Create subsequences-with-a-unique-middle-mode-ii.py

Author: kamyu104

Python/subsequences-with-a-unique-middle-mode-ii.py

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+# Time:  O(n)
+# Space: O(n)
+
+import collections
+
+
+# freq table, prefix sum, combinatorics
+class Solution(object):
+    def subsequencesWithMiddleMode(self, nums):
+        """
+        :type nums: List[int]
+        :rtype: int
+        """
+        def nC2(x):
+            return x*(x-1)//2
+
+        MOD = 10**9+7
+        result = 0
+        left = collections.defaultdict(int)
+        right = collections.defaultdict(int)
+        for x in nums:
+            right[x] += 1
+        left_x_sq = 0  # sum(left[x]^2 for x != v)
+        right_x_sq = sum(v**2 for v in right.itervalues())  # sum(right[x]^2 for x != v)
+        left_x_right_x = 0 # sum(left[x]*right[x] for x != v)
+        left_x_sq_right_x = 0 # sum(left[x]^2*right[x] for x != v)
+        left_x_right_x_sq = 0 # sum(left[x]*right[x]^2 for x != v)
+        for i, v in enumerate(nums):
+            left_x_sq -= left[v]**2
+            right_x_sq -= right[v]**2
+            left_x_right_x -= left[v]*right[v]
+            left_x_sq_right_x -= left[v]**2*right[v]
+            left_x_right_x_sq -= left[v]*right[v]**2
+            right[v] -= 1
+
+            l, r = i, len(nums)-(i+1)
+            # all possibles
+            result += nC2(l)*nC2(r)
+            # only mid is a
+            result -= nC2(l-left[v])*nC2(r-right[v])
+            # bb/a/ac
+            # sum((left[x]*(left[x]-1)//2)*right[v]*((r-right[v])-right[x]) for x != v)
+            result -= ((left_x_sq-(l-left[v]))*(r-right[v])-(left_x_sq_right_x-left_x_right_x))*right[v]//2
+            # ac/a/bb
+            # sum(left[v]*((l-left[v])-left[x])*(right[x]*(right[x]-1)//2) for x != v)
+            result -= ((right_x_sq-(r-right[v]))*(l-left[v])-(left_x_right_x_sq-left_x_right_x))*left[v]//2
+            # ab/a/bc
+            # sum(left[v]*left[x]*right[x]*((r-right[v])-right[x]) for x != v)
+            result -= left[v]*left_x_right_x*(r-right[v])-left[v]*left_x_right_x_sq
+            # bc/a/ab
+            # sum(left[x]*((l-left[v])-left[x])*right[v]*right[x] for x != v)
+            result -= right[v]*left_x_right_x*(l-left[v])-right[v]*left_x_sq_right_x
+            # bb/a/ab
+            # sum((left[x]*(left[x]-1)//2)*right[v]*right[x] for x != v)
+            result -= right[v]*(left_x_sq_right_x-left_x_right_x)//2
+            # ab/a/bb
+            # sum((right[x]*(right[x]-1)//2)*left[v]*left[x] for x != v)
+            result -= left[v]*(left_x_right_x_sq-left_x_right_x)//2
+
+            left[v] += 1
+            left_x_sq += left[v]**2
+            right_x_sq += right[v]**2
+            left_x_right_x += left[v]*right[v]
+            left_x_sq_right_x += left[v]**2*right[v]
+            left_x_right_x_sq += left[v]*right[v]**2
+        return result % MOD