leetcode-problemset/leetcode/originData/count-the-number-of-houses-at-a-certain-distance-ii.json

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            "title": "Count the Number of Houses at a Certain Distance II",
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            "content": "<p>You are given three <strong>positive</strong> integers <code>n</code>, <code>x</code>, and <code>y</code>.</p>\n\n<p>In a city, there exist houses numbered <code>1</code> to <code>n</code> connected by <code>n</code> streets. There is a street connecting the house numbered <code>i</code> with the house numbered <code>i + 1</code> for all <code>1 &lt;= i &lt;= n - 1</code> . An additional street connects the house numbered <code>x</code> with the house numbered <code>y</code>.</p>\n\n<p>For each <code>k</code>, such that <code>1 &lt;= k &lt;= n</code>, you need to find the number of <strong>pairs of houses</strong> <code>(house<sub>1</sub>, house<sub>2</sub>)</code> such that the <strong>minimum</strong> number of streets that need to be traveled to reach <code>house<sub>2</sub></code> from <code>house<sub>1</sub></code> is <code>k</code>.</p>\n\n<p>Return <em>a <strong>1-indexed</strong> array </em><code>result</code><em> of length </em><code>n</code><em> where </em><code>result[k]</code><em> represents the <strong>total</strong> number of pairs of houses such that the <strong>minimum</strong> streets required to reach one house from the other is </em><code>k</code>.</p>\n\n<p><strong>Note</strong> that <code>x</code> and <code>y</code> can be <strong>equal</strong>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/12/20/example2.png\" style=\"width: 474px; height: 197px;\" />\n<pre>\n<strong>Input:</strong> n = 3, x = 1, y = 3\n<strong>Output:</strong> [6,0,0]\n<strong>Explanation:</strong> Let&#39;s look at each pair of houses:\n- For the pair (1, 2), we can go from house 1 to house 2 directly.\n- For the pair (2, 1), we can go from house 2 to house 1 directly.\n- For the pair (1, 3), we can go from house 1 to house 3 directly.\n- For the pair (3, 1), we can go from house 3 to house 1 directly.\n- For the pair (2, 3), we can go from house 2 to house 3 directly.\n- For the pair (3, 2), we can go from house 3 to house 2 directly.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/12/20/example3.png\" style=\"width: 668px; height: 174px;\" />\n<pre>\n<strong>Input:</strong> n = 5, x = 2, y = 4\n<strong>Output:</strong> [10,8,2,0,0]\n<strong>Explanation:</strong> For each distance k the pairs are:\n- For k == 1, the pairs are (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (3, 4), (4, 3), (4, 5), and (5, 4).\n- For k == 2, the pairs are (1, 3), (3, 1), (1, 4), (4, 1), (2, 5), (5, 2), (3, 5), and (5, 3).\n- For k == 3, the pairs are (1, 5), and (5, 1).\n- For k == 4 and k == 5, there are no pairs.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/12/20/example5.png\" style=\"width: 544px; height: 130px;\" />\n<pre>\n<strong>Input:</strong> n = 4, x = 1, y = 1\n<strong>Output:</strong> [6,4,2,0]\n<strong>Explanation:</strong> For each distance k the pairs are:\n- For k == 1, the pairs are (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), and (4, 3).\n- For k == 2, the pairs are (1, 3), (3, 1), (2, 4), and (4, 2).\n- For k == 3, the pairs are (1, 4), and (4, 1).\n- For k == 4, there are no pairs.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= x, y &lt;= n</code></li>\n</ul>\n",
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                "If there were no additional street connecting house <code>x</code> to house <code>y</code>, there would be <code>2 * (n - i)</code> pairs of houses at distance <code>i</code>.",
                "The shortest distance between house <code>i</code> and house <code>j</code> (<code>j < i</code>) is along one of these paths:\r\n- <code>i -> j</code>\r\n- <code>i -> y---x -> j</code>",
                "Try to change the distances calculated by path <code>i ->j</code> to the other path.",
                "Can we use prefix sums to compute the answer?"
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