leetcode-problemset/leetcode-cn/originData/inverse-coin-change.json

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        "question": {
            "questionId": "3903",
            "questionFrontendId": "3592",
            "categoryTitle": "Algorithms",
            "boundTopicId": 3704999,
            "title": "Inverse Coin Change",
            "titleSlug": "inverse-coin-change",
            "content": "<p>You are given a <strong>1-indexed</strong> integer array <code>numWays</code>, where <code>numWays[i]</code> represents the number of ways to select a total amount <code>i</code> using an <strong>infinite</strong> supply of some <em>fixed</em> coin denominations. Each denomination is a <strong>positive</strong> integer with value <strong>at most</strong> <code>numWays.length</code>.</p>\n\n<p>However, the exact coin denominations have been <em>lost</em>. Your task is to recover the set of denominations that could have resulted in the given <code>numWays</code> array.</p>\n\n<p>Return a <strong>sorted</strong> array containing <strong>unique</strong> integers which represents this set of denominations.</p>\n\n<p>If no such set exists, return an <strong>empty</strong> array.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">numWays = [0,1,0,2,0,3,0,4,0,5]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">[2,4,6]</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<table style=\"border: 1px solid black;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"border: 1px solid black;\">Amount</th>\n\t\t\t<th style=\"border: 1px solid black;\">Number of ways</th>\n\t\t\t<th style=\"border: 1px solid black;\">Explanation</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">There is no way to select coins with total value 1.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">The only way is <code>[2]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">There is no way to select coins with total value 3.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[2, 2]</code> and <code>[4]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">There is no way to select coins with total value 5.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">6</td>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[2, 2, 2]</code>, <code>[2, 4]</code>, and <code>[6]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">7</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">There is no way to select coins with total value 7.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">8</td>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[2, 2, 2, 2]</code>, <code>[2, 2, 4]</code>, <code>[2, 6]</code>, and <code>[4, 4]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">9</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">There is no way to select coins with total value 9.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">10</td>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[2, 2, 2, 2, 2]</code>, <code>[2, 2, 2, 4]</code>, <code>[2, 4, 4]</code>, <code>[2, 2, 6]</code>, and <code>[4, 6]</code>.</td>\n\t\t</tr>\n\t</tbody>\n</table>\n<strong class=\"example\">Example 2:</strong>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">numWays = [1,2,2,3,4]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">[1,2,5]</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<table style=\"border: 1px solid black;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"border: 1px solid black;\">Amount</th>\n\t\t\t<th style=\"border: 1px solid black;\">Number of ways</th>\n\t\t\t<th style=\"border: 1px solid black;\">Explanation</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">The only way is <code>[1]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[1, 1]</code> and <code>[2]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[1, 1, 1]</code> and <code>[1, 2]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[1, 1, 1, 1]</code>, <code>[1, 1, 2]</code>, and <code>[2, 2]</code>.</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">The ways are <code>[1, 1, 1, 1, 1]</code>, <code>[1, 1, 1, 2]</code>, <code>[1, 2, 2]</code>, and <code>[5]</code>.</td>\n\t\t</tr>\n\t</tbody>\n</table>\n</div>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">numWays = [1,2,3,4,15]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">[]</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>No set of denomination satisfies this array.</p>\n</div>\n\n<table style=\"border: 1px solid black;\">\n</table>\n</div>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= numWays.length &lt;= 100</code></li>\n\t<li><code>0 &lt;= numWays[i] &lt;= 2 * 10<sup>8</sup></code></li>\n</ul>\n",
            "translatedTitle": "硬币面值还原",
