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    `,n}}}var y=S,k='# 示例文章：Google 搜索的即时自动补全功能究竟是如何“工作”的？\n\n> Google 搜索**自动补全功能**的强大，相信不少朋友都能感受到，它帮助我们更快地“补全”我们所要输入的搜索关键字。那么，它怎么知道我们要输入什么内容？它又是如何工作的？在这篇文章里，我们一起来看看。\n\n## 使用自动补全\n\nGoogle 搜索的自动补全功能可以在 Google 搜索应用的大多数位置使用，包括 [Google](https://www.google.com/) 主页、适用于 IOS 和 Android 的 Google 应用，我们只需要在 Google 搜索框上开始键入关键字，就可以看到联想词了。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303019705-c161ce00-d245-446a-b81c-42ec91474a40.gif)\n\n在上图示例中，我们可以看到，输入关键字 `juej`，Google 搜索会联想到“掘金”、“掘金小册”、“绝句”等等，好处就是，我们无须输入完整的关键字即可轻松完成针对这些 topics 的搜索。\n\n谷歌搜索的自动补全功能对于使用移动设备的用户来说特别有用，用户可以轻松在难以键入的小屏幕上完成搜索。当然，对于移动设备用户和台式机用户而言，这都节省了大量的时间。根据 Google 官方报告，自动补全功能可以减少大约 25% 的打字，累积起来，预计每天可以节省 200 多年的打字时间。是的，每天！\n\n> 注意，本文所提到的“**联想词**”与“**预测**”，是同一个意思。\n\n## 基于“预测”而非“建议”\n\nGoogle 官方将自动补全功能称之为“预测”，而不是“建议”，为什么呢？其实是有充分理由的。自动补全功能是为了**帮助用户完成他们打算进行的搜索**，而不是建议用户要执行什么搜索。\n\n那么，Google 是如何确定这些“预测”的？其实，Google 会根据趋势搜索 [trends](https://trends.google.com/trends/?geo=US) 给到我们这些“预测”。简单来说，哪个热门、哪个搜索频率高，就更可能推给我们。当然，这也与我们当前所处的位置以及我们的搜索历史相关。\n\n另外，这些“预测”也会随着我们键入的关键字的变更而更改。例如，当我们把键入的关键字从 `juej` 更改为 `juex` 时，与“掘金”相关的预测会“消失”，同时，与“觉醒”、“决心”相关联的词会出现。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303068169-386a99cb-143b-4ded-a859-1b7a4c4b5bd3.gif)\n\n## 为什么看不到某些联想词？\n\n如果我们在输入某个关键字时看不到联想词，那么表明 Google 的算法可能检测到：\n\n- 这个关键字不是热门字词；\n- 搜索的字词太新了，我们可能需要等待几天或几周才能看到联想词；\n- 这是一个侮辱性或敏感字词，这个搜索字词违反了 Google 的相关政策。更加详细的情况，可以了解 [Google 搜索自动补全政策](https://support.google.com/websearch/answer/7368877)。\n\n## 为什么会看到某些不当的联想词？\n\nGoogle 拥有专门设计的系统，可以自动捕获不适当的预测结果而不显示出来。然而，Google 每天需要处理数十亿次搜索，这意味着 Google 每天会显示数十亿甚至上百亿条预测。再好的系统，也可能存在缺陷，不正确的预测也可能随时会出现。\n\n我们作为 Google 搜索的用户，如果认定某条预测违反了相关的搜索自动补全政策，可以进行举报反馈，点击右下角“**举报不当的联想查询**”并勾选相关选项即可。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303098026-cac215dc-42c9-462a-a359-dcfb12ed3234.gif)\n\n## 如何实现自动补全算法？\n\n目前，Google 官方似乎并没有公开搜索自动补全的算法实现，但是业界在这方面已经有了不少研究。\n\n一个好的自动补全器必须是快速的，并且在用户键入下一个字符后立即更新联想词列表。**自动补全器的核心是一个函数，它接受输入的前缀，并搜索以给定前缀开头的词汇或语句列表**。通常来说，只需要返回少量的数目即可。\n\n接下来，我们先从一个简单且低效的实现开始，并在此基础上逐步构建更高效的方法。\n\n### 词汇表实现\n\n一个**简单粗暴的实现方式**是：顺序查找词汇表，依次检查每个词汇，看它是否以给定的前缀开头。\n\n但是，此方法需要将前缀与每个词汇进行匹配检查，若词汇量较少，这种方式可能勉强行得通。但是，如果词汇量规模较大，效率就太低了。\n\n一个**更好的实现方式是**：让词汇按字典顺序排序。借助二分搜索算法，可以快速搜索有序词汇表中的前缀。由于二分搜索的每一步都会将搜索的范围减半，因此，总的搜索时间与词汇表中单词数量的对数成正比，即时间复杂度是 `O(log N)`。二分搜索的性能很好，但有没有更好的实现呢？当然有，往下看。\n\n### 前缀树实现\n\n通常来说，许多词汇都以相同的前缀开头，比如 `need`、`nested` 都以 `ne` 开头，`seed`、`speed` 都以 `s` 开头。要是为每个单词分别存储公共前缀似乎很浪费。\n\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303128008-93cf798d-2662-4eec-8f80-2e07436aebfe.png)\n\n\n前缀树是一种利用公共前缀来加速补全速度的数据结构。前缀树在节点树中排列一组单词，单词沿着从根节点到叶子节点的路径存储，树的层次对应于前缀的字母位置。\n\n前缀的补全是顺着前缀定义的路径来查找的。例如，在上图的前缀树中，前缀 `ne` 对应于从子节点取左边缘 `N` 和唯一边缘 `E` 的路径。然后可以通过继续遍历从 `E` 节点可以达到的所有叶节点来生成补全列表。在图中，`ne` 的补全可以是两个分支：`-ed` 和 `-sted`。如果在数中找不到由前缀定义的路径，则说明词汇表中不包含以该前缀开头的单词。\n\n### 有限状态自动机(DFA)实现\n\n前缀树可以有效处理公共前缀，但是，对于其他共享词部分，仍会分别存储在每个分支中。比如，后缀 `ed`、`ing`、`tion` 在英文单词中特别常见。在上一个例子中，`e`、`d` 分别存放在了每一个分支上。\n\n有没有一种方法可以更加节省存储空间呢？有的，那就是 DFA。\n\n
    \n
    \n\n在上面的例子中，单词 `need`、`nested`、`seed` 和 `speed` 仅由 9 个节点组成，而上一张图中的前缀树包含了 17 个节点。\n\n可以看出，最小化前缀树 DFA 可以在很大程度上减少数据结构的大小。即使词汇量很大，最小化 DFA 通常也适合在内存中存储，避免昂贵的磁盘访问是实现快速自动补全的关键。\n\n### 一些扩展\n\n上面介绍了如何利用合理的数据结构实现基本的自动补全功能。这些数据结构可以通过多种方式进行扩展，从而改善用户体验。\n\n通常，满足特定前缀的词汇可能很多，而用户界面上能够显示的却不多，我们更希望能显示最常搜索或者最有价值的词汇。这通常可以通过为词汇表中的每个单词增加一个代表单词值的**权重** `weight`，并且按照权重高低来排序自动补全列表。\n\n- 对于排序后的词汇表来说，在词汇表每个元素上增加 `weight` 属性并不难；\n- 对于前缀树来说，将 `weight` 存储在叶子节点中，也是很简单的一个实现；\n- 对于 `DFA` 来说，则较为复杂。因为一个叶子节点可以通过多条路径到达。一种解决方案是将权重关联到路径而不是叶子节点。\n\n目前有不少开源库都提供了这个功能，比如主流的搜索引擎框架 [Elasticsearch](https://www.elastic.co/products/elasticsearch)、[Solr](https://lucene.apache.org/solr/) 等，基于此，我们可以实现高效而强大的自动补全功能。\n\n#### 推荐阅读\n\n- [阿里又一个 20k+ stars 开源项目诞生，恭喜 fastjson！](https://mp.weixin.qq.com/s/RNKDCK2KoyeuMeEs6GUrow)\n- [刷掉 90% 候选人的互联网大厂海量数据面试题（附题解 + 方法总结）](https://mp.weixin.qq.com/s/rjGqxUvrEqJNlo09GrT1Dw)\n- [好用！期待已久的文本块功能究竟如何在 Java 13 中发挥作用？](https://mp.weixin.qq.com/s/kalGv5T8AZGxTnLHr2wDsA)\n- [2019 GitHub 开源贡献排行榜新鲜出炉！微软谷歌领头，阿里跻身前 12！](https://mp.weixin.qq.com/s/_q812aGD1b9QvZ2WFI0Qgw)\n\n---\n\n欢迎关注我的公众号“**Doocs**”，原创技术文章第一时间推送。\n\n
