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"question": {
"questionId": "3364",
"questionFrontendId": "3117",
"categoryTitle": "Algorithms",
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"title": "Minimum Sum of Values by Dividing Array",
"titleSlug": "minimum-sum-of-values-by-dividing-array",
"content": "<p>You are given two arrays <code>nums</code> and <code>andValues</code> of length <code>n</code> and <code>m</code> respectively.</p>\n\n<p>The <strong>value</strong> of an array is equal to the <strong>last</strong> element of that array.</p>\n\n<p>You have to divide <code>nums</code> into <code>m</code> <strong>disjoint contiguous</strong> <span data-keyword=\"subarray-nonempty\">subarrays</span> such that for the <code>i<sup>th</sup></code> subarray <code>[l<sub>i</sub>, r<sub>i</sub>]</code>, the bitwise <code>AND</code> of the subarray elements is equal to <code>andValues[i]</code>, in other words, <code>nums[l<sub>i</sub>] &amp; nums[l<sub>i</sub> + 1] &amp; ... &amp; nums[r<sub>i</sub>] == andValues[i]</code> for all <code>1 &lt;= i &lt;= m</code>, where <code>&amp;</code> represents the bitwise <code>AND</code> operator.</p>\n\n<p>Return <em>the <strong>minimum</strong> possible sum of the <strong>values</strong> of the </em><code>m</code><em> subarrays </em><code>nums</code><em> is divided into</em>. <em>If it is not possible to divide </em><code>nums</code><em> into </em><code>m</code><em> subarrays satisfying these conditions, return</em> <code>-1</code>.</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\">nums = [1,4,3,3,2], andValues = [0,3,3,2]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">12</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>The only possible way to divide <code>nums</code> is:</p>\n\n<ol>\n\t<li><code>[1,4]</code> as <code>1 &amp; 4 == 0</code>.</li>\n\t<li><code>[3]</code> as the bitwise <code>AND</code> of a single element subarray is that element itself.</li>\n\t<li><code>[3]</code> as the bitwise <code>AND</code> of a single element subarray is that element itself.</li>\n\t<li><code>[2]</code> as the bitwise <code>AND</code> of a single element subarray is that element itself.</li>\n</ol>\n\n<p>The sum of the values for these subarrays is <code>4 + 3 + 3 + 2 = 12</code>.</p>\n</div>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">nums = [2,3,5,7,7,7,5], andValues = [0,7,5]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">17</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>There are three ways to divide <code>nums</code>:</p>\n\n<ol>\n\t<li><code>[[2,3,5],[7,7,7],[5]]</code> with the sum of the values <code>5 + 7 + 5 == 17</code>.</li>\n\t<li><code>[[2,3,5,7],[7,7],[5]]</code> with the sum of the values <code>7 + 7 + 5 == 19</code>.</li>\n\t<li><code>[[2,3,5,7,7],[7],[5]]</code> with the sum of the values <code>7 + 7 + 5 == 19</code>.</li>\n</ol>\n\n<p>The minimum possible sum of the values is <code>17</code>.</p>\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\">nums = [1,2,3,4], andValues = [2]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">-1</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>The bitwise <code>AND</code> of the entire array <code>nums</code> is <code>0</code>. As there is no possible way to divide <code>nums</code> into a single subarray to have the bitwise <code>AND</code> of elements <code>2</code>, return <code>-1</code>.</p>\n</div>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= n == nums.length &lt;= 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= m == andValues.length &lt;= min(n, 10)</code></li>\n\t<li><code>1 &lt;= nums[i] &lt; 10<sup>5</sup></code></li>\n\t<li><code>0 &lt;= andValues[j] &lt; 10<sup>5</sup></code></li>\n</ul>\n",
