mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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183 lines
20 KiB
JSON
183 lines
20 KiB
JSON
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"questionId": "3224",
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"title": "Count the Number of Infection Sequences",
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"content": "<p>You are given an integer <code>n</code> and a <strong>0-indexed</strong><strong> </strong>integer array <code>sick</code> which is <strong>sorted</strong> in <strong>increasing</strong> order.</p>\n\n<p>There are <code>n</code> children standing in a queue with positions <code>0</code> to <code>n - 1</code> assigned to them. The array <code>sick</code> contains the positions of the children who are infected with an infectious disease. An infected child at position <code>i</code> can spread the disease to either of its immediate neighboring children at positions <code>i - 1</code> and <code>i + 1</code> <strong>if</strong> they exist and are currently not infected. <strong>At most one</strong> child who was previously not infected can get infected with the disease in one second.</p>\n\n<p>It can be shown that after a finite number of seconds, all the children in the queue will get infected with the disease. An <strong>infection sequence</strong> is the sequential order of positions in which <strong>all</strong> of the non-infected children get infected with the disease. Return <em>the total number of possible infection sequences</em>.</p>\n\n<p>Since the answer may be large, return it modulo <code>10<sup>9</sup> + 7</code>.</p>\n\n<p><strong>Note</strong> that an infection sequence <strong>does not</strong> contain positions of children who were already infected with the disease in the beginning.</p>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> n = 5, sick = [0,4]\n<strong>Output:</strong> 4\n<strong>Explanation:</strong> Children at positions 1, 2, and 3 are not infected in the beginning. There are 4 possible infection sequences:\n- The children at positions 1 and 3 can get infected since their positions are adjacent to the infected children 0 and 4. The child at position 1 gets infected first.\nNow, the child at position 2 is adjacent to the child at position 1 who is infected and the child at position 3 is adjacent to the child at position 4 who is infected, hence either of them can get infected. The child at position 2 gets infected.\nFinally, the child at position 3 gets infected because it is adjacent to children at positions 2 and 4 who are infected. The infection sequence is [1,2,3].\n- The children at positions 1 and 3 can get infected because their positions are adjacent to the infected children 0 and 4. The child at position 1 gets infected first.\nNow, the child at position 2 is adjacent to the child at position 1 who is infected and the child at position 3 is adjacent to the child at position 4 who is infected, hence either of them can get infected. The child at position 3 gets infected.\nFinally, the child at position 2 gets infected because it is adjacent to children at positions 1 and 3 who are infected. The infection sequence is [1,3,2].\n- The infection sequence is [3,1,2]. The order of infection of disease in the children can be seen as: [<u>0</u>,1,2,3,<u>4</u>] => [<u>0</u>,1,2,<u>3</u>,<u>4</u>] => [<u>0</u>,<u>1</u>,2,<u>3</u>,<u>4</u>] => [<u>0</u>,<u>1</u>,<u>2</u>,<u>3</u>,<u>4</u>].\n- The infection sequence is [3,2,1]. The order of infection of disease in the children can be seen as: [<u>0</u>,1,2,3,<u>4</u>] => [<u>0</u>,1,2,<u>3</u>,<u>4</u>] => [<u>0</u>,1,<u>2</u>,<u>3</u>,<u>4</u>] => [<u>0</u>,<u>1</u>,<u>2</u>,<u>3</u>,<u>4</u>].\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> n = 4, sick = [1]\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> Children at positions 0, 2, and 3 are not infected in the beginning. There are 3 possible infection sequences:\n- The infection sequence is [0,2,3]. The order of infection of disease in the children can be seen as: [0,<u>1</u>,2,3] => [<u>0</u>,<u>1</u>,2,3] => [<u>0</u>,<u>1</u>,<u>2</u>,3] => [<u>0</u>,<u>1</u>,<u>2</u>,<u>3</u>].\n- The infection sequence is [2,0,3]. The order of infection of disease in the children can be seen as: [0,<u>1</u>,2,3] => [0,<u>1</u>,<u>2</u>,3] => [<u>0</u>,<u>1</u>,<u>2</u>,3] => [<u>0</u>,<u>1</u>,<u>2</u>,<u>3</u>].\n- The infection sequence is [2,3,0]. The order of infection of disease in the children can be seen as: [0,<u>1</u>,2,3] => [0,<u>1</u>,<u>2</u>,3] => [0,<u>1</u>,<u>2</u>,<u>3</u>] => [<u>0</u>,<u>1</u>,<u>2</u>,<u>3</u>].\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 <= n <= 10<sup>5</sup></code></li>\n\t<li><code>1 <= sick.length <= n - 1</code></li>\n\t<li><code>0 <= sick[i] <= n - 1</code></li>\n\t<li><code>sick</code> is sorted in increasing order.</li>\n</ul>\n",
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"Consider infected children as <code>0</code> and non-infected as <code>1</code>, then divide the array into segments with the same value.",
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"For each segment of non-infected children whose indices are <code>[i, j]</code> and indices <code>(i - 1)</code> and <code>(j + 1)</code>, if they exist, are already infected. Then if <code>i == 0</code> or <code>j == n - 1</code>, each second there is only one kid that can be infected (which is at the other endpoint).",
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"If <code>i > 0</code> and <code>j < n - 1</code>, we have two choices per second since the children at the two endpoints can both be the infect candidates. So there are <code>2<sup>j - i + 1</sup></code> orders to infect all children in the segment.",
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"Each second we can select a segment and select one endpoint from it.",
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"The answer is: \r\n<code>S! / (len[1]! * len[2]! * ... * len[m]!) * 2<sup>k</sup></code> \r\nwhere <code>len[1], len[2], ..., len[m]</code> are the lengths of each segment of non-infected children that have an infected child at both endpoints, <code>S</code> is the total length of all segments of non-infected children, and <code>k = (len[1] - 1) + (len[2] - 1) + ... + (len[m] - 1)</code>."
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