## Description

There are two sorted arrays nums1 and nums2 of size m and n respectively.

Find the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)).

You may assume nums1 and nums2 cannot be both empty.

Example 1:

nums1 = [1, 3]
nums2 = [2]

The median is 2.0


Example 2:

nums1 = [1, 2]
nums2 = [3, 4]

The median is (2 + 3)/2 = 2.5


## Solutions

### 1. O(m+n)

如果不考虑 $O(m+n)$ 的要求，那么还是比较简单的。

# Time: O(m+n)
# space: O(m+n)
class Solution:
def findMedianSortedArrays(self, nums1: List[int], nums2: List[int]) -> float:
if not nums1 and nums2:
n2 = len(nums2)
if n2 % 2 == 1:
return nums2[n2 // 2]
else:
return (nums2[n2 // 2 - 1] + nums2[n2 // 2]) / 2
elif not nums2 and nums1:
n1 = len(nums1)
if n1 % 2 == 1:
return nums1[n1 // 2]
else:
return (nums1[n1 // 2 - 1] + nums1[n1 // 2]) / 2
else:
nums = []
n1, n2 = len(nums1), len(nums2)
i, j = 0, 0
while i < n1 and j < n2:
if nums1[i] <= nums2[j]:
nums.append(nums1[i])
i += 1
else:
nums.append(nums2[j])
j += 1
if i < n1:
nums += nums1[i:]
elif j < n2:
nums += nums2[j:]
# print('nums', nums)
# print(n1, n2, n1 + n2, len(nums), i, j)
if (n1 + n2) % 2 == 1:
return nums[(n1+n2) // 2]
else:
return (nums[(n1+n2) // 2 - 1] + nums[(n1+n2) // 2]) / 2
# 2085/2085 cases passed (92 ms)
# Your runtime beats 84.05 % of python3 submissions
# Your memory usage beats 100 % of python3 submissions (12.8 MB)


### 2. 二分法 O(log(m+n))

# Time: O(log(m+n))
# space: O(1)
class Solution:
def findMedianSortedArrays(self, nums1: List[int], nums2: List[int]) -> float:
n1, n2 = len(nums1), len(nums2)
# Make sure n1 <= n2
if n1 > n2:
return self.findMedianSortedArrays(nums2, nums1)
k = (n1 + n2 + 1) // 2
l, r = 0, n1
while l < r:
m1 = (l+r) >> 1
m2 = k - m1
if nums1[m1] <= nums2[m2-1]:
l = m1 + 1
else:
r = m1
m1, m2 = l, k - l
c1 = max(nums1[m1-1] if m1 > 0 else float('-inf'), nums2[m2-1] if m2 > 0 else float('-inf'))
if (n1 + n2) % 2 == 1:
return c1

c2 = min(nums1[m1] if m1 < n1 else float('inf'), nums2[m2] if m2 < n2 else float('inf'))
return (c1 + c2) / 2
# 2085/2085 cases passed (80 ms)
# Your runtime beats 99.41 % of python3 submissions
# Your memory usage beats 100 % of python3 submissions (12.7 MB)