4.21

the general form of this problem is to find the kth largest value. finding the median is when k = n/2.

to find the kth largest value,

• select a partition element and split the array into 2 sub-arrays - one with the elements smaller than the partition and one with the elements larger than the partition. O(n)
• if the array with the elements larger than the partition has k - 1 elements, the partition is the kth largest element
• if the array with the elements larger than the partition has >= k elements, recurse with the same value of k using the larger elements as the new array. O(n/2) (average case)
• else the median is in the array with elements smaller than the partition so adjust k to account for the large elements being discarded and recurse using the smaller elements as the new array O(n/2) (average case)

the overall complexity is O(n) since

O(n) + O(n/2) + O(n/4) + ... approaches O(2n) which is just O(n) since 2 is a constant.

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