4. Median of Two Sorted Arrays
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To solve this problem, we need to understand "What is the use of median". In statistics, the median is used fordividing a set into two equal length subsets, that one subset is always greater than the other
. If we understand the use of median for dividing, we are very close to the answer.
First let's cut A into two parts at a random position i:
left_A | right_AA[0], A[1], ..., A[i-1] | A[i], A[i+1], ..., A[m-1]
Since A has m elements, so there are m+1 kinds of cutting( i = 0 ~ m ). And we know: len(left_A) = i, len(right_A) = m - i . Note: when i = 0 ,left_A is empty, and wheni = m , right_A is empty.
With the same way, cut B into two parts at a random position j:
left_B | right_BB[0], B[1], ..., B[j-1] | B[j], B[j+1], ..., B[n-1]
Put left_A and left_B into one set, and put right_A and right_B into another set. Let's name them left_part and right_part :
left_part | right_partA[0], A[1], ..., A[i-1] | A[i], A[i+1], ..., A[m-1]B[0], B[1], ..., B[j-1] | B[j], B[j+1], ..., B[n-1]
If we can ensure:
1) len(left_part) == len(right_part)2) max(left_part) <= min(right_part)
then we divide all elements in {A, B} into two parts with equal length, and one part is always greater than the other. Thenmedian = (max(left_part) + min(right_part))/2.
To ensure these two conditions, we just need to ensure:
(1) i + j == m - i + n - j (or: m - i + n - j + 1) if n >= m, we just need to set: i = 0 ~ m, j = (m + n + 1)/2 - i(2) B[j-1] <= A[i] and A[i-1] <= B[j]
ps.1 For simplicity, I presume A[i-1],B[j-1],A[i],B[j] are always valid even ifi=0/i=m/j=0/j=n . I will talk about how to deal with these edge values at last.
ps.2 Why n >= m? Because I have to make sure j is non-nagative since 0 <= i <= m and j = (m + n + 1)/2 - i. If n < m , then j may be nagative, that will lead to wrong result.
So, all we need to do is:
Searching i in [0, m], to find an object `i` that: B[j-1] <= A[i] and A[i-1] <= B[j], ( where j = (m + n + 1)/2 - i )
And we can do a binary search following steps described below:
<1> Set imin = 0, imax = m, then start searching in [imin, imax]<2> Set i = (imin + imax)/2, j = (m + n + 1)/2 - i<3> Now we have len(left_part)==len(right_part). And there are only 3 situations that we may encounter: <a> B[j-1] <= A[i] and A[i-1] <= B[j] Means we have found the object `i`, so stop searching. <b> B[j-1] > A[i] Means A[i] is too small. We must `ajust` i to get `B[j-1] <= A[i]`. Can we `increase` i? Yes. Because when i is increased, j will be decreased. So B[j-1] is decreased and A[i] is increased, and `B[j-1] <= A[i]` may be satisfied. Can we `decrease` i? `No!` Because when i is decreased, j will be increased. So B[j-1] is increased and A[i] is decreased, and B[j-1] <= A[i] will be never satisfied. So we must `increase` i. That is, we must ajust the searching range to [i+1, imax]. So, set imin = i+1, and goto <2>. <c> A[i-1] > B[j] Means A[i-1] is too big. And we must `decrease` i to get `A[i-1]<=B[j]`. That is, we must ajust the searching range to [imin, i-1]. So, set imax = i-1, and goto <2>.
When the object i is found, the median is:
max(A[i-1], B[j-1]) (when m + n is odd)or (max(A[i-1], B[j-1]) + min(A[i], B[j]))/2 (when m + n is even)
Now let's consider the edges values i=0,i=m,j=0,j=n where A[i-1],B[j-1],A[i],B[j] may not exist. Actually this situation is easier than you think.
What we need to do is ensuring that max(left_part) <= min(right_part)
. So, ifi andj are not edges values(means A[i-1],B[j-1],A[i],B[j] all exist), then we must check bothB[j-1] <= A[i] andA[i-1] <= B[j]. But if some ofA[i-1],B[j-1],A[i],B[j] don't exist, then we don't need to check one(or both) of these two conditions. For example, ifi=0, thenA[i-1] doesn't exist, then we don't need to checkA[i-1] <= B[j]. So, what we need to do is:
Searching i in [0, m], to find an object `i` that: (j == 0 or i == m or B[j-1] <= A[i]) and (i == 0 or j == n or A[i-1] <= B[j]) where j = (m + n + 1)/2 - i
And in a searching loop, we will encounter only three situations:
<a> (j == 0 or i == m or B[j-1] <= A[i]) and (i == 0 or j = n or A[i-1] <= B[j]) Means i is perfect, we can stop searching.<b> j > 0 and i < m and B[j - 1] > A[i] Means i is too small, we must increase it.<c> i > 0 and j < n and A[i - 1] > B[j] Means i is too big, we must decrease it.
pointed out that: i < m ==> j > 0
and i > 0 ==> j < n
. Because:
m <= n, i < m ==> j = (m+n+1)/2 - i > (m+n+1)/2 - m >= (2*m+1)/2 - m >= 0 m <= n, i > 0 ==> j = (m+n+1)/2 - i < (m+n+1)/2 <= (2*n+1)/2 <= n
So in situation <b> and <c>, we don't need to check whether j > 0
and whetherj < n
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