Update unique-paths.py

Author: kamyu104

Python/unique-paths.py

@@ -1,16 +1,39 @@
-# Time:  O(m * n)
-# Space: O(m + n)
+# Time:  O(m + n)
+# Space: O(1)
 
 class Solution(object):
-    # @return an integer
     def uniquePaths(self, m, n):
+        """
+        :type m: int
+        :type n: int
+        :rtype: int
+        """
+        def nCr(n, r):  # Time: O(n), Space: O(1)
+            if n-r < r:
+                r = n-r
+            c = 1
+            for k in xrange(1, r+1):
+                c *= n-k+1
+                c //= k
+            return c
+
+        return nCr((m-1)+(n-1), n-1)
+
+
+# Time:  O(m * n)
+# Space: O(m + n)
+class Solution2(object):
+    def uniquePaths(self, m, n):
+        """
+        :type m: int
+        :type n: int
+        :rtype: int
+        """
         if m < n:
-            return self.uniquePaths(n, m)
-        ways = [1] * n
+            m, n  = n, m
 
+        dp = [1]*n
         for i in xrange(1, m):
             for j in xrange(1, n):
-                ways[j] += ways[j - 1]
-
-        return ways[n - 1]
-
+                dp[j] += dp[j-1]
+        return dp[n-1]