# Java 8ms O(log min(n, m)), but beats less than 5%, why?

• The core ideas:

1. Use binary search only on the smallest array. If we find the median, we will know it by looking at the second array and figuring out that there are exactly as many elements less than or equal / greater than or equal.

2. If we don't find the median, we will know its insertion point. That gives us how many elements less than the median there are in the first array, so we know where the median is in the second one.

3. For even total count we don't need to search for the second median. It's the minimum of the next numbers (assuming there are two, otherwise it's the only one).

I can't think how it can get any faster than this. So why beating only 4.95%? Is it the margin of timing error or am I missing something here?

``````public double findMedianSortedArrays(int[] nums1, int[] nums2) {
if (nums1.length == 0) {
return singleArrayMedian(nums2);
} else if (nums2.length == 0) {
return singleArrayMedian(nums1);
}
final int k = nums1.length + nums2.length;
int[] smaller = nums1.length <= nums2.length ? nums1 : nums2;
int[] bigger = smaller == nums1 ? nums2 : nums1;
int[] median = findMedian(smaller, bigger, (k - 1) / 2 + 1);
int m1 = median[0] >= 0 ? smaller[median[0]] : bigger[median[1]];
if (k % 2 == 1)
return m1;
// For even lengths, we need to find the next median.
// It should be somewhere nearby!
int i1 = median[0] >= 0 ? median[0] + 1 : -median[0] - 1;
int i2 = median[1] >= 0 ? median[1] + 1: -median[1] - 1;
int m2;
if (i1 == smaller.length) {
m2 = bigger[i2];
} else if (i2 == bigger.length) {
m2 = smaller[i1];
} else {
m2 = Math.min(smaller[i1], bigger[i2]);
}
return ((double) m1 + m2) / 2.0;
}

/**
* Finds median or one of the medians. The median coordinate is given
* by the {@code leCount} parameter which can be though of as 1-based
* index of the median (like in the find kth element problem).
* The returning value contains two values: first for the {@code nums1}
* array, second for the {@code nums2} one. One of the values is a
* non-negative index indicating where the median was found. The other one
* is an insertion point like in binary search, that is, {@code -index - 1}.
* Unlike binary search, however, the median may actually be present at
* that point too, the algorithm just doesn't check that.
*
* @param nums1 the first array
* @param nums2 the second array
* @param leCount the number of the elements in the merged array that
* are less than or equal to the median (including itself, but not beyond)
* @return an array of one index and one insertion point
*/
private int[] findMedian(int[] nums1, int[] nums2, int leCount) {
int s1 = 0, e1 = nums1.length - 1;
while (s1 <= e1) {
int mid1 = (s1 + e1) / 2;
// We have exactly mid1 + 1 elements in the first array that
// are less than or equal to nums1[mid1] (including itself).
// If mid1 is the median, then there must be exactly
// leCount - (mid1 + 1) elements in the second array that
// are less than or equal to the would-be median.
// All subsequent elements must be greater than or equal,
// so they can be put into the right half during the merge.
int le2 = leCount - (mid1 + 1);
assert le2 >= 0 && le2 <= nums2.length;
if ((le2 == 0 || nums2[le2 - 1] <= nums1[mid1])
&& (le2 == nums2.length || nums2[le2] >= nums1[mid1])) {
return new int[] {mid1, -le2 - 1};
} else if (le2 > 0 && nums2[le2 - 1] > nums1[mid1]) {
// missed, the median must be to the right
s1 = mid1 + 1;
} else {
// or to the left
e1 = mid1 - 1;
}
}
// Haven't found the median in the first array, but we now know
// where it would be if it was there. That means we know exactly
// how many elements less than the median are here. That gives
// us the exact answer where the median is in the second array!
int mid2 = leCount - s1 - 1;
assert mid2 >= 0 && mid2 < nums2.length;
return new int[] {-s1 - 1, mid2};
}

private static double singleArrayMedian(int[] nums) {
if (nums.length % 2 == 0) {
return ((double) nums[nums.length / 2 - 1] + nums[nums.length / 2]) / 2.0;
} else {
return nums[nums.length / 2];
}
}``````

• I just did a test with O(n) merge and find the median by O(1) which takes 6ms.

• So it's just that the test cases are too small?

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