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    `,n}}}var y=S,k='# 示例文章：Google 搜索的即时自动补全功能究竟是如何“工作”的？\n\n> Google 搜索**自动补全功能**的强大，相信不少朋友都能感受到，它帮助我们更快地“补全”我们所要输入的搜索关键字。那么，它怎么知道我们要输入什么内容？它又是如何工作的？在这篇文章里，我们一起来看看。\n\n## 使用自动补全\n\nGoogle 搜索的自动补全功能可以在 Google 搜索应用的大多数位置使用，包括 [Google](https://www.google.com/) 主页、适用于 IOS 和 Android 的 Google 应用，我们只需要在 Google 搜索框上开始键入关键字，就可以看到联想词了。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303019705-c161ce00-d245-446a-b81c-42ec91474a40.gif)\n\n在上图示例中，我们可以看到，输入关键字 `juej`，Google 搜索会联想到“掘金”、“掘金小册”、“绝句”等等，好处就是，我们无须输入完整的关键字即可轻松完成针对这些 topics 的搜索。\n\n谷歌搜索的自动补全功能对于使用移动设备的用户来说特别有用，用户可以轻松在难以键入的小屏幕上完成搜索。当然，对于移动设备用户和台式机用户而言，这都节省了大量的时间。根据 Google 官方报告，自动补全功能可以减少大约 25% 的打字，累积起来，预计每天可以节省 200 多年的打字时间。是的，每天！\n\n> 注意，本文所提到的“**联想词**”与“**预测**”，是同一个意思。\n\n## 基于“预测”而非“建议”\n\nGoogle 官方将自动补全功能称之为“预测”，而不是“建议”，为什么呢？其实是有充分理由的。自动补全功能是为了**帮助用户完成他们打算进行的搜索**，而不是建议用户要执行什么搜索。\n\n那么，Google 是如何确定这些“预测”的？其实，Google 会根据趋势搜索 [trends](https://trends.google.com/trends/?geo=US) 给到我们这些“预测”。简单来说，哪个热门、哪个搜索频率高，就更可能推给我们。当然，这也与我们当前所处的位置以及我们的搜索历史相关。\n\n另外，这些“预测”也会随着我们键入的关键字的变更而更改。例如，当我们把键入的关键字从 `juej` 更改为 `juex` 时，与“掘金”相关的预测会“消失”，同时，与“觉醒”、“决心”相关联的词会出现。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303068169-386a99cb-143b-4ded-a859-1b7a4c4b5bd3.gif)\n\n## 为什么看不到某些联想词？\n\n如果我们在输入某个关键字时看不到联想词，那么表明 Google 的算法可能检测到：\n\n- 这个关键字不是热门字词；\n- 搜索的字词太新了，我们可能需要等待几天或几周才能看到联想词；\n- 这是一个侮辱性或敏感字词，这个搜索字词违反了 Google 的相关政策。更加详细的情况，可以了解 [Google 搜索自动补全政策](https://support.google.com/websearch/answer/7368877)。\n\n## 为什么会看到某些不当的联想词？\n\nGoogle 拥有专门设计的系统，可以自动捕获不适当的预测结果而不显示出来。然而，Google 每天需要处理数十亿次搜索，这意味着 Google 每天会显示数十亿甚至上百亿条预测。再好的系统，也可能存在缺陷，不正确的预测也可能随时会出现。\n\n我们作为 Google 搜索的用户，如果认定某条预测违反了相关的搜索自动补全政策，可以进行举报反馈，点击右下角“**举报不当的联想查询**”并勾选相关选项即可。\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303098026-cac215dc-42c9-462a-a359-dcfb12ed3234.gif)\n\n## 如何实现自动补全算法？\n\n目前，Google 官方似乎并没有公开搜索自动补全的算法实现，但是业界在这方面已经有了不少研究。\n\n一个好的自动补全器必须是快速的，并且在用户键入下一个字符后立即更新联想词列表。**自动补全器的核心是一个函数，它接受输入的前缀，并搜索以给定前缀开头的词汇或语句列表**。通常来说，只需要返回少量的数目即可。\n\n接下来，我们先从一个简单且低效的实现开始，并在此基础上逐步构建更高效的方法。\n\n### 词汇表实现\n\n一个**简单粗暴的实现方式**是：顺序查找词汇表，依次检查每个词汇，看它是否以给定的前缀开头。\n\n但是，此方法需要将前缀与每个词汇进行匹配检查，若词汇量较少，这种方式可能勉强行得通。但是，如果词汇量规模较大，效率就太低了。\n\n一个**更好的实现方式是**：让词汇按字典顺序排序。借助二分搜索算法，可以快速搜索有序词汇表中的前缀。由于二分搜索的每一步都会将搜索的范围减半，因此，总的搜索时间与词汇表中单词数量的对数成正比，即时间复杂度是 `O(log N)`。二分搜索的性能很好，但有没有更好的实现呢？当然有，往下看。\n\n### 前缀树实现\n\n通常来说，许多词汇都以相同的前缀开头，比如 `need`、`nested` 都以 `ne` 开头，`seed`、`speed` 都以 `s` 开头。要是为每个单词分别存储公共前缀似乎很浪费。\n\n\n![](https://cdn-doocs.oss-cn-shenzhen.aliyuncs.com/gh/doocs/md/images/1648303128008-93cf798d-2662-4eec-8f80-2e07436aebfe.png)\n\n\n前缀树是一种利用公共前缀来加速补全速度的数据结构。前缀树在节点树中排列一组单词，单词沿着从根节点到叶子节点的路径存储，树的层次对应于前缀的字母位置。\n\n前缀的补全是顺着前缀定义的路径来查找的。例如，在上图的前缀树中，前缀 `ne` 对应于从子节点取左边缘 `N` 和唯一边缘 `E` 的路径。然后可以通过继续遍历从 `E` 节点可以达到的所有叶节点来生成补全列表。在图中，`ne` 的补全可以是两个分支：`-ed` 和 `-sted`。如果在数中找不到由前缀定义的路径，则说明词汇表中不包含以该前缀开头的单词。\n\n### 有限状态自动机(DFA)实现\n\n前缀树可以有效处理公共前缀，但是，对于其他共享词部分，仍会分别存储在每个分支中。比如，后缀 `ed`、`ing`、`tion` 在英文单词中特别常见。在上一个例子中，`e`、`d` 分别存放在了每一个分支上。\n\n有没有一种方法可以更加节省存储空间呢？有的，那就是 DFA。\n\n
    \n
    \n\n在上面的例子中，单词 `need`、`nested`、`seed` 和 `speed` 仅由 9 个节点组成，而上一张图中的前缀树包含了 17 个节点。\n\n可以看出，最小化前缀树 DFA 可以在很大程度上减少数据结构的大小。即使词汇量很大，最小化 DFA 通常也适合在内存中存储，避免昂贵的磁盘访问是实现快速自动补全的关键。\n\n### 一些扩展\n\n上面介绍了如何利用合理的数据结构实现基本的自动补全功能。这些数据结构可以通过多种方式进行扩展，从而改善用户体验。\n\n通常，满足特定前缀的词汇可能很多，而用户界面上能够显示的却不多，我们更希望能显示最常搜索或者最有价值的词汇。这通常可以通过为词汇表中的每个单词增加一个代表单词值的**权重** `weight`，并且按照权重高低来排序自动补全列表。\n\n- 对于排序后的词汇表来说，在词汇表每个元素上增加 `weight` 属性并不难；\n- 对于前缀树来说，将 `weight` 存储在叶子节点中，也是很简单的一个实现；\n- 对于 `DFA` 来说，则较为复杂。因为一个叶子节点可以通过多条路径到达。一种解决方案是将权重关联到路径而不是叶子节点。\n\n目前有不少开源库都提供了这个功能，比如主流的搜索引擎框架 [Elasticsearch](https://www.elastic.co/products/elasticsearch)、[Solr](https://lucene.apache.org/solr/) 等，基于此，我们可以实现高效而强大的自动补全功能。\n\n#### 推荐阅读\n\n- [阿里又一个 20k+ stars 开源项目诞生，恭喜 fastjson！](https://mp.weixin.qq.com/s/RNKDCK2KoyeuMeEs6GUrow)\n- [刷掉 90% 候选人的互联网大厂海量数据面试题（附题解 + 方法总结）](https://mp.weixin.qq.com/s/rjGqxUvrEqJNlo09GrT1Dw)\n- [好用！期待已久的文本块功能究竟如何在 Java 13 中发挥作用？](https://mp.weixin.qq.com/s/kalGv5T8AZGxTnLHr2wDsA)\n- [2019 GitHub 开源贡献排行榜新鲜出炉！微软谷歌领头，阿里跻身前 12！](https://mp.weixin.qq.com/s/_q812aGD1b9QvZ2WFI0Qgw)\n\n---\n\n欢迎关注我的公众号“**Doocs**”，原创技术文章第一时间推送。\n\n
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graph6_export graph6_import graph_center graph_charpoly graph_eigenvalues graph_flow graph_order graph_periphery graph_product graph_size graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grind grobner_basis grotzch_graph hamilton_cycle hamilton_path hankel hankel_1 hankel_2 harmonic harmonic_mean hav heawood_graph hermite hessian hgfred hilbertmap hilbert_matrix hipow histogram histogram_description hodge horner hypergeometric i0 i1 %ibes ic1 ic2 ic_convert ichr1 ichr2 icosahedron_graph icosidodecahedron_graph icurvature ident identfor identity idiff idim idummy ieqn %if ifactors iframes ifs igcdex igeodesic_coords ilt image imagpart imetric implicit implicit_derivative implicit_plot indexed_tensor indices induced_subgraph inferencep inference_result infix info_display init_atensor init_ctensor in_neighbors innerproduct inpart inprod inrt integerp integer_partitions integrate intersect intersection intervalp intopois intosum invariant1 invariant2 inverse_fft inverse_jacobi_cd inverse_jacobi_cn inverse_jacobi_cs inverse_jacobi_dc inverse_jacobi_dn inverse_jacobi_ds inverse_jacobi_nc inverse_jacobi_nd inverse_jacobi_ns inverse_jacobi_sc inverse_jacobi_sd inverse_jacobi_sn invert invert_by_adjoint