Mindset model

In binary search, key points:

• if right is defined as len(nums) - 1, then
  • search space is between [left, right]
  • termination condition: while (left <= right)
  • in each shrinking iteration: left = mid + 1 and right = mid - 1
  • final state: left == right + 1
• if right is defined as len(nums), then
  • search space is between [left, right)
  • termination condition: while (left < right)
  • in each shrinking iteration: left = mid + 1 but right = mid
  • final state: left == right
• for left bound searching, right = mid (-1) if f(mid) == target, return left (this will be immediate ceiling item if target is not in f(x))
• for right bound searching, left = mid + 1 if f(mid) == target, return left - 1 (this will be immediate floor item if target is not in f(x))

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# monotone function
def f(x: int) -> int:
    pass

# binary search to get target
def binary_search(nums: List[int], target: int) -> int:
    if len(nums) == 0:
        return -1
    left = ... # min of x
    right = ... + 1 # max of x

    while left < right:
        mid = left + (right - left) // 2
        if f(mid) == target:
            # left bound or right bound
        elif f(mid) < target:
            # how to make f(x) larger
        elif f(mid) > target:
            # how to make f(x) smaller
    return left # or left - 1