{ "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.

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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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