invert_by_lu inv_mod irr is is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph ishow is_isomorphic isolate isomorphism is_planar isqrt isreal_p is_sconnected is_tree is_vertex_in_graph items_inference %j j0 j1 jacobi jacobian jacobi_cd jacobi_cn jacobi_cs jacobi_dc jacobi_dn jacobi_ds jacobi_nc jacobi_nd jacobi_ns jacobi_p jacobi_sc jacobi_sd jacobi_sn JF jn join jordan julia julia_set julia_sin %k kdels kdelta kill killcontext kostka kron_delta kronecker_product kummer_m kummer_u kurtosis kurtosis_bernoulli kurtosis_beta kurtosis_binomial kurtosis_chi2 kurtosis_continuous_uniform kurtosis_discrete_uniform kurtosis_exp kurtosis_f kurtosis_gamma kurtosis_general_finite_discrete kurtosis_geometric kurtosis_gumbel kurtosis_hypergeometric kurtosis_laplace kurtosis_logistic kurtosis_lognormal kurtosis_negative_binomial kurtosis_noncentral_chi2 kurtosis_noncentral_student_t kurtosis_normal kurtosis_pareto kurtosis_poisson kurtosis_rayleigh kurtosis_student_t kurtosis_weibull label labels lagrange laguerre lambda lambert_w laplace laplacian_matrix last lbfgs lc2kdt lcharp lc_l lcm lc_u ldefint ldisp ldisplay legendre_p legendre_q leinstein length let letrules letsimp levi_civita lfreeof lgtreillis lhs li liediff limit Lindstedt linear linearinterpol linear_program linear_regression line_graph linsolve listarray list_correlations listify list_matrix_entries list_nc_monomials listoftens listofvars listp lmax lmin load loadfile local locate_matrix_entry log logcontract log_gamma lopow lorentz_gauge lowercasep lpart lratsubst lreduce lriemann lsquares_estimates lsquares_estimates_approximate lsquares_estimates_exact lsquares_mse lsquares_residual_mse lsquares_residuals lsum ltreillis lu_backsub lucas lu_factor %m macroexpand macroexpand1 make_array makebox makefact makegamma make_graph make_level_picture makelist makeOrders make_poly_continent make_poly_country make_polygon make_random_state make_rgb_picture makeset make_string_input_stream make_string_output_stream make_transform mandelbrot mandelbrot_set map mapatom maplist matchdeclare matchfix mat_cond mat_fullunblocker mat_function mathml_display mat_norm matrix matrixmap matrixp matrix_size mattrace mat_trace mat_unblocker max max_clique max_degree max_flow maximize_lp max_independent_set max_matching maybe md5sum mean mean_bernoulli mean_beta mean_binomial mean_chi2 mean_continuous_uniform mean_deviation mean_discrete_uniform mean_exp mean_f mean_gamma mean_general_finite_discrete mean_geometric mean_gumbel mean_hypergeometric mean_laplace mean_logistic mean_lognormal mean_negative_binomial mean_noncentral_chi2 mean_noncentral_student_t mean_normal mean_pareto mean_poisson mean_rayleigh mean_student_t mean_weibull median median_deviation member mesh metricexpandall mgf1_sha1 min min_degree min_edge_cut minfactorial minimalPoly minimize_lp minimum_spanning_tree minor minpack_lsquares minpack_solve min_vertex_cover min_vertex_cut mkdir mnewton mod mode_declare mode_identity ModeMatrix moebius mon2schur mono monomial_dimensions multibernstein_poly multi_display_for_texinfo multi_elem multinomial multinomial_coeff multi_orbit multiplot_mode multi_pui multsym multthru mycielski_graph nary natural_unit nc_degree ncexpt ncharpoly negative_picture neighbors new newcontext newdet new_graph newline newton new_variable next_prime nicedummies niceindices ninth nofix nonarray noncentral_moment nonmetricity nonnegintegerp nonscalarp nonzeroandfreeof notequal nounify nptetrad npv nroots nterms ntermst nthroot nullity nullspace num numbered_boundaries numberp number_to_octets num_distinct_partitions numerval numfactor num_partitions nusum nzeta nzetai nzetar octets_to_number octets_to_oid odd_girth oddp ode2 ode_check odelin oid_to_octets op opena opena_binary openr openr_binary openw openw_binary operatorp opsubst optimize %or orbit orbits ordergreat ordergreatp orderless orderlessp orthogonal_complement orthopoly_recur orthopoly_weight outermap out_neighbors outofpois pade parabolic_cylinder_d parametric parametric_surface parg parGosper parse_string parse_timedate part part2cont partfrac partition partition_set partpol path_digraph path_graph pathname_directory pathname_name pathname_type pdf_bernoulli pdf_beta pdf_binomial pdf_cauchy pdf_chi2 pdf_continuous_uniform pdf_discrete_uniform pdf_exp pdf_f pdf_gamma pdf_general_finite_discrete pdf_geometric pdf_gumbel pdf_hypergeometric pdf_laplace pdf_logistic pdf_lognormal pdf_negative_binomial pdf_noncentral_chi2 pdf_noncentral_student_t pdf_normal pdf_pareto pdf_poisson pdf_rank_sum pdf_rayleigh pdf_signed_rank pdf_student_t pdf_weibull pearson_skewness permanent permut permutation permutations petersen_graph petrov pickapart picture_equalp picturep piechart piechart_description