Update number-of-sets-of-k-non-overlapping-line-segments.py

Author: kamyu104

Python/number-of-sets-of-k-non-overlapping-line-segments.py

@@ -1,7 +1,41 @@
-# Time:  O(min(k, min(n - k)))
-# Space: O(1)
+# Time:  O(1), excluding precomputation time
+# Space: O(n)
 
 class Solution(object):
+    def __init__(self):
+        MOD = 10**9+7
+        MAX_N = 1000
+        self.__fact = [0]*(2*MAX_N-1+1)
+        self.__inv = [0]*(2*MAX_N-1+1)
+        self.__inv_fact = [0]*(2*MAX_N-1+1)
+        self.__fact[0] = self.__inv_fact[0] = self.__fact[1] = self.__inv_fact[1] = self.__inv[1] = 1
+        for i in xrange(2, len(self.__fact)):
+            self.__fact[i] = self.__fact[i-1]*i % MOD
+            self.__inv[i] = self.__inv[MOD%i]*(MOD-MOD//i) % MOD  # https://cp-algorithms.com/algebra/module-inverse.html
+            self.__inv_fact[i] = self.__inv_fact[i-1]*self.__inv[i] % MOD
+
+    def numberOfSets(self, n, k):
+        """
+        :type n: int
+        :type k: int
+        :rtype: int
+        """
+        MOD = 10**9+7
+        def nCr(n, k, mod):
+            return (self.__fact[n]*self.__inv_fact[n-k] % mod) * self.__inv_fact[k] % mod
+    
+        # find k segments with 1+ length and (k+1) spaces with 0+ length s.t. total length is n-1
+        # => find k segments with 0+ length and (k+1) spaces with 0+ length s.t. total length is n-k-1
+        # => find the number of combinations of 2k+1 variables with total sum n-k-1
+        # => H(2k+1, n-k-1)
+        # => C((2k+1) + (n-k-1) - 1, n-k-1)
+        # => C(n+k-1, n-k-1) = C(n+k-1, 2k)
+        return nCr(n+k-1, 2*k, MOD)
+
+
+# Time:  O(min(k, min(n - k)))
+# Space: O(1)
+class Solution2(object):
     def numberOfSets(self, n, k):
         """
         :type n: int