Implement sampling, subarray scan, and percentile estimate

You will solve three independent coding tasks.

Problem 1: Generate a 2D uniform sample in a square

You are given access to a function rand01() that returns an i.i.d. sample from a continuous Uniform(0, 1) distribution.

Write a function samplePoint() that returns a 2D point (x, y) that is uniformly distributed over the square:

• $(-1, 1) \times (-1, 1)$

Constraints/requirements:

• You may call rand01() as many times as needed.
• The output must be uniform over the area of the square.

Problem 2: Longest increasing continuous subarray

Given an integer array nums, compute the length of the longest contiguous subarray that is strictly increasing.

Example:

• nums = [1, 3, 5, 4, 7] → answer is 3 (subarray [1, 3, 5] )

Clarifications:

• “Continuous” means contiguous (a subarray), not a subsequence.
• Strictly increasing means nums[i] < nums[i+1] .

Problem 3: Approximate an n-th percentile from bucketed counts

You have search queries aggregated into K buckets (a histogram). Each bucket i has:

• left_i : left boundary (inclusive)
• right_i : right boundary (exclusive, unless it is the last bucket)
• count_i : number of observations in that bucket

You do not know the individual values inside buckets—only bucket boundaries and counts.

Write a function to return an approximation of the p-th percentile (e.g., p=0.90 for 90th percentile).

Required behavior/assumptions (state clearly in your answer):

• Buckets are ordered by boundary.
• Total count is N = sum(count_i) .
• You may assume values are uniformly distributed within the bucket used to locate the percentile (to enable interpolation).

Output:

• A single numeric estimate of the percentile value.