416. Partition Equal Subset Sum

Approach 1: 2D DP

• Time: $O(nk)$, where $k = \Sigma |\texttt{nums[i]}| / 2$
• Space: $O(nk)$

C++JavaPython

class Solution {
 public:
  bool canPartition(vector<int>& nums) {
    const int sum = accumulate(nums.begin(), nums.end(), 0);
    if (sum % 2 == 1)
      return false;
    return knapsack(nums, sum / 2);
  }

 private:
  bool knapsack(const vector<int>& nums, int subsetSum) {
    const int n = nums.size();
    // dp[i][j] := true if j can be formed by nums[0..i)
    vector<vector<bool>> dp(n + 1, vector<bool>(subsetSum + 1));
    dp[0][0] = true;

    for (int i = 1; i <= n; ++i) {
      const int num = nums[i - 1];
      for (int j = 0; j <= subsetSum; ++j)
        if (j < num)
          dp[i][j] = dp[i - 1][j];
        else
          dp[i][j] = dp[i - 1][j] || dp[i - 1][j - num];
    }

    return dp[n][subsetSum];
  }
};

class Solution {
  public boolean canPartition(int[] nums) {
    final int sum = Arrays.stream(nums).sum();
    if (sum % 2 == 1)
      return false;
    return knapsack(nums, sum / 2);
  }

  private boolean knapsack(int[] nums, int subsetSum) {
    final int n = nums.length;
    // dp[i][j] := true if j can be formed by nums[0..i)
    boolean[][] dp = new boolean[n + 1][subsetSum + 1];
    dp[0][0] = true;

    for (int i = 1; i <= n; ++i) {
      final int num = nums[i - 1];
      for (int j = 0; j <= subsetSum; ++j)
        if (j < num)
          dp[i][j] = dp[i - 1][j];
        else
          dp[i][j] = dp[i - 1][j] || dp[i - 1][j - num];
    }

    return dp[n][subsetSum];
  }
}

class Solution:
  def canPartition(self, nums: list[int]) -> bool:
    summ = sum(nums)
    if summ % 2 == 1:
      return False
    return self.knapsack_(nums, summ // 2)

  def knapsack_(self, nums: list[int], subsetSum: int) -> bool:
    n = len(nums)
    # dp[i][j] := True if j can be formed by nums[0..i)
    dp = [[False] * (subsetSum + 1) for _ in range(n + 1)]
    dp[0][0] = True

    for i in range(1, n + 1):
      num = nums[i - 1]
      for j in range(subsetSum + 1):
        if j < num:
          dp[i][j] = dp[i - 1][j]
        else:
          dp[i][j] = dp[i - 1][j] or dp[i - 1][j - num]

    return dp[n][subsetSum]

Approach 2: 1D DP

• Time: $O(nk)$
• Space: $O(k)$

C++JavaPython

class Solution {
 public:
  bool canPartition(vector<int>& nums) {
    const int sum = accumulate(nums.begin(), nums.end(), 0);
    if (sum % 2 == 1)
      return false;
    return knapsack(nums, sum / 2);
  }

 private:
  bool knapsack(const vector<int>& nums, int subsetSum) {
    // dp[i] := true if i can be formed by nums so far
    vector<bool> dp(subsetSum + 1);
    dp[0] = true;

    for (const int num : nums)
      for (int i = subsetSum; i >= num; --i)
        dp[i] = dp[i] || dp[i - num];

    return dp[subsetSum];
  }
};

class Solution {
  public boolean canPartition(int[] nums) {
    final int sum = Arrays.stream(nums).sum();
    if (sum % 2 == 1)
      return false;

    return knapsack(nums, sum / 2);
  }

  private boolean knapsack(int[] nums, int subsetSum) {
    // dp[i] := true if i can be formed by nums so far
    boolean[] dp = new boolean[subsetSum + 1];
    dp[0] = true;

    for (final int num : nums)
      for (int i = subsetSum; i >= num; --i)
        dp[i] = dp[i] || dp[i - num];

    return dp[subsetSum];
  }
}

class Solution:
  def canPartition(self, nums: list[int]) -> bool:
    summ = sum(nums)
    if summ % 2 == 1:
      return False
    return self.knapsack_(nums, summ // 2)

  def knapsack_(self, nums: list[int], subsetSum: int) -> bool:
    # dp[i] := True if i can be formed by nums so far
    dp = [False] * (subsetSum + 1)
    dp[0] = True

    for num in nums:
      for i in range(subsetSum, num - 1, -1):
        dp[i] = dp[i] or dp[i - num]

    return dp[subsetSum]