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December 2010

PrefixSet edition (2010-10)

Find the longest path down the Cartesian tree.

A non-empty array A consisting of N different integers is given. We are looking for the longest possible sequence built from elements of A, A[B[0]], A[B[1]], ..., A[B[K]], satisfying the following conditions:

- The sequence must be decreasing; that is, A[B[0]] > A[B[1]] > ... > A[B[K]].
- For any two consecutive elements of the sequence, A[B[I]] and A[B[I+1]], all the elements of A between them must be smaller than them; that is, for any J = MIN(B[I], B[I+1]) + 1, ..., MAX(B[I], B[I+1]) − 1, we have A[J] < A[B[I+1]].

Write a function:

class Solution { public int solution(int[] A); }

that, given an array A containing N different integers, computes the maximum length of a sequence satisfying the above conditions.

For example, for the following array A:

the function should return 6.

A sequence of length 6 satisfying the given conditions can be as follows:

Write an ** efficient** algorithm for the following assumptions:

- the elements of A are all distinct;
- N is an integer within the range [1..100,000];
- each element of array A is an integer within the range [−1,000,000,000..1,000,000,000].

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