Binary Search

개요

• 정렬된 배열에서 특정 값을 찾는 알고리즘.
• 중간 값을 기준으로 탐색 범위를 절반씩 줄여가며 효율적으로 탐색. ( 시간 복잡도: $O(logN)$ )

사용처

1. upper bound/lower bound 찾기
2. 특정 범위 내에 있는 값들의 개수를 효율적으로 계산.
3. 조건을 만족하는 최소 또는 최대 값을 찾는 문제에 적용.

Template

Lower Bound

while left < right:
    mid = left + (right - left) // 2

    if arr[mid] > target:
        right = mid
    else:
        left = mid + 1

Upper Bound

while left < right:
    mid = left + (right - left) // 2
    
    if arr[mid] <= target:
        left = mid + 1
    else:
        right = mid

1760. Minimum Limit of Balls in a Bag

1760. Minimum Limit of Balls in a Bag

( 3. 조건을 만족하는 최소 또는 최대 값을 찾는 문제. )

You are given an integer array nums where the ith bag contains nums[i] balls. You are also given an integer maxOperations.

You can perform the following operation at most maxOperations times:

Take any bag of balls and divide it into two new bags with a positive number of balls.
For example, a bag of 5 balls can become two new bags of 1 and 4 balls, or two new bags of 2 and 3 balls.
Your penalty is the maximum number of balls in a bag. You want to minimize your penalty after the operations.

Return the minimum possible penalty after performing the operations.

Input: nums = [9], maxOperations = 2
Output: 3
Explanation: 
- Divide the bag with 9 balls into two bags of sizes 6 and 3. [9] -> [6,3].
- Divide the bag with 6 balls into two bags of sizes 3 and 3. [6,3] -> [3,3,3].
The bag with the most number of balls has 3 balls, so your penalty is 3 and you should return 3.

Input: nums = [2,4,8,2], maxOperations = 4
Output: 2
Explanation:
- Divide the bag with 8 balls into two bags of sizes 4 and 4. [2,4,8,2] -> [2,4,4,4,2].
- Divide the bag with 4 balls into two bags of sizes 2 and 2. [2,4,4,4,2] -> [2,2,2,4,4,2].
- Divide the bag with 4 balls into two bags of sizes 2 and 2. [2,2,2,4,4,2] -> [2,2,2,2,2,4,2].
- Divide the bag with 4 balls into two bags of sizes 2 and 2. [2,2,2,2,2,4,2] -> [2,2,2,2,2,2,2,2].
The bag with the most number of balls has 2 balls, so your penalty is 2, and you should return 2.

1574. Shortest Subarray to be Removed to Make Array Sorted

1574. Shortest Subarray to be Removed to Make Array Sorted

( 1. upper bound/lower bound 찾기 )

Given an integer array arr, remove a subarray (can be empty) from arr such that the remaining elements in arr are non-decreasing.

Return the length of the shortest subarray to remove.

Input: arr = [1,2,3,10,4,2,3,5]
Output: 3

Input: arr = [5,4,3,2,1]
Output: 4