975 Odd Even Jump

You are given an integer array arr. From some starting index, you can make a series of jumps. The (1^st, 3^rd, 5^th, ...) jumps in the series are called odd-numbered jumps, and the (2^nd, 4^th, 6^th, ...) jumps in the series are called even-numbered jumps. Note that the jumps are numbered, not the indices.

You may jump forward from index i to index j (with i < j) in the following way:

• During odd-numbered jumps (i.e., jumps 1, 3, 5, ...), you jump to the index j such that arr[i] <= arr[j] and arr[j] is the smallest possible value. If there are multiple such indices j, you can only jump to the smallest such index j.
• During even-numbered jumps (i.e., jumps 2, 4, 6, ...), you jump to the index j such that arr[i] >= arr[j] and arr[j] is the largest possible value. If there are multiple such indices j, you can only jump to the smallest such index j.
• It may be the case that for some index i, there are no legal jumps.

A starting index is good if, starting from that index, you can reach the end of the array (index arr.length - 1) by jumping some number of times (possibly 0 or more than once).

Return the number of good starting indices.

Example 1:

Input: arr = [10,13,12,14,15]
Output: 2
Explanation: 
From starting index i = 0, we can make our 1st jump to i = 2
  (since arr[2] is the smallest among arr[1], arr[2], arr[3], arr[4]
  that is greater or equal to arr[0]), then we cannot jump any more.
From starting index i = 1 and i = 2, we can make our 1st jump to i = 3,
  then we cannot jump any more.
From starting index i = 3, we can make our 1st jump to i = 4,
  so we have reached the end.
From starting index i = 4, we have reached the end already.
In total, there are 2 different starting indices i = 3 and i = 4,
where we can reach the end with some number of jumps.

Example 2:

Input: arr = [2,3,1,1,4]
Output: 3
Explanation: 
From starting index i = 0, we make jumps to i = 1, i = 2, i = 3:
During our 1st jump (odd-numbered), we first jump to i = 1 because arr[1] is the
  smallest value in [arr[1], arr[2], arr[3], arr[4]] that is greater than or
  equal to arr[0].
During our 2nd jump (even-numbered), we jump from i = 1 to i = 2 because arr[2]
  is the largest value in [arr[2], arr[3], arr[4]] that is less than or equal
  to arr[1]. arr[3] is also the largest value, but 2 is a smaller index, so we
  can only jump to i = 2 and not i = 3
During our 3rd jump (odd-numbered), we jump from i = 2 to i = 3 because arr[3] is
  the smallest value in [arr[3], arr[4]] that is greater than or equal to arr[2].
We can't jump from i = 3 to i = 4, so the starting index i = 0 is not good.
In a similar manner, we can deduce that:
From starting index i = 1, we jump to i = 4, so we reach the end.
From starting index i = 2, we jump to i = 3, and then we can't jump anymore.
From starting index i = 3, we jump to i = 4, so we reach the end.
From starting index i = 4, we are already at the end.
In total, there are 3 different starting indices i = 1, i = 3, and i = 4,
where we can reach the end with some number of jumps.

Example 3:

Input: arr = [5,1,3,4,2]
Output: 3
Explanation: We can reach the end from starting indices 1, 2, and 4.

Constraints:

• 1 <= arr.length <= 2 * 104
• 0 <= arr[i] < 105

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

class Solution:
    def oddEvenJumps(self, arr: List[int]) -> int:
        n = len(arr)
        next_higher, next_lower = [0] * n, [0] * n

        stack = []
        for a, i in sorted([a, i] for i, a in enumerate(arr)):
            while stack and stack[-1] < i:
                next_higher[stack.pop()] = i
            stack.append(i)

        stack = []
        for a, i in sorted([-a, i] for i, a in enumerate(arr)):
            while stack and stack[-1] < i:
                next_lower[stack.pop()] = i
            stack.append(i)

        higher, lower = [0] * n, [0] * n
        higher[-1] = lower[-1] = 1
        for i in range(n - 1)[::-1]:
            higher[i] = lower[next_higher[i]]
            lower[i] = higher[next_lower[i]]
        return sum(higher)