            "translatedContent": "<p>给你一个&nbsp;<strong>从 1 开始计数&nbsp;</strong>的整数数组 <code>numWays</code>，其中 <code>numWays[i]</code> 表示使用某些&nbsp;<strong>固定&nbsp;</strong>面值的硬币（每种面值可以使用无限次）凑出总金额 <code>i</code> 的方法数。每种面值都是一个&nbsp;<strong>正整数&nbsp;</strong>，并且其值&nbsp;<strong>最多&nbsp;</strong>为 <code>numWays.length</code>。</p>\n\n<p>然而，具体的硬币面值已经&nbsp;<strong>丢失&nbsp;</strong>。你的任务是还原出可能生成这个 <code>numWays</code> 数组的面值集合。</p>\n\n<p>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&nbsp;<strong>唯一&nbsp;</strong>整数面值。</p>\n\n<p>如果不存在这样的集合，返回一个&nbsp;<strong>空&nbsp;</strong>数组。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<div class=\"example-block\">\n<p><strong>输入：</strong> <span class=\"example-io\">numWays = [0,1,0,2,0,3,0,4,0,5]</span></p>\n\n<p><strong>输出：</strong> <span class=\"example-io\">[2,4,6]</span></p>\n\n<p><strong>解释：</strong></p>\n\n<table style=\"border: 1px solid black;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"border: 1px solid black;\">金额</th>\n\t\t\t<th style=\"border: 1px solid black;\">方法数</th>\n\t\t\t<th style=\"border: 1px solid black;\">解释</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">无法用硬币凑出总金额 1。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">唯一的方法是 <code>[2]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">无法用硬币凑出总金额 3。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[2, 2]</code> 或 <code>[4]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">无法用硬币凑出总金额 5。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">6</td>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[2, 2, 2]</code>、<code>[2, 4]</code> 或 <code>[6]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">7</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">无法用硬币凑出总金额 7。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">8</td>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[2, 2, 2, 2]</code>、<code>[2, 2, 4]</code>、<code>[2, 6]</code> 或 <code>[4, 4]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">9</td>\n\t\t\t<td style=\"border: 1px solid black;\">0</td>\n\t\t\t<td style=\"border: 1px solid black;\">无法用硬币凑出总金额 9。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">10</td>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[2, 2, 2, 2, 2]</code>、<code>[2, 2, 2, 4]</code>、<code>[2, 4, 4]</code>、<code>[2, 2, 6]</code> 或 <code>[4, 6]</code>。</td>\n\t\t</tr>\n\t</tbody>\n</table>\n</div>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<div class=\"example-block\">\n<p><strong>输入：</strong> <span class=\"example-io\">numWays = [1,2,2,3,4]</span></p>\n\n<p><strong>输出：</strong> <span class=\"example-io\">[1,2,5]</span></p>\n\n<p><strong>解释：</strong></p>\n\n<table style=\"border: 1px solid black;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"border: 1px solid black;\">金额</th>\n\t\t\t<th style=\"border: 1px solid black;\">方法数</th>\n\t\t\t<th style=\"border: 1px solid black;\">解释</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">1</td>\n\t\t\t<td style=\"border: 1px solid black;\">唯一的方法是 <code>[1]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[1, 1]</code> 或 <code>[2]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">2</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[1, 1, 1]</code> 或 <code>[1, 2]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">3</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[1, 1, 1, 1]</code>、<code>[1, 1, 2]</code> 或 <code>[2, 2]</code>。</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"border: 1px solid black;\">5</td>\n\t\t\t<td style=\"border: 1px solid black;\">4</td>\n\t\t\t<td style=\"border: 1px solid black;\">可以用 <code>[1, 1, 1, 1, 1]</code>、<code>[1, 1, 1, 2]</code>、<code>[1, 2, 2]</code> 或 <code>[5]</code>。</td>\n\t\t</tr>\n\t</tbody>\n</table>\n</div>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<div class=\"example-block\">\n<p><strong>输入：</strong> <span class=\"example-io\">numWays = [1,2,3,4,15]</span></p>\n\n<p><strong>输出：</strong> <span class=\"example-io\">[]</span></p>\n\n<p><strong>解释：</strong></p>\n\n<p>没有任何面值集合可以生成该数组。</p>\n</div>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= numWays.length &lt;= 100</code></li>\n\t<li><code>0 &lt;= numWays[i] &lt;= 2 * 10<sup>8</sup></code></li>\n</ul>\n",
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                "Observe that for the smallest denomination <code>c</code>, you must have <code>numWays[c] == 1</code>.",
                "Find the smallest <code>c > 0</code> with <code>numWays[c] == 1</code> and append <code>c</code> to your <code>ans</code> list.",
                "\"Remove\" that coin’s contribution by doing, for each <code>s</code> from <code>c</code> up to <code>n</code>: numWays[s] -= numWays[s - c]",
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