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Mud Walls

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Problem Statement
A child likes to build mud walls by placing mud between sticks positioned on a number line. The gap between sticks will be referred to as a cell, and each cell will contain one segment of wall. The height of mud in a segment cannot exceed 1 unit above an adjacent stick or mud segment. Given the placement of a number of sticks and their heights, determine the maximum height segment of mud that can be built. If no mud segment can be built, return 0.

For example, there are n = 3 sticks at stickPositions = [1, 2, 4, 7] with stickHeights = [4, 5, 7, 11]. There is no space between the first two sticks, so there is no cell for mud. Between positions 2 and 4, there is one cell. Heights of the surrounding sticks are 5 and 7, so the maximum height of mud is 5 + 1 = 6. Between positions 4 and 7 there are two cells. The heights of surrounding sticks are 7 and 11. The maximum height mud segment next to the stick of height 7 is 8. The maximum height mud next to a mud segment of height 8 and a stick of height 11 is 9. Mud segment heights are 6, 8 and 9, and the maximum height is 9. In the table below, digits are in the columns of sticks and M is in the mud segments.

Function Description
Complete the function maxHeight in the editor below. The function must return an integer, the maximum height mud segment that can be built.

maxHeight has the following parameter(s):
stickPositions[stickPositions[0],…stickPositions[n-1]]: an array of integers
stickHeights[stickHeights[0],…stickHeights[n-1]]: an array of integers

1 ≤ n ≤ 105
1 ≤ stickPositions[i] ≤ 109 (where 0 ≤ i < n)
1 ≤ stickHeights[i] ≤ 109 (where 0 ≤ i < n)

Sample Input For Custom Testing

Sample Output


Here stickPositions = [1, 3, 7] and stickHeights = [4, 3, 3]. There can be a segment of height 4 at position 2 supported by sticks of heights 4 and 3. Between positions 3 and 7, there can be a segment of height 4 at positions 4 and 6. Between them, a segment can be built of height 5 at position 5.