adding algo

Author: ivan1016017

src/my_project/interviews/amazon_high_frequency_23/round_2/k_free_subsets.py

@@ -0,0 +1,57 @@
+from typing import List, Union, Collection, Mapping, Optional
+from collections import defaultdict
+
+class Solution:
+    def countTheNumOfKFreeSubsets(self, nums: List[int], k: int) -> int:
+        """
+        Count k-Free subsets using dynamic programming.
+        
+        Approach:
+        1. Group elements by (num % k) to find independent groups
+        2. Within each group, sort and build chains where elements differ by k
+        3. For each chain, use House Robber DP to count valid subsets
+        4. Multiply results across all independent chains
+        
+        Time: O(n log n), Space: O(n)
+        """
+        # Group numbers by their remainder when divided by k
+        groups = defaultdict(list)
+        for num in nums:
+            groups[num % k].append(num)
+        
+        res = 1
+        
+        # Process each group independently
+        for group in groups.values():
+            group.sort()
+            
+            # Build chains within this group
+            i = 0
+            while i < len(group):
+                chain = [group[i]]
+                j = i + 1
+                
+                # Build chain where each element is exactly k more than previous
+                while j < len(group) and group[j] == chain[-1] + k:
+                    chain.append(group[j])
+                    j += 1
+                
+                # House Robber DP for this chain
+                m = len(chain)
+                if m == 1:
+                    chain_res = 2  # {} or {chain[0]}
+                else:
+                    take = 1  # Take first element
+                    skip = 1  # Skip first element
+                    
+                    for idx in range(1, m):
+                        new_take = skip  # Can only take current if we skipped previous
+                        new_skip = take + skip  # Can skip current regardless
+                        take, skip = new_take, new_skip
+                    
+                    chain_res = take + skip
+                
+                res *= chain_res
+                i = j
+        
+        return res