planar_embedding playback plog plot2d plot3d plotdf ploteq plsquares pochhammer points poisdiff poisexpt poisint poismap poisplus poissimp poissubst poistimes poistrim polar polarform polartorect polar_to_xy poly_add poly_buchberger poly_buchberger_criterion poly_colon_ideal poly_content polydecomp poly_depends_p poly_elimination_ideal poly_exact_divide poly_expand poly_expt poly_gcd polygon poly_grobner poly_grobner_equal poly_grobner_member poly_grobner_subsetp poly_ideal_intersection poly_ideal_polysaturation poly_ideal_polysaturation1 poly_ideal_saturation poly_ideal_saturation1 poly_lcm poly_minimization polymod poly_multiply polynome2ele polynomialp poly_normal_form poly_normalize poly_normalize_list poly_polysaturation_extension poly_primitive_part poly_pseudo_divide poly_reduced_grobner poly_reduction poly_saturation_extension poly_s_polynomial poly_subtract polytocompanion pop postfix potential power_mod powerseries powerset prefix prev_prime primep primes principal_components print printf printfile print_graph printpois printprops prodrac product properties propvars psi psubst ptriangularize pui pui2comp pui2ele pui2polynome pui_direct puireduc push put pv qput qrange qty quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws quadrilateral quantile quantile_bernoulli quantile_beta quantile_binomial quantile_cauchy quantile_chi2 quantile_continuous_uniform quantile_discrete_uniform quantile_exp quantile_f quantile_gamma quantile_general_finite_discrete quantile_geometric quantile_gumbel quantile_hypergeometric quantile_laplace quantile_logistic quantile_lognormal quantile_negative_binomial quantile_noncentral_chi2 quantile_noncentral_student_t quantile_normal quantile_pareto quantile_poisson quantile_rayleigh quantile_student_t quantile_weibull quartile_skewness quit qunit quotient racah_v racah_w radcan radius random random_bernoulli random_beta random_binomial random_bipartite_graph random_cauchy random_chi2 random_continuous_uniform random_digraph random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_graph random_graph1 random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_network random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_permutation random_poisson random_rayleigh random_regular_graph random_student_t random_tournament random_tree random_weibull range rank rat ratcoef ratdenom ratdiff ratdisrep ratexpand ratinterpol rational rationalize ratnumer ratnump ratp ratsimp ratsubst ratvars ratweight read read_array read_binary_array read_binary_list read_binary_matrix readbyte readchar read_hashed_array readline read_list read_matrix read_nested_list readonly read_xpm real_imagpart_to_conjugate realpart realroots rearray rectangle rectform rectform_log_if_constant recttopolar rediff reduce_consts reduce_order region region_boundaries region_boundaries_plus rem remainder remarray rembox remcomps remcon remcoord remfun remfunction remlet remove remove_constvalue remove_dimensions remove_edge remove_fundamental_dimensions remove_fundamental_units remove_plot_option remove_vertex rempart remrule remsym remvalue rename rename_file reset reset_displays residue resolvante resolvante_alternee1 resolvante_bipartite resolvante_diedrale resolvante_klein resolvante_klein3 resolvante_produit_sym resolvante_unitaire resolvante_vierer rest resultant return reveal reverse revert revert2 rgb2level rhs ricci riemann rinvariant risch rk rmdir rncombine romberg room rootscontract round row rowop rowswap rreduce run_testsuite %s save saving scalarp scaled_bessel_i scaled_bessel_i0 scaled_bessel_i1 scalefactors scanmap scatterplot scatterplot_description scene schur2comp sconcat scopy scsimp scurvature sdowncase sec sech second sequal sequalignore set_alt_display setdifference set_draw_defaults set_edge_weight setelmx setequalp setify setp set_partitions set_plot_option set_prompt set_random_state set_tex_environment set_tex_environment_default setunits setup_autoload set_up_dot_simplifications set_vertex_label seventh sexplode sf sha1sum sha256sum shortest_path shortest_weighted_path show showcomps showratvars sierpinskiale sierpinskimap sign signum similaritytransform simp_inequality simplify_sum simplode simpmetderiv simtran sin sinh sinsert sinvertcase sixth skewness skewness_bernoulli skewness_beta skewness_binomial skewness_chi2 skewness_continuous_uniform skewness_discrete_uniform skewness_exp skewness_f skewness_gamma skewness_general_finite_discrete skewness_geometric skewness_gumbel skewness_hypergeometric skewness_laplace skewness_logistic skewness_lognormal skewness_negative_binomial skewness_noncentral_chi2 skewness_noncentral_student_t skewness_normal skewness_pareto skewness_poisson skewness_rayleigh skewness_student_t skewness_weibull