num < dp.length ? dc[j + num] : 0);
            }
        }

        return dp[target];
    }
}

    JavaScript # 

    var findTargetSumWays = function (nums, target) {
   target = Math.abs(target)

   const dp = Array(Math.max(_.sum(nums), target) + 1).fill(0)
   dp[0] = 1

   for (const num of nums) {
      const dc = [...dp]

      for (let j = 0; j < dp.length; j++) {
         dp[j] = dc[Math.abs(j - num)] + (j + num < dp.length ? dc[j + num] : 0)
      }
   }

   return dp[target]
};

    Go # 

    func findTargetSumWays(nums []int, target int) int {
    sum := 0
    for _, num := range nums {
        sum += num
    }
    target = int(math.Abs(float64(target)))

    dp := make([]int, max(sum, target) + 1)
    dp[0] = 1

    for _, num := range nums {
        dc := slices.Clone(dp)

        for j := 0; j < len(dp); j++ {
            addition := 0
            if j + num < len(dp) {
                addition = dc[j + num]
            }
            dp[j] = dc[int(math.Abs(float64((j - num))))] + addition
        }
    }

    return dp[target]
}

    Ruby # 

    def find_target_sum_ways(nums, target)
  target = target.abs

  dp = Array.new([ nums.sum, target ].max + 1, 0)
  dp[0] = 1

  nums.each do |num|
    dc = dp.clone

    dp.each_with_index do |_, j|
      dp[j] = dc[(j - num).abs] + (j + num < dp.size ? dc[j + num] : 0)
    end
  end

  dp[target]
end
    Solution in languages: C# Python C++ Java JavaScript Go Ruby

    Other languages

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