Refined Buneman Trees

Algorithm 9 The RANDOMIZED-PARTITION algorithm
Require: A is an array of length n.
Require: p, r are indices in A
Ensure: returns q such that elements in A[p..q] are all less than or equal to all
elements in A[q +1..r].
1: i = RANDOM(p, r)
2: SWAP(A[p],A[i])
3: return PARTITION(A, p, r)
Algorithm 10 The PARTITION algorithm
Require: A is an array of length n.
Require: p, r are indices in A
Ensure: returns j such that elements in A[p..j − 1] are all less than or equal
to all elements in A[j..r].
1: x = A[p]
2: i = p − 1
3: j = r +1
4: while true do
5: repeat
6: j = j − 1
7: until A[j] ≤ x
8: repeat
9: i = i +1
10: until A[i] ≥ x
11: if i