slength smake small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph smax smin smismatch snowmap snub_cube_graph snub_dodecahedron_graph solve solve_rec solve_rec_rat some somrac sort sparse6_decode sparse6_encode sparse6_export sparse6_import specint spherical spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2 spherical_harmonic spherical_to_xyz splice split sposition sprint sqfr sqrt sqrtdenest sremove sremovefirst sreverse ssearch ssort sstatus ssubst ssubstfirst staircase standardize standardize_inverse_trig starplot starplot_description status std std1 std_bernoulli std_beta std_binomial std_chi2 std_continuous_uniform std_discrete_uniform std_exp std_f std_gamma std_general_finite_discrete std_geometric std_gumbel std_hypergeometric std_laplace std_logistic std_lognormal std_negative_binomial std_noncentral_chi2 std_noncentral_student_t std_normal std_pareto std_poisson std_rayleigh std_student_t std_weibull stemplot stirling stirling1 stirling2 strim striml strimr string stringout stringp strong_components struve_h struve_l sublis sublist sublist_indices submatrix subsample subset subsetp subst substinpart subst_parallel substpart substring subvar subvarp sum sumcontract summand_to_rec supcase supcontext symbolp symmdifference symmetricp system take_channel take_inference tan tanh taylor taylorinfo taylorp taylor_simplifier taytorat tcl_output tcontract tellrat tellsimp tellsimpafter tentex tenth test_mean test_means_difference test_normality test_proportion test_proportions_difference test_rank_sum test_sign test_signed_rank test_variance test_variance_ratio tex tex1 tex_display texput %th third throw time timedate timer timer_info tldefint tlimit todd_coxeter toeplitz tokens to_lisp topological_sort to_poly to_poly_solve totaldisrep totalfourier totient tpartpol trace tracematrix trace_options transform_sample translate translate_file transpose treefale tree_reduce treillis treinat triangle triangularize trigexpand trigrat trigreduce trigsimp trunc truncate truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tr_warnings_get tube tutte_graph ueivects uforget ultraspherical underlying_graph undiff union unique uniteigenvectors unitp units unit_step unitvector unorder unsum untellrat untimer untrace uppercasep uricci uriemann uvect vandermonde_matrix var var1 var_bernoulli var_beta var_binomial var_chi2 var_continuous_uniform var_discrete_uniform var_exp var_f var_gamma var_general_finite_discrete var_geometric var_gumbel var_hypergeometric var_laplace var_logistic var_lognormal var_negative_binomial var_noncentral_chi2 var_noncentral_student_t var_normal var_pareto var_poisson var_rayleigh var_student_t var_weibull vector vectorpotential vectorsimp verbify vers vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertices vertices_to_cycle vertices_to_path %w weyl wheel_graph wiener_index wigner_3j wigner_6j wigner_9j with_stdout write_binary_data writebyte write_data writefile wronskian xreduce xthru %y Zeilberger zeroequiv zerofor zeromatrix zeromatrixp zeta zgeev zheev zlange zn_add_table zn_carmichael_lambda zn_characteristic_factors zn_determinant zn_factor_generators zn_invert_by_lu zn_log zn_mult_table absboxchar activecontexts adapt_depth additive adim aform algebraic algepsilon algexact aliases allbut all_dotsimp_denoms allocation allsym alphabetic animation antisymmetric arrays askexp assume_pos assume_pos_pred assumescalar asymbol atomgrad atrig1 axes axis_3d axis_bottom axis_left axis_right axis_top azimuth background background_color backsubst berlefact bernstein_explicit besselexpand beta_args_sum_to_integer beta_expand bftorat bftrunc bindtest border boundaries_array box boxchar breakup %c capping cauchysum cbrange cbtics center cflength cframe_flag cnonmet_flag color color_bar color_bar_tics colorbox columns commutative complex cone context contexts contour contour_levels cosnpiflag ctaypov ctaypt ctayswitch ctayvar ct_coords ctorsion_flag ctrgsimp cube current_let_rule_package cylinder data_file_name debugmode decreasing default_let_rule_package delay dependencies derivabbrev derivsubst detout diagmetric diff dim dimensions dispflag display2d|10 display_format_internal distribute_over doallmxops domain domxexpt domxmxops domxnctimes dontfactor doscmxops doscmxplus dot0nscsimp dot0simp dot1simp dotassoc dotconstrules dotdistrib dotexptsimp dotident dotscrules draw_graph_program draw_realpart edge_color edge_coloring edge_partition edge_type edge_width %edispflag elevation %emode endphi endtheta engineering_format_floats enhanced3d %enumer epsilon_lp erfflag erf_representation errormsg error_size error_syms error_type %e_to_numlog eval even evenfun evflag evfun ev_point expandwrt_denom expintexpand expintrep expon expop exptdispflag