From dde4adc4d827fbaa497cd6a8146a73bc4e0aff01 Mon Sep 17 00:00:00 2001 From: xiongraorao Date: Fri, 17 Aug 2018 14:17:10 +0800 Subject: [PATCH] =?UTF-8?q?=E6=95=B0=E6=8D=AE=E7=BB=93=E6=9E=84?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- interview/struct/datastruct.md | 159 ++++++++++++++++++++++++++++++++- interview/struct/img/0.JPG | Bin 0 -> 18634 bytes interview/struct/img/4.jpg | Bin 0 -> 27258 bytes interview/struct/img/5.JPG | Bin 0 -> 39595 bytes interview/struct/img/6.JPG | Bin 0 -> 42380 bytes 5 files changed, 158 insertions(+), 1 deletion(-) create mode 100644 interview/struct/img/0.JPG create mode 100644 interview/struct/img/4.jpg create mode 100644 interview/struct/img/5.JPG create mode 100644 interview/struct/img/6.JPG diff --git a/interview/struct/datastruct.md b/interview/struct/datastruct.md index 1d9a0a8e..c80b8c92 100644 --- a/interview/struct/datastruct.md +++ b/interview/struct/datastruct.md @@ -8,6 +8,12 @@ - [完全二叉树](#完全二叉树) - [二叉查找树](#二叉查找树) - [AVL树](#avl树) + - [RBT 红黑树](#rbt-红黑树) + - [B-树](#b-树) + - [B+树](#b树) + - [B*树](#b树) + - [B B- B+ B* 树总结](#b-b--b-b-树总结) + - [B+/B*Tree应用](#bbtree应用) - [参考链接](#参考链接) @@ -75,7 +81,158 @@ ## AVL树 -定义:自平衡二叉查找树 +定义:自平衡二叉查找树,平衡二叉树或为空树,或为如下性质的二叉排序树: + +(1)左右子树深度之差的绝对值不超过1; +(2)左右子树仍然为平衡二叉树. +平衡因子BF=左子树深度-右子树深度. + +平衡二叉树每个结点的平衡因子只能是1,0,-1。若其绝对值超过1,则该二叉排序树就是不平衡的。 + +## RBT 红黑树 + +AVL是严格平衡树,因此在增加或者删除节点的时候,根据不同情况,旋转的次数比红黑树要多; +红黑是**弱平衡**的,用非严格的平衡来换取增删节点时候旋转次数的降低; + +所以简单说,**搜索的次数远远大于插入和删除,那么选择AVL树,如果搜索,插入删除次数几乎差不多,应该选择RB树。** + +红黑树上每个结点内含五个域,color,key,left,right,p。如果相应的指针域没有,则设为NIL。 +一般的,红黑树,满足以下性质,即只有满足以下全部性质的树,我们才称之为红黑树: +1)每个结点要么是红的,要么是黑的。 +2)根结点是黑的。 +3)每个叶结点,即空结点(NIL)是黑的。 +4)如果一个结点是红的,那么它的俩个儿子都是黑的。 +5)对每个结点,从该结点到其子孙结点的所有路径上包含相同数目的黑结点。 +下图所示,即是一颗红黑树: + +![](http://hi.csdn.net/attachment/201012/29/8394323_1293613306CGzE.jpg) + +## B-树 + +定义:B树是一种平衡多路搜索树(并不是二叉的),满足如下性质: + +1.定义任意非叶子结点最多只有M个儿子;且M>2; + +2.根结点的儿子数为[2, M]; + +3.除根结点以外的非叶子结点的儿子数为[M/2, M]; + +4.每个结点存放至少M/2-1(取上整)和至多M-1个关键字;(至少2个关键字) + +5.非叶子结点的关键字个数=指向儿子的指针个数-1; + +6.非叶子结点的关键字:K[1], K[2], …, K[M-1];且K[i] < K[i+1]; + +7.非叶子结点的指针:P[1], P[2], …, P[M];其中P[1]指向关键字小于K[1]的 + +子树,P[M]指向关键字大于K[M-1]的子树,其它P[i]指向关键字属于(K[i-1], K[i])的子树; + +8.所有叶子结点位于同一层; + +如M=3: + +![](img/4.jpg) + +B-树的搜索,从根结点开始,对结点内的关键字(有序)序列进行二分查找,如果 + +命中则结束,否则进入查询关键字所属范围的儿子结点;重复,直到所对应的儿子指针为 + +空,或已经是叶子结点; + +B-树的特性: + +1.关键字集合分布在整颗树中; + +2.任何一个关键字出现且只出现在一个结点中; + +3.搜索有可能在非叶子结点结束; + +4.其搜索性能等价于在关键字全集内做一次二分查找; + +5.