feat: 416

Author: Alonza0314

Medium/416 Partition Equal Subset Sum.md

@@ -66,4 +66,4 @@ func canPartition(nums []int) bool {
     }
     return dp[sum]
 }
-```
+```

Week12/416 AC.png

@@ -0,0 +1,69 @@
+# 416. Partition Equal Subset Sum
+
+## Intuition
+
+The problem can be transformed into finding if we can select some numbers from the array that sum up to half of the total sum. This is essentially a 0/1 knapsack problem where we need to determine if we can fill a knapsack of capacity sum/2 using the given numbers.
+
+## Approach
+
+1. First, calculate the total sum of the array
+2. If the sum is odd, return false as equal partition is impossible
+3. Divide the sum by 2 to get our target sum
+4. Use dynamic programming to solve:
+   - Create a dp array where dp[j] represents if we can make sum j using the numbers
+   - For each number, we can either include it or not include it
+   - Use a temporary array to avoid affecting previous calculations
+   - If at any point we can make the target sum, return true
+5. As an optimization, we also:
+   - Sort the array first
+   - Use binary search to check if target sum exists directly in array
+
+## Complexity
+
+- Time complexity: O(n * sum)
+- Space complexity: O(sum)
+
+## Keywords
+
+- Dynamic Programming
+- 0/1 Knapsack
+- Binary Search
+
+## Code
+
+```go
+func canPartition(nums []int) bool {
+    sum := 0
+    for _, num := range nums {
+        sum += num
+    }
+    if sum & 1 == 1 {
+        return false
+    }
+    sum >>= 1
+    sort.Ints(nums)
+    _, flag := slices.BinarySearch(nums, sum)
+    if flag {
+        return true
+    }
+    dp := make([]bool, sum + 1)
+    dp[0] = true
+    for _, num := range nums {
+        tmp := make([]bool, sum + 1)
+        tmp[0] = true
+        for j := 1; j <= sum ; j += 1 {
+            tmp[j] = dp[j]
+            if j >= num {
+                tmp[j] = (tmp[j] || dp[j - num])
+            }
+        }
+
+        if tmp[sum] {
+            return true
+        }
+
+        copy(dp, tmp)
+    }
+    return dp[sum]
+}
+```