leetcode-problemset/leetcode-cn/originData/minimum-number-of-coins-to-be-added.json

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            "categoryTitle": "Algorithms",
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            "title": "Minimum Number of Coins to be Added",
            "titleSlug": "minimum-number-of-coins-to-be-added",
            "content": "<p>You are given a <strong>0-indexed</strong> integer array <code>coins</code>, representing the values of the coins available, and an integer <code>target</code>.</p>\n\n<p>An integer <code>x</code> is <strong>obtainable</strong> if there exists a subsequence of <code>coins</code> that sums to <code>x</code>.</p>\n\n<p>Return <em>the<strong> minimum</strong> number of coins <strong>of any value</strong> that need to be added to the array so that every integer in the range</em> <code>[1, target]</code><em> is <strong>obtainable</strong></em>.</p>\n\n<p>A <strong>subsequence</strong> of an array is a new <strong>non-empty</strong> array that is formed from the original array by deleting some (<strong>possibly none</strong>) of the elements without disturbing the relative positions of the remaining elements.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> coins = [1,4,10], target = 19\n<strong>Output:</strong> 2\n<strong>Explanation:</strong> We need to add coins 2 and 8. The resulting array will be [1,2,4,8,10].\nIt can be shown that all integers from 1 to 19 are obtainable from the resulting array, and that 2 is the minimum number of coins that need to be added to the array. \n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> coins = [1,4,10,5,7,19], target = 19\n<strong>Output:</strong> 1\n<strong>Explanation:</strong> We only need to add the coin 2. The resulting array will be [1,2,4,5,7,10,19].\nIt can be shown that all integers from 1 to 19 are obtainable from the resulting array, and that 1 is the minimum number of coins that need to be added to the array. \n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<pre>\n<strong>Input:</strong> coins = [1,1,1], target = 20\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> We need to add coins 4, 8, and 16. The resulting array will be [1,1,1,4,8,16].\nIt can be shown that all integers from 1 to 20 are obtainable from the resulting array, and that 3 is the minimum number of coins that need to be added to the array.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= target &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= coins.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= coins[i] &lt;= target</code></li>\n</ul>\n",
            "translatedTitle": "需要添加的硬币的最小数量",
            "translatedContent": "<p>给你一个下标从 <strong>0 </strong>开始的整数数组 <code>coins</code>，表示可用的硬币的面值，以及一个整数 <code>target</code> 。</p>\n\n<p>如果存在某个 <code>coins</code> 的子序列总和为 <code>x</code>，那么整数 <code>x</code> 就是一个 <strong>可取得的金额 </strong>。</p>\n\n<p>返回需要添加到数组中的<strong> 任意面值 </strong>硬币的 <strong>最小数量 </strong>，使范围 <code>[1, target]</code> 内的每个整数都属于 <strong>可取得的金额</strong> 。</p>\n\n<p>数组的 <strong>子序列</strong> 是通过删除原始数组的一些（<strong>可能不删除</strong>）元素而形成的新的 <strong>非空</strong> 数组，删除过程不会改变剩余元素的相对位置。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<pre>\n<strong>输入：</strong>coins = [1,4,10], target = 19\n<strong>输出：</strong>2\n<strong>解释：</strong>需要添加面值为 2 和 8 的硬币各一枚，得到硬币数组 [1,2,4,8,10] 。\n可以证明从 1 到 19 的所有整数都可由数组中的硬币组合得到，且需要添加到数组中的硬币数目最小为 2 。\n</pre>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<pre>\n<strong>输入：</strong>coins = [1,4,10,5,7,19], target = 19\n<strong>输出：</strong>1\n<strong>解释：</strong>只需要添加一枚面值为 2 的硬币，得到硬币数组 [1,2,4,5,7,10,19] 。\n可以证明从 1 到 19 的所有整数都可由数组中的硬币组合得到，且需要添加到数组中的硬币数目最小为 1 。</pre>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<pre>\n<strong>输入：</strong>coins = [1,1,1], target = 20\n<strong>输出：</strong>3\n<strong>解释：</strong>\n需要添加面值为 4 、8 和 16 的硬币各一枚，得到硬币数组 [1,1,1,4,8,16] 。 \n可以证明从 1 到 20 的所有整数都可由数组中的硬币组合得到，且需要添加到数组中的硬币数目最小为 3 。</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= target &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= coins.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= coins[i] &lt;= target</code></li>\n</ul>\n",
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                "Sort the coins array and maintain the smallest sum that is unobtainable by induction.",
                "If we don’t use any coins, the smallest integer that we cannot obtain by sum is <code>1</code>. Suppose currently, for a fixed set of the first several coins the smallest integer that we cannot obtain is <code>x + 1</code>, namely we can form all integers in the range <code>[1, x]</code> but not <code>x + 1</code>.",
                "If the next unused coin’s value is NOT <code>x + 1</code> (note the array is sorted), we have to add <code>x + 1</code> to the array. After this addition, we can form all values from <code>x + 1</code> to <code>2 * x + 1</code> by adding <code>x + 1</code> in <code>[1, x]</code>'s formations. So now we can form all the numbers of <code>[1, 2 * x + 1]</code>. After this iteration the new value of <code>x</code> becomes <code>2 * x + 1</code>."
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