"translatedTitle": "划分数组得到最小的值之和",
"translatedContent": "<p>给你两个数组 <code>nums</code> 和 <code>andValues</code>,长度分别为 <code>n</code> 和 <code>m</code>。</p>\n\n<p>数组的 <strong>值 </strong>等于该数组的 <strong>最后一个 </strong>元素。</p>\n\n<p>你需要将 <code>nums</code> 划分为 <code>m</code> 个 <strong>不相交的连续 </strong>子数组,对于第 <code>i<sup>th</sup></code> 个子数组 <code>[l<sub>i</sub>, r<sub>i</sub>]</code>,子数组元素的按位<code>AND</code>运算结果等于 <code>andValues[i]</code>,换句话说,对所有的 <code>1 &lt;= i &lt;= m</code><code>nums[l<sub>i</sub>] &amp; nums[l<sub>i</sub> + 1] &amp; ... &amp; nums[r<sub>i</sub>] == andValues[i]</code> ,其中 <code>&amp;</code> 表示按位<code>AND</code>运算符。</p>\n\n<p>返回将 <code>nums</code> 划分为 <code>m</code> 个子数组所能得到的可能的 <strong>最小 </strong>子数组 <strong>值</strong> 之和。如果无法完成这样的划分,则返回 <code>-1</code> 。</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\">nums = [1,4,3,3,2], andValues = [0,3,3,2]</span></p>\n\n<p><strong>输出:</strong> <span class=\"example-io\">12</span></p>\n\n<p><strong>解释:</strong></p>\n\n<p>唯一可能的划分方法为:</p>\n\n<ol>\n\t<li><code>[1,4]</code> 因为 <code>1 &amp; 4 == 0</code></li>\n\t<li><code>[3]</code> 因为单元素子数组的按位 <code>AND</code> 结果就是该元素本身</li>\n\t<li><code>[3]</code> 因为单元素子数组的按位 <code>AND</code> 结果就是该元素本身</li>\n\t<li><code>[2]</code> 因为单元素子数组的按位 <code>AND</code> 结果就是该元素本身</li>\n</ol>\n\n<p>这些子数组的值之和为 <code>4 + 3 + 3 + 2 = 12</code></p>\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\">nums = [2,3,5,7,7,7,5], andValues = [0,7,5]</span></p>\n\n<p><strong>输出:</strong> <span class=\"example-io\">17</span></p>\n\n<p><strong>解释:</strong></p>\n\n<p>划分 <code>nums</code> 的三种方式为:</p>\n\n<ol>\n\t<li><code>[[2,3,5],[7,7,7],[5]]</code> 其中子数组的值之和为 <code>5 + 7 + 5 = 17</code></li>\n\t<li><code>[[2,3,5,7],[7,7],[5]]</code> 其中子数组的值之和为 <code>7 + 7 + 5 = 19</code></li>\n\t<li><code>[[2,3,5,7,7],[7],[5]]</code> 其中子数组的值之和为 <code>7 + 7 + 5 = 19</code></li>\n</ol>\n\n<p>子数组值之和的最小可能值为 <code>17</code></p>\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\">nums = [1,2,3,4], andValues = [2]</span></p>\n\n<p><strong>输出:</strong> <span class=\"example-io\">-1</span></p>\n\n<p><strong>解释:</strong></p>\n\n<p>整个数组 <code>nums</code> 的按位 <code>AND</code> 结果为 <code>0</code>。由于无法将 <code>nums</code> 划分为单个子数组使得元素的按位 <code>AND</code> 结果为 <code>2</code>,因此返回 <code>-1</code>。</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;= n == nums.length &lt;= 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= m == andValues.length &lt;= min(n, 10)</code></li>\n\t<li><code>1 &lt;= nums[i] &lt; 10<sup>5</sup></code></li>\n\t<li><code>0 &lt;= andValues[j] &lt; 10<sup>5</sup></code></li>\n</ul>\n",
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"Let <code>dp[i][j]</code> be the optimal answer to split <code>nums[0..(i - 1)]</code> into the first <code>j</code> andValues.",
"<code>dp[i][j] = min(dp[(i - z)][j - 1]) + nums[i - 1]</code> over all <code>x <= z <= y</code> and <code>dp[0][0] = 0</code>, where <code>x</code> and <code>y</code> are the longest and shortest subarrays ending with <code>nums[i - 1]</code> and the bitwise-and of all the values in it is <code>andValues[j - 1]</code>.",
"The answer is <code>dp[n][m]</code>.",
"To calculate <code>x</code> and <code>y</code>, we can use binary search (or sliding window). Note that the more values we have, the smaller the <code>AND</code> value is.",
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