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Given a sorted array of positive integers nums, and a positive integer n. Select any number from the interval [1, n] and add it to nums, so that
any number in the interval [1, n] can be represented by the sum of certain numbers in nums. Please output the minimum number of digits that meet the above requirements.Example 1:
Input: nums = [1,3], n = 6 Output: 1 Explanation: According to the existing combination [1], [3], [1,3] in nums,
1, 3, 4 can be obtained . Now if we add 2 to nums, the combination becomes: [1], [2], [3], [1,3], [2,3],
[1,2,3]. The sum can represent the numbers 1, 2, 3, 4, 5, 6, which can cover all the numbers in the interval [1, 6].
So we need to add at least one number.Example 2:
Input: nums = [1,5,10], n = 20 Output: 2 Explanation: We need to add [2, 4].
Example 3:
Input: nums = [1,2,2], n = 5 Output: 0
Problem-solving ideas
how are you
Features of the filled array:
假设数组 arr添加一个元素即可覆盖 [1,n)内所有数字，那么添加的数字 mmm 一定满足m <= n。 假设数组 arr 可以覆盖 [1,n) 的所有数字，则给 arr内加元素 m： 若m <= n，新数组可以覆盖[1, m + n) = [1, n) ∪ [m, m + n)内所有数字；Greedy rule: For an array arr covering [1,n), adding the number n has the largest continuous expansion range (expanding to [1,2n)). Idea: Set an initial range [1,1), and finally calculate the minimum number that needs to be added by continuously confirming and expanding the range that the array can cover.
当i < len(nums)且nums[i] <= add时：不需要加入新数字，循环确认并更新数组可以覆盖的范围[1, add + nums[i])，直到找到大于确认范围 add 的 nums[i] 或索引越界。 否则：无法根据现有数字构建更大的连续范围，因此需要使用贪心策略向数组加入数字 add ，将数组从覆盖 [1,add) 扩容至可覆盖 [1,2add)。 直到确认的范围add > n，说明此时已经覆盖 [1,n] ，退出迭代并返回。
Code
class Solution:
def minPatches(self, nums: List[int], n: int) -> int:
patches, x = 0, 1
length, index = len(nums), 0
while x <= n:
if index < length and nums[index] <= x:
x += nums[index]
index += 1
else:
x *= 2
patches += 1
return patches