exptisolate exptsubst facexpand facsum_combine factlim factorflag factorial_expand factors_only fb feature features file_name file_output_append file_search_demo file_search_lisp file_search_maxima|10 file_search_tests file_search_usage file_type_lisp file_type_maxima|10 fill_color fill_density filled_func fixed_vertices flipflag float2bf font font_size fortindent fortspaces fpprec fpprintprec functions gamma_expand gammalim gdet genindex gensumnum GGFCFMAX GGFINFINITY globalsolve gnuplot_command gnuplot_curve_styles gnuplot_curve_titles gnuplot_default_term_command gnuplot_dumb_term_command gnuplot_file_args gnuplot_file_name gnuplot_out_file gnuplot_pdf_term_command gnuplot_pm3d gnuplot_png_term_command gnuplot_postamble gnuplot_preamble gnuplot_ps_term_command gnuplot_svg_term_command gnuplot_term gnuplot_view_args Gosper_in_Zeilberger gradefs grid grid2d grind halfangles head_angle head_both head_length head_type height hypergeometric_representation %iargs ibase icc1 icc2 icounter idummyx ieqnprint ifb ifc1 ifc2 ifg ifgi ifr iframe_bracket_form ifri igeowedge_flag ikt1 ikt2 imaginary inchar increasing infeval infinity inflag infolists inm inmc1 inmc2 intanalysis integer integervalued integrate_use_rootsof integration_constant integration_constant_counter interpolate_color intfaclim ip_grid ip_grid_in irrational isolate_wrt_times iterations itr julia_parameter %k1 %k2 keepfloat key key_pos kinvariant kt label label_alignment label_orientation labels lassociative lbfgs_ncorrections lbfgs_nfeval_max leftjust legend letrat let_rule_packages lfg lg lhospitallim limsubst linear linear_solver linechar linel|10 linenum line_type linewidth line_width linsolve_params linsolvewarn lispdisp listarith listconstvars listdummyvars lmxchar load_pathname loadprint logabs logarc logcb logconcoeffp logexpand lognegint logsimp logx logx_secondary logy logy_secondary logz lriem m1pbranch macroexpansion macros mainvar manual_demo maperror mapprint matrix_element_add matrix_element_mult matrix_element_transpose maxapplydepth 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poly_grobner_debug poly_monomial_order poly_primary_elimination_order poly_return_term_list poly_secondary_elimination_order poly_top_reduction_only posfun position powerdisp pred prederror primep_number_of_tests product_use_gamma program programmode promote_float_to_bigfloat prompt proportional_axes props psexpand ps_file radexpand radius radsubstflag rassociative ratalgdenom ratchristof ratdenomdivide rateinstein ratepsilon ratfac rational ratmx ratprint ratriemann ratsimpexpons ratvarswitch ratweights ratweyl ratwtlvl real realonly redraw refcheck resolution restart resultant ric riem rmxchar %rnum_list rombergabs rombergit rombergmin rombergtol rootsconmode rootsepsilon run_viewer same_xy same_xyz savedef savefactors scalar scalarmatrixp scale scale_lp setcheck setcheckbreak setval show_edge_color show_edges show_edge_type show_edge_width show_id show_label showtime show_vertex_color show_vertex_size show_vertex_type show_vertices show_weight simp simplified_output 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V1759 v"+t+" v1759 h84z";case"vert":return"M145 15 v585 v"+t+" v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v"+-t+" v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v"+t+" v585 h43z";case"doublevert":return"M145 15 v585 v"+t+" v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v"+-t+" v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v"+t+" v585 h43z\nM367 15 v585 v"+t+" v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v"+-t+" v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M410 15 H367 v585 v"+t+" v585 h43z";case"lfloor":return"M319 602 V0 H403 V602 v"+t+" v1715 h263 v84 H319z\nMM319 602 V0 H403 V602 v"+t+" v1715 H319z";case"rfloor":return"M319 602 V0 H403 V602 v"+t+" v1799 H0 v-84 H319z\nMM319 602 V0 H403 V602 v"+t+" v1715 H319z";case"lceil":return"M403 1759 V84 H666 V0 H319 V1759 v"+t+" v602 h84z\nM403 1759 V0 H319 V1759 v"+t+" v602 h84z";case"rceil":return"M347 1759 V0 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Se=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Se>>>26)|0,Se&=67108863,r=Math.imul(P,z),i=Math.imul(P,q),i=i+Math.imul(L,z)|0,o=Math.imul(L,q),r=r+Math.imul(I,$)|0,i=i+Math.imul(I,G)|0,i=i+Math.imul(M,$)|0,o=o+Math.imul(M,G)|0,r=r+Math.imul(k,Y)|0,i=i+Math.imul(k,W)|0,i=i+Math.imul(O,Y)|0,o=o+Math.imul(O,W)|0,r=r+Math.imul(T,Q)|0,i=i+Math.imul(T,X)|0,i=i+Math.imul(D,Q)|0,o=o+Math.imul(D,X)|0,r=r+Math.imul(S,J)|0,i=i+Math.imul(S,ee)|0,i=i+Math.imul(x,J)|0,o=o+Math.imul(x,ee)|0,r=r+Math.imul(_,ne)|0,i=i+Math.imul(_,re)|0,i=i+Math.imul(E,ne)|0,o=o+Math.imul(E,re)|0,r=r+Math.imul(v,oe)|0,i=i+Math.imul(v,ae)|0,i=i+Math.imul(b,oe)|0,o=o+Math.imul(b,ae)|0,r=r+Math.imul(f,le)|0,i=i+Math.imul(f,ue)|0,i=i+Math.imul(m,le)|0,o=o+Math.imul(m,ue)|0,r=r+Math.imul(d,de)|0,i=i+Math.imul(d,pe)|0,i=i+Math.imul(p,de)|0,o=o+Math.imul(p,pe)|0;var