{ "data": { "question": { "questionId": "3903", "questionFrontendId": "3592", "categoryTitle": "Algorithms", "boundTopicId": 3704999, "title": "Inverse Coin Change", "titleSlug": "inverse-coin-change", "content": "

You are given a 1-indexed integer array numWays, where numWays[i] represents the number of ways to select a total amount i using an infinite supply of some fixed coin denominations. Each denomination is a positive integer with value at most numWays.length.

\n\n

However, the exact coin denominations have been lost. Your task is to recover the set of denominations that could have resulted in the given numWays array.

\n\n

Return a sorted array containing unique integers which represents this set of denominations.

\n\n

If no such set exists, return an empty array.

\n\n

 

\n

Example 1:

\n\n
\n

Input: numWays = [0,1,0,2,0,3,0,4,0,5]

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Output: [2,4,6]

\n\n

Explanation:

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
AmountNumber of waysExplanation
10There is no way to select coins with total value 1.
21The only way is [2].
30There is no way to select coins with total value 3.
42The ways are [2, 2] and [4].
50There is no way to select coins with total value 5.
63The ways are [2, 2, 2], [2, 4], and [6].
70There is no way to select coins with total value 7.
84The ways are [2, 2, 2, 2], [2, 2, 4], [2, 6], and [4, 4].
90There is no way to select coins with total value 9.
105The ways are [2, 2, 2, 2, 2], [2, 2, 2, 4], [2, 4, 4], [2, 2, 6], and [4, 6].
\nExample 2:\n\n
\n

Input: numWays = [1,2,2,3,4]

\n\n

Output: [1,2,5]

\n\n

Explanation:

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
AmountNumber of waysExplanation
11The only way is [1].
22The ways are [1, 1] and [2].
32The ways are [1, 1, 1] and [1, 2].
43The ways are [1, 1, 1, 1], [1, 1, 2], and [2, 2].
54The ways are [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [5].
\n
\n\n

Example 3:

\n\n
\n

Input: numWays = [1,2,3,4,15]

\n\n

Output: []

\n\n

Explanation:

\n\n

No set of denomination satisfies this array.

\n
\n\n\n
\n
\n\n

 

\n

Constraints:

\n\n\n", "translatedTitle": "硬币面值还原", "translatedContent": "

给你一个 从 1 开始计数 的整数数组 numWays，其中 numWays[i] 表示使用某些 固定 面值的硬币（每种面值可以使用无限次）凑出总金额 i 的方法数。每种面值都是一个 正整数 ，并且其值 最多 为 numWays.length。

\n\n

然而，具体的硬币面值已经 丢失 。你的任务是还原出可能生成这个 numWays 数组的面值集合。

\n\n

返回一个按从小到大顺序排列的数组，其中包含所有可能的 唯一 整数面值。

\n\n

如果不存在这样的集合，返回一个 空 数组。

\n\n

 

\n\n

示例 1：

\n\n
\n

输入： numWays = [0,1,0,2,0,3,0,4,0,5]

\n\n

输出： [2,4,6]

\n\n

解释：

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
金额方法数解释
10无法用硬币凑出总金额 1。
21唯一的方法是 [2]。
30无法用硬币凑出总金额 3。
42可以用 [2, 2] 或 [4]。
50无法用硬币凑出总金额 5。
63可以用 [2, 2, 2]、[2, 4] 或 [6]。
70无法用硬币凑出总金额 7。
84可以用 [2, 2, 2, 2]、[2, 2, 4]、[2, 6] 或 [4, 4]。
90无法用硬币凑出总金额 9。
105可以用 [2, 2, 2, 2, 2]、[2, 2, 2, 4]、[2, 4, 4]、[2, 2, 6] 或 [4, 6]。
\n
\n\n

示例 2：

\n\n
\n

输入： numWays = [1,2,2,3,4]

\n\n

输出： [1,2,5]

\n\n

解释：

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
金额方法数解释
11唯一的方法是 [1]。
22可以用 [1, 1] 或 [2]。
32可以用 [1, 1, 1] 或 [1, 2]。
43可以用 [1, 1, 1, 1]、[1, 1, 2] 或 [2, 2]。
54可以用 [1, 1, 1, 1, 1]、[1, 1, 1, 2]、[1, 2, 2] 或 [5]。
\n
\n\n

示例 3：

\n\n
\n

输入： numWays = [1,2,3,4,15]

\n\n

输出： []

\n\n

解释：

\n\n

没有任何面值集合可以生成该数组。

\n
\n\n

 

\n\n

提示：

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