自动层次控制; + +由于限制了除根结点以外的非叶子结点,至少含有M/2个儿子,确保了结点的至少 + +利用率,其最底搜索性能为: + +![](img/0.JPG) + +其中,M为设定的非叶子结点最多子树个数,N为关键字总数; + +所以B-树的性能总是等价于二分查找(与M值无关),也就没有B树平衡的问题; + +由于M/2的限制,在插入结点时,如果结点已满,需要将结点分裂为两个各占 + +M/2的结点;删除结点时,需将两个不足M/2的兄弟结点合并; + +## B+树 + +定义:B+树是B-树的变体,也是一种多路搜索树: + +1.其定义基本与B-树同,除了: + +2.非叶子结点的子树指针与关键字个数相同; + +3.非叶子结点的子树指针P[i],指向关键字值属于[K[i], K[i+1])的子树 + +(B-树是开区间); + +5.为所有叶子结点增加一个链指针; + +6.所有关键字都在叶子结点出现; + +如:(M=3) + +![](img/5.JPG) + +B+的搜索与B-树也基本相同,区别是B+树只有达到叶子结点才命中(B-树可以在 +非叶子结点命中),其性能也等价于在关键字全集做一次二分查找; + +B+的特性: + +1.所有关键字都出现在叶子结点的链表中(稠密索引),且链表中的关键字恰好是有序的; + +2.不可能在非叶子结点命中; + +3.非叶子结点相当于是叶子结点的索引(稀疏索引),叶子结点相当于是存储(关键字)数据的数据层; + +4.更适合文件索引系统;比如对已经建立索引的数据库记录,查找10<=id<=20,那么只要通过根节点搜索到id=10的叶节点,之后只要根据叶节点的链表找到第一个大于20的就行了,比B-树在查找10到20内的每一个时每次都从根节点出发查找提高了不少效率。 + +## B*树 + +定义:B*树是B+树的变体,在B+树的非根和非叶子结点再增加指向兄弟的指针; + +![](img/6.JPG) + +B\*树定义了非叶子结点关键字个数至少为(2/3)\*M,即块的最低使用率为2/3(代替B+树的1/2); + +B+树的分裂:当一个结点满时,分配一个新的结点,并将原结点中1/2的数据复制到新结点,最后在父结点中增加新结点的指针;B+树的分裂只影响原结点和父结点,而不会影响兄弟结点,所以它不需要指向兄弟的指针; + +B*树的分裂:当一个结点满时,如果它的下一个兄弟结点未满,那么将一部分数据移到兄弟结点中,再在原结点插入关键字,最后修改父结点中兄弟结点的关键字(因为兄弟结点的关键字范围改变了);如果兄弟也满了,则在原结点与兄弟结点之间增加新结点,并各复制1/3的数据到新结点,最后在父结点增加新结点的指针; + +**所以,B*树分配新结点的概率比B+树要低,空间使用率更高;** + +## B B- B+ B* 树总结 + +B树:二叉树,每个结点只存储一个关键字,等于则命中,小于走左结点,大于走右结点; + +B-树:多路搜索树,每个结点存储M/2到M个关键字,非叶子结点存储指向关键字范围的子结点;所有关键字在整颗树中出现,且只出现一次,非叶子结点可以命中; + +B+树:在B-树基础上,为叶子结点增加链表指针,所有关键字都在叶子结点中出现,非叶子结点作为叶子结点的索引;B+树总是到叶子结点才命中; + +B*树:在B+树基础上,为非叶子结点也增加链表指针,将结点的最低利用率从1/2提高到2/3; + +## B+/B*Tree应用 + +数据库索引--索引文件和数据文件是分离的,索引文件仅保存数据记录的地址。 + + +数据库索引--表数据文件本身就是按B+Tree组织的一个索引结构,这棵树的叶节点data域保存了完整的数据记录。这个索引的key是数据表的主键。 + + +倒排索引--也可以由B树及其变种实现但不一定非要B树及其变种实现,如lucene没有使用B树结构,因此lucene可以用二分搜索算法快速定位关键词。实现时,lucene将下面三列分别作为词典文件(Term Dictionary)、频率文件(frequencies)、位置文件 (positions)保存。其中词典文件不仅保存有每个关键词,还保留了指向频率文件和位置文件的指针,通过指针可以找到该关键字的频率信息和位置信息。    + +参考链接:[B-树和B+树的应用:数据搜索和数据库索引](http://blog.csdn.net/hguisu/article/details/7786014) # 参考链接 diff --git a/interview/struct/img/0.JPG b/interview/struct/img/0.JPG new file mode 100644 index 0000000000000000000000000000000000000000..d2107961195f408affe083f1e38e35fd128e31dc GIT binary patch literal 18634 zcmb@tWmFu|wk=#Z!QCxLAh=8L;2}r^*93{X97`bXE;RkL^1UTe)c_tX5-IzaG7QAH6zLP7#e5g*`b10YiX+gJjC ziV8b`3H;mfv;@cjnCR#j=xCT27#PoH@~pBvAMOqv%B|q z{{VJ&esOtq4Zpel2QDN4<)5(r9oc`uMTo$KjEahait!IzNXQN8I z6Ns3R|0^cR>xA6u?q^H_@1dlhohPx#m<2ajVE=&jKal-D0~YxI3E95``=7WL0UQ)0 z#N?q60@A?s?GE*(Qi#ZEIaNlg&F8NtMb&j_yKEmTkLnWg*(fkFuB1Yn*5ZsAW;k6X 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