xe=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(xe>>>26)|0,xe&=67108863,r=Math.imul(B,z),i=Math.imul(B,q),i=i+Math.imul(j,z)|0,o=Math.imul(j,q),r=r+Math.imul(P,$)|0,i=i+Math.imul(P,G)|0,i=i+Math.imul(L,$)|0,o=o+Math.imul(L,G)|0,r=r+Math.imul(I,Y)|0,i=i+Math.imul(I,W)|0,i=i+Math.imul(M,Y)|0,o=o+Math.imul(M,W)|0,r=r+Math.imul(k,Q)|0,i=i+Math.imul(k,X)|0,i=i+Math.imul(O,Q)|0,o=o+Math.imul(O,X)|0,r=r+Math.imul(T,J)|0,i=i+Math.imul(T,ee)|0,i=i+Math.imul(D,J)|0,o=o+Math.imul(D,ee)|0,r=r+Math.imul(S,ne)|0,i=i+Math.imul(S,re)|0,i=i+Math.imul(x,ne)|0,o=o+Math.imul(x,re)|0,r=r+Math.imul(_,oe)|0,i=i+Math.imul(_,ae)|0,i=i+Math.imul(E,oe)|0,o=o+Math.imul(E,ae)|0,r=r+Math.imul(v,le)|0,i=i+Math.imul(v,ue)|0,i=i+Math.imul(b,le)|0,o=o+Math.imul(b,ue)|0,r=r+Math.imul(f,de)|0,i=i+Math.imul(f,pe)|0,i=i+Math.imul(m,de)|0,o=o+Math.imul(m,pe)|0,r=r+Math.imul(d,fe)|0,i=i+Math.imul(d,me)|0,i=i+Math.imul(p,fe)|0,o=o+Math.imul(p,me)|0;var Ce=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Ce>>>26)|0,Ce&=67108863,r=Math.imul(B,$),i=Math.imul(B,G),i=i+Math.imul(j,$)|0,o=Math.imul(j,G),r=r+Math.imul(P,Y)|0,i=i+Math.imul(P,W)|0,i=i+Math.imul(L,Y)|0,o=o+Math.imul(L,W)|0,r=r+Math.imul(I,Q)|0,i=i+Math.imul(I,X)|0,i=i+Math.imul(M,Q)|0,o=o+Math.imul(M,X)|0,r=r+Math.imul(k,J)|0,i=i+Math.imul(k,ee)|0,i=i+Math.imul(O,J)|0,o=o+Math.imul(O,ee)|0,r=r+Math.imul(T,ne)|0,i=i+Math.imul(T,re)|0,i=i+Math.imul(D,ne)|0,o=o+Math.imul(D,re)|0,r=r+Math.imul(S,oe)|0,i=i+Math.imul(S,ae)|0,i=i+Math.imul(x,oe)|0,o=o+Math.imul(x,ae)|0,r=r+Math.imul(_,le)|0,i=i+Math.imul(_,ue)|0,i=i+Math.imul(E,le)|0,o=o+Math.imul(E,ue)|0,r=r+Math.imul(v,de)|0,i=i+Math.imul(v,pe)|0,i=i+Math.imul(b,de)|0,o=o+Math.imul(b,pe)|0,r=r+Math.imul(f,fe)|0,i=i+Math.imul(f,me)|0,i=i+Math.imul(m,fe)|0,o=o+Math.imul(m,me)|0;var Te=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Te>>>26)|0,Te&=67108863,r=Math.imul(B,Y),i=Math.imul(B,W),i=i+Math.imul(j,Y)|0,o=Math.imul(j,W),r=r+Math.imul(P,Q)|0,i=i+Math.imul(P,X)|0,i=i+Math.imul(L,Q)|0,o=o+Math.imul(L,X)|0,r=r+Math.imul(I,J)|0,i=i+Math.imul(I,ee)|0,i=i+Math.imul(M,J)|0,o=o+Math.imul(M,ee)|0,r=r+Math.imul(k,ne)|0,i=i+Math.imul(k,re)|0,i=i+Math.imul(O,ne)|0,o=o+Math.imul(O,re)|0,r=r+Math.imul(T,oe)|0,i=i+Math.imul(T,ae)|0,i=i+Math.imul(D,oe)|0,o=o+Math.imul(D,ae)|0,r=r+Math.imul(S,le)|0,i=i+Math.imul(S,ue)|0,i=i+Math.imul(x,le)|0,o=o+Math.imul(x,ue)|0,r=r+Math.imul(_,de)|0,i=i+Math.imul(_,pe)|0,i=i+Math.imul(E,de)|0,o=o+Math.imul(E,pe)|0,r=r+Math.imul(v,fe)|0,i=i+Math.imul(v,me)|0,i=i+Math.imul(b,fe)|0,o=o+Math.imul(b,me)|0;var De=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(De>>>26)|0,De&=67108863,r=Math.imul(B,Q),i=Math.imul(B,X),i=i+Math.imul(j,Q)|0,o=Math.imul(j,X),r=r+Math.imul(P,J)|0,i=i+Math.imul(P,ee)|0,i=i+Math.imul(L,J)|0,o=o+Math.imul(L,ee)|0,r=r+Math.imul(I,ne)|0,i=i+Math.imul(I,re)|0,i=i+Math.imul(M,ne)|0,o=o+Math.imul(M,re)|0,r=r+Math.imul(k,oe)|0,i=i+Math.imul(k,ae)|0,i=i+Math.imul(O,oe)|0,o=o+Math.imul(O,ae)|0,r=r+Math.imul(T,le)|0,i=i+Math.imul(T,ue)|0,i=i+Math.imul(D,le)|0,o=o+Math.imul(D,ue)|0,r=r+Math.imul(S,de)|0,i=i+Math.imul(S,pe)|0,i=i+Math.imul(x,de)|0,o=o+Math.imul(x,pe)|0,r=r+Math.imul(_,fe)|0,i=i+Math.imul(_,me)|0,i=i+Math.imul(E,fe)|0,o=o+Math.imul(E,me)|0;var Ae=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Ae>>>26)|0,Ae&=67108863,r=Math.imul(B,J),i=Math.imul(B,ee),i=i+Math.imul(j,J)|0,o=Math.imul(j,ee),r=r+Math.imul(P,ne)|0,i=i+Math.imul(P,re)|0,i=i+Math.imul(L,ne)|0,o=o+Math.imul(L,re)|0,r=r+Math.imul(I,oe)|0,i=i+Math.imul(I,ae)|0,i=i+Math.imul(M,oe)|0,o=o+Math.imul(M,ae)|0,r=r+Math.imul(k,le)|0,i=i+Math.imul(k,ue)|0,i=i+Math.imul(O,le)|0,o=o+Math.imul(O,ue)|0,r=r+Math.imul(T,de)|0,i=i+Math.imul(T,pe)|0,i=i+Math.imul(D,de)|0,o=o+Math.imul(D,pe)|0,r=r+Math.imul(S,fe)|0,i=i+Math.imul(S,me)|0,i=i+Math.imul(x,fe)|0,o=o+Math.imul(x,me)|0;var ke=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(ke>>>26)|0,ke&=67108863,r=Math.imul(B,ne),i=Math.imul(B,re),i=i+Math.imul(j,ne)|0,o=Math.imul(j,re),r=r+Math.imul(P,oe)|0,i=i+Math.imul(P,ae)|0,i=i+Math.imul(L,oe)|0,o=o+Math.imul(L,ae)|0,r=r+Math.imul(I,le)|0,i=i+Math.imul(I,ue)|0,i=i+Math.imul(M,le)|0,o=o+Math.imul(M,ue)|0,r=r+Math.imul(k,de)|0,i=i+Math.imul(k,pe)|0,i=i+Math.imul(O,de)|0,o=o+Math.imul(O,pe)|0,r=r+Math.imul(T,fe)|0,i=i+Math.imul(T,me)|0,i=i+Math.imul(D,fe)|0,o=o+Math.imul(D,me)|0;var Oe=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Oe>>>26)|0,Oe&=67108863,r=Math.imul(B,oe),i=Math.imul(B,ae),i=i+Math.imul(j,oe)|0,o=Math.imul(j,ae),r=r+Math.imul(P,le)|0,i=i+Math.imul(P,ue)|0,i=i+Math.imul(L,le)|0,o=o+Math.imul(L,ue)|0,r=r+Math.imul(I,de)|0,i=i+Math.imul(I,pe)|0,i=i+Math.imul(M,de)|0,o=o+Math.imul(M,pe)|0,r=r+Math.imul(k,fe)|0,i=i+Math.imul(k,me)|0,i=i+Math.imul(O,fe)|0,o=o+Math.imul(O,me)|0;var