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Finding the Median of Two Sorted Arrays Efficiently


Parallel Merge and Parallel In-Place Merge algorithms merge two already sorted arrays (blocks) into a single sorted array using a similar divide-and-conquer strategy. The strategy picks the mid-element (X) within the larger of the two sorted blocks and uses binary search to split the smaller block into two sections: one with elements that are all smaller than (X), and the other with elements larger or equal to X, as shown in Figure 1:

Sort
Figure 1: Merging two already sorted arrays.

The block on the left (between l and m indexes) is sorted and has 13 elements, with indexes 0 to 12. The mid-element X is at index 6. The block on the right (between m+1 and r indexes) has 7 elements. The value of X is used in the binary search to find an index where all elements to the left are smaller than X.

In the above example, there are 9 elements that are smaller or equal to X: 6 from the left block, 2 from the right block, and X itself. Also, 11 elements are larger or equal to X: 6 from the left block and 5 from the right block. Thus, using the mid-element X of the left block as the partition resulted in an uneven split: 9 elements smaller and 11 elements larger.

In the worst case, this "uneven-ness" of the split can be significantly worse, with one quarter of the elements being smaller and three quarters being larger, or vice versa. The worst case occurs when the two blocks are of equal size, and the split is such that either all elements of the right block are smaller than X, or all elements are larger or equal to X. For instance, if both blocks have M elements, and M is odd, then within the left block, M/2 elements will be smaller than or equal to X, and M/2 elements will be greater than or equal X. If the split is such that all elements of the right block are smaller than X, then overall we have M + M/2 elements that are smaller or equal to X and M/2 elements that are larger or equal to X. The opposite worst case split would occur if all elements of the right array are larger than or equal to X.

If one of these two worst cases occurred every time, then one side of the divide-and-conquer algorithm would be working on M/2 elements, while the other side would be working on M + M/2 elements — that is, one quarter of all elements and three quarters of overall elements, respectively. This uneven split will result in an unbalanced recursive binary tree during divide-and-conquer, performing deeper recursion than with an even split.

In this article, a better strategy is developed that always splits the two sorted blocks in a more optimal way by finding an overall median of the two blocks. Using this median in the same divide-and-conquer algorithm creates a more balanced recursive binary tree (as suggested in [1]).

Searching for the Median

The median of a single sorted array is trivial to find and is O(1) constant time. For example, a sorted array A = [5, 7, 9, 11, 15], which has an odd number of elements, has a unique median element 9 at index 2. Elements [5, 7] to the left are smaller than or equal to the median. Elements [11, 15] to the right are bigger than or equal to the median. The array has 5 elements with indexes in the range of 0 to 4, and the median element is at index (5-1)/2 = 2. In general, the median is at index (n-1)/2 if the number of elements in an array (n) is odd.

For a sorted array with an even number of elements, two elements in the middle are medians. For example, A = [5, 7, 9, 11] has medians of 7 and 9 at indexes of 1 and 2. In general, the medians are at index floor((n-1)/2) and at n/2. We could just pick the lower median, which leads to a single index computation no matter whether the array has an odd or an even number of elements, to wit, floor((n-1)/2).

Finding a median of two sorted arrays is more difficult and is no longer constant time. For example, two sorted arrays A = [5, 7, 9, 11, 15] and B = [1, 8] could be merged into a single sorted array in O(N) time to produce C = [1, 5, 7, 8, 9, 11, 15]. This resulting array has 7 elements, with a median of 8 at index floor((7-1)/2) = 3. It's possible to do better than O(N), by using a modified binary search leading to O(lgN) performance.

Modified Binary Search

In the case above, the median of value 8 came from the smaller of the two arrays. If we use different array values, A = [1, 2, 3, 4, 7] and B = [0, 5, 6, 9], then the combined sorted array C = [0, 1, 2, 3, 4, 5, 6, 7, 9] has a median of 4 at index 4. In this case, the median came from the larger array. This shows that the median can come from either array, no matter the size of either. Even if one of the arrays has a single element, that element could be the overall median. Thus, when we search for the median, both arrays must be involved in the search, otherwise we could potentially miss the true overall median.

Because both arrays are sorted, binary search can be used to search each of the arrays quickly, in O(lgN) time. The trick is to develop a test to determine quickly whether the selected element is the median or not — ideally, in constant time. Using a binary search, the mid-element of the sorted array A is selected, which is 3 at index 2. Is this element an overall median? What kind of a test is necessary?


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