Ne=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Ne>>>26)|0,Ne&=67108863,r=Math.imul(B,le),i=Math.imul(B,ue),i=i+Math.imul(j,le)|0,o=Math.imul(j,ue),r=r+Math.imul(P,de)|0,i=i+Math.imul(P,pe)|0,i=i+Math.imul(L,de)|0,o=o+Math.imul(L,pe)|0,r=r+Math.imul(I,fe)|0,i=i+Math.imul(I,me)|0,i=i+Math.imul(M,fe)|0,o=o+Math.imul(M,me)|0;var Ie=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Ie>>>26)|0,Ie&=67108863,r=Math.imul(B,de),i=Math.imul(B,pe),i=i+Math.imul(j,de)|0,o=Math.imul(j,pe),r=r+Math.imul(P,fe)|0,i=i+Math.imul(P,me)|0,i=i+Math.imul(L,fe)|0,o=o+Math.imul(L,me)|0;var Me=(u+r|0)+((8191&i)<<13)|0;u=(o+(i>>>13)|0)+(Me>>>26)|0,Me&=67108863,r=Math.imul(B,fe),i=Math.imul(B,me),i=i+Math.imul(j,fe)|0,o=Math.imul(j,me);var Re=(u+r|0)+((8191&i)<<13)|0;return u=(o+(i>>>13)|0)+(Re>>>26)|0,Re&=67108863,l[0]=ge,l[1]=ve,l[2]=be,l[3]=ye,l[4]=_e,l[5]=Ee,l[6]=we,l[7]=Se,l[8]=xe,l[9]=Ce,l[10]=Te,l[11]=De,l[12]=Ae,l[13]=ke,l[14]=Oe,l[15]=Ne,l[16]=Ie,l[17]=Me,l[18]=Re,0!==u&&(l[19]=u,n.length++),n};function g(e,t,n){n.negative=t.negative^e.negative,n.length=e.length+t.length;for(var r=0,i=0,o=0;o>>26)|0,i+=a>>>26,a&=67108863}n.words[o]=s,r=a,a=i}return 0!==r?n.words[o]=r:n.length--,n.strip()}function v(e,t,n){var r=new b;return r.mulp(e,t,n)}function b(e,t){this.x=e,this.y=t}Math.imul||(m=f),o.prototype.mulTo=function(e,t){var n,r=this.length+e.length;return n=10===this.length&&10===e.length?m(this,e,t):r<63?f(this,e,t):r<1024?g(this,e,t):v(this,e,t),n},b.prototype.makeRBT=function(e){for(var t=new Array(e),n=o.prototype._countBits(e)-1,r=0;r>=1;return r},b.prototype.permute=function(e,t,n,r,i,o){for(var a=0;a>>=1)i++;return 1<>>=13,n[2*a+1]=8191&o,o>>>=13;for(a=2*t;a>=26,t+=i/67108864|0,t+=o>>>26,this.words[n]=67108863&o}return 0!==t&&(this.words[n]=t,this.length++),this},o.prototype.muln=function(e){return this.clone().imuln(e)},o.prototype.sqr=function(){return this.mul(this)},o.prototype.isqr=function(){return this.imul(this.clone())},o.prototype.pow=function(e){var t=h(e);if(0===t.length)return new o(1);for(var n=this,r=0;r=0);var t,n=e%26,i=(e-n)/26,o=67108863>>>26-n<<26-n;if(0!==n){var a=0;for(t=0;t>>26-n}a&&(this.words[t]=a,this.length++)}if(0!==i){for(t=this.length-1;t>=0;t--)this.words[t+i]=this.words[t];for(t=0;t=0),i=t?(t-t%26)/26:0;var o=e%26,a=Math.min((e-o)/26,this.length),s=67108863^67108863>>>o<a)for(this.length-=a,u=0;u=0&&(0!==c||u>=i);u--){var d=0|this.words[u];this.words[u]=c<<26-o|d>>>o,c=d&s}return l&&0!==c&&(l.words[l.length++]=c),0===this.length&&(this.words[0]=0,this.length=1),this.strip()},o.prototype.ishrn=function(e,t,n){return r(0===this.negative),this.iushrn(e,t,n)},o.prototype.shln=function(e){return this.clone().ishln(e)},o.prototype.ushln=function(e){return this.clone().iushln(e)},o.prototype.shrn=function(e){return this.clone().ishrn(e)},o.prototype.ushrn=function(e){return this.clone().iushrn(e)},o.prototype.testn=function(e){r("number"===typeof e&&e>=0);var t=e%26,n=(e-t)/26,i=1<=0);var t=e%26,n=(e-t)/26;if(r(0===this.negative,"imaskn works only with positive numbers"),this.length<=n)return this;if(0!==t&&n++,this.length=Math.min(n,this.length),0!==t){var i=67108863^67108863>>>t<=67108864;t++)this.words[t]-=67108864,t===this.length-1?this.words[t+1]=1:this.words[t+1]++;return this.length=Math.max(this.length,t+1),this},o.prototype.isubn=function(e){if(r("number"===typeof e),r(e<67108864),e<0)return this.iaddn(-e);if(0!==this.negative)return this.negative=0,this.iaddn(e),this.negative=1,this;if(this.words[0]-=e,1===this.length&&this.words[0]<0)this.words[0]=-this.words[0],this.negative=1;else for(var t=0;t>26)-(l/67108864|0),this.words[i+n]=67108863&o}for(;i>26,this.words[i+n]=67108863&o;if(0===s)return this.strip();for(r(-1===s),s=0,i=0;i>26,this.words[i]=67108863&o;return this.negative=1,this.strip()},o.prototype._wordDiv=function(e,t){var n=this.length-e.length,r=this.clone(),i=e,a=0|i.words[i.length-1],s=this._countBits(a);n=26-s,0!==n&&(i=i.ushln(n),r.iushln(n),a=0|i.words[i.length-1]);var l,u=r.length-i.length;if("mod"!==t){l=new o(null),l.length=u+1,l.words=new Array(l.length);for(var c=0;c=0;p--){var h=67108864*(0|r.words[i.length+p])+(0|r.words[i.length+p-1]);h=Math.min(h/a|0,67108863),r._ishlnsubmul(i,h,p);while(0!==r.negative)h--,r.negative=0,r._ishlnsubmul(i,1,p),r.isZero()||(r.negative^=1);l&&(l.words[p]=h)}return l&&l.strip(),r.strip(),"div"!==t&&0!==n&&r.iushrn(n),{div:l||null,mod:r}},o.prototype.divmod=function(e,t,n){return r(!e.isZero()),this.isZero()?{div:new o(0),mod:new 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