From 65521633543604685b4105f0c649150b17c090fa Mon Sep 17 00:00:00 2001 From: Shine wOng <1551885@tongji.edu.cn> Date: Sat, 15 Jun 2019 10:53:45 +0800 Subject: [PATCH] Create a conclusion on SplayTree , and add some images. --- thu_dsa/chp7/AVL.md | 1 - thu_dsa/chp8/B-tree.md | 22 +++ thu_dsa/chp8/SplayTree.md | 173 ++++++++++++++++-- thu_dsa/chp8/images/SplayTreeInsert.png | Bin 0 -> 49151 bytes thu_dsa/chp8/images/SplayTreeRemove.png | Bin 0 -> 58984 bytes thu_dsa/chp8/images/one-step-splay.png | Bin 0 -> 79901 bytes .../chp8/images/onestep_versus_twostep.png | Bin 0 -> 56678 bytes 7 files changed, 182 insertions(+), 14 deletions(-) create mode 100644 thu_dsa/chp8/B-tree.md create mode 100644 thu_dsa/chp8/images/SplayTreeInsert.png create mode 100644 thu_dsa/chp8/images/SplayTreeRemove.png create mode 100644 thu_dsa/chp8/images/one-step-splay.png create mode 100644 thu_dsa/chp8/images/onestep_versus_twostep.png diff --git a/thu_dsa/chp7/AVL.md b/thu_dsa/chp7/AVL.md index 492d305..0c305ad 100644 --- a/thu_dsa/chp7/AVL.md +++ b/thu_dsa/chp7/AVL.md @@ -1,7 +1,6 @@ Conclusion on AVL ================= - ## 平衡树 > 为什么需要平衡树?BST哪里不好吗? diff --git a/thu_dsa/chp8/B-tree.md b/thu_dsa/chp8/B-tree.md new file mode 100644 index 0000000..34b585e --- /dev/null +++ b/thu_dsa/chp8/B-tree.md @@ -0,0 +1,22 @@ +B树知识总结 +========= + +## B树的概念 + +> 为什么B树又宽又矮? + +> 为什么需要B树? + +B树和二叉搜索树完全等效,将多个结点合成一个超级结点的多路平衡搜索树而已。所以,为什么还需要B树 + +> B树的定义 + +m阶B树,即m路平衡搜索树。分支数上界为m,下界为[m/2]取上整 + +叶节点,外部结点的区别与联系。B树高度的定义 + +## B树的查找 + +## B树的插入 + +B树高度增加的唯一情况,与B树根节点可以只拥有两个分支,之间的关系 diff --git a/thu_dsa/chp8/SplayTree.md b/thu_dsa/chp8/SplayTree.md index b4ff6d8..389f30a 100644 --- a/thu_dsa/chp8/SplayTree.md +++ b/thu_dsa/chp8/SplayTree.md @@ -1,18 +1,165 @@ -伸展树知识总结 -============= +Conclusion on Splay Tree +======================== -> 相对于AVL树的比较。 -AVL树更像是循规蹈矩,如履薄冰。而伸展树则更加潇洒,不顾小节。 +## 伸展树基本概念 -> 局部性原理(locality) -类比于列表 +> 伸展树的基本思想。 -> 逐层伸展策略与最坏情况 +伸展树完全是基于<局部性原理>(locality)的。 -双层伸展策略 -+ 描述 -+ 效果 -+ 时间复杂度 +局部性原理是计算机科学中非常重要的原理,很多设计,比如说多级存储器,缓存,都是基于局部性原理。简单说来就是<刚访问过的数据,在一段时间内极有可能再次被访问>。因此,对于物理存储器而言,会将刚刚访问过的数据,转移存储到更高级的存储介质中,比如内存或者缓存,以便下次访问这些数据时可以高效地进行。而对于数据结构而言,则可以把刚访问过的数据移动到更容易被访问到的区域,比如列表可以把每次被访问的数据都交换到列表的头部,这就构成了自调整列表。 -> 伸展树的实现 -为了保证局部性,插入结点也需要插入到根节点。删除结点后,需要将被删除结点附近的结点移到根节点。这两个操作都可以通过伸展树的查找快速实现。 +伸展树也是同样的思想,即对于一棵二叉搜索树而言,将刚刚被访问的数据移动到树根结点,从而使得后续的访问可以高效地找到该数据。 + +> 伸展树和AVL树的比较。 + +伸展树也是一棵平衡二叉搜索树,但是可以看到,伸展树和AVL树有很大的差异。 + +前面我们说,对于任意一种平衡二叉搜索树,都是关注两个方面的问题,即平衡条件,以及失衡后的调整算法。因此,AVL树引入了一个平衡因子来作为它的平衡条件,并且失衡调整算法也都是围绕这个平衡因子而展开。但是,从前面伸展树的思想可以看出,伸展树没有所谓的平衡条件,因此它不需要维护任何额外的信息,包括树的高度。它只是在每次对某结点的访问后,将该结点移动到树根,并且同时对树的结构进行调整,就这样实现了它的平衡。 + +因此相较而言,AVL树更像是循规蹈矩,如履薄冰。而伸展树则更加潇洒,不顾小节。 + +## 伸展策略 + +那么实现伸展树就比较简单了,只需要将被访问结点移动到根结点就可以了,所以应该怎么移动呢? + +> 逐层伸展策略。 + +一个很简单的思想是找到被访问的结点后,不断地将该结点与它的父结点进行单旋转,直到该结点被旋转到根结点。 + +这种策略固然是可行的,但是在最坏情况下,它拥有比较差的时间性能,如下图所示: + +![one-step splay](images/one-step-splay.png) + +对于这种一开始就呈现单链结构的搜索树,设它具有n个结点,如果每次都访问它深度最深的那个结点,那么每次被访问结点的深度为$n-1, n-2, ...., 3, 2, 1, 0$,所以一个周期这样的访问累计需要$\Omega(n^2)$的时间,分摊时间复杂度为$\Omega(n)$。这个结果已经与列表这种线性结构相当了,所以很难可以让人满意。 + +> 双层伸展策略。 + +双层伸展策略只是逐层伸展策略的一个推广,它们之间的差别其实非常小,但是对性能却有一个质的提升。据邓公所说,双层伸展策略其实就是逐层伸展这条龙上点出的<睛>,而在这之前,逐层伸展这样一个整体的龙的结构是已经具备了的。 + +双层伸展策略,顾名思义,就是每次伸展时都以两个单位向上层追溯,而不是逐层伸展的一个单位。所以,这就需要反复考察祖孙三代$g, p ,v$,通过两次旋转将$v$上升两层,使之成为新的树根。关于这个前面我们已经进行过讨论,根据祖孙三代的拓扑结构可以分为`zigzig`, `zigzag`, `zagzig`, `zagzag`四种情况。 + ++ `zigzag`和`zagzig`。优先讨论着两种情况是因为,在这两种拓扑结构下,以$v$为新的根结点的树是没有异构情况的,无非是$p, g$分别是$v$的左右孩子结点。所以伸展树的`zigzag`与前面AVL的`zigzag`没有任何区别,可以直接调用前面的`connect34`来进行重构。 ++ `zigzig`和`zagzag`。这两种情况是有重构的,以`zigzig`为例,调整后以$v$为根结点的树有`zagzag`和`zagzig`两种构型,如果将前面的两次单旋转看做这里的一次旋转的话,逐层伸展策略对应的`zagzig`构型,而我们这里的双层伸展策略将要采用的`zagzag`构型。一会儿我们可以看到这点细微的差距给整棵伸展树带来的不同。 + +下图展示了单层伸展和双层伸展策略的不同。 + +![onestep_versus_twostep](images/onestep_versus_twostep.png) + +可以看到,双层伸展策略具有某种<折叠>的效果。即一旦访问最坏的结点,双层伸展就会通过调整将整棵树的高度降低到大约原来的一般,从而使得最坏情况不至于持续发生。 + +这里,也可以看出两种策略本质上的区别。在最坏情况下,单层伸展策略每次伸展只能将树高降低1,而双层伸展策略每次伸展可以将树高降低到原来的一般。所以,这也是单层伸展的分摊时间复杂度为$O(n)$而双层伸展的分摊时间复杂度为$O(logn)$的原因,详细的证明可以查看邓公习题集。 + +最后,由于双层伸展是以两个单位向上层追溯,要是当前结点只有父亲而没有祖父时(即父结点是根结点),可以相应的将当前结点与其父结点做一次单旋转,从而可以将当前结点移动到树根。 + +## 伸展树的实现 + +前面提到,伸展树完全是基于局部性原理构造的一种数据结构,因此,其算法的具体实现中也要处处体现局部性原理,才能使伸展树发挥最大的作用。 + +伸展树的基本操作同样只有搜索,插入,删除三种。对于这三种基本操作,都要应用局部性原理。 + +> 搜索操作。 + +首先调用BST常规的搜索操作。每次需要将被查找的结点移动到树根,或者没有找到目标关键码,则将失败处的父结点移动到树根,这是因为失败处的父结点是和目标关键码差距最小的结点,根据局部性原理,该结点在后续的访问中也极有可能出现。无论如何,都返回树根结点。 + +搜索操作的代码如下: + +```cpp +template +BinNodePosi(T) SplayTree::search(T const &key){ + BinNodePosi(T) x = searchIn(__root, key, __hot = nullptr); + if(x) splay(x); + else if(__hot) splay(__hot); + return __root; +} +``` + +> 插入操作。 + +一种思路是首先调用BST的常规插入算法,然后为了保证局部性,将被插入结点移动到根结点。这个方法的确是简明可行的。 + +另一种思路是,直接调用上面伸展树的搜索算法,这样一次搜索必然会失败,但是此时`__hot`会被移动到根结点。我们可以直接比较待插入关键码与`__hot`关键码的大小,从而直接将被插入结点插入到根结点。整个过程如下图所示: + +![SplayTreeInsert](images/SplayTreeInsert.png) + +具体的插入算法如下: + +```cpp +template +bool SplayTree::insert(T const &key){ + BinNodePosi(T) x = search(key); + if(x && x->data == key) return false; + + //else + __root = new BinNode(key); + ++__size; + if (!x) return true; + + if(key < x->data){ + __root->leftChild = x->leftChild; + __root->rightChild = x; + x->leftChild = nullptr; + x->parent = __root; + }else{ + __root->leftChild = x; + __root->rightChild = x->rightChild; + x->rightChild = nullptr; + x->parent = __root; + } + updateHeight(x); + updateHeight(__root); + return true; +} +``` + +> 删除操作。 + +删除操作会比较复杂。一种简明的方法也是首先调用BST的删除算法,然后将`__hot`伸展到根结点。这个方法同样是可行的。 + +另一种思路是,首先调用伸展树的搜索算法,这样被删除结点就已经被移动到了根结点,因此可以直接将根结点删除。接下来的问题是,应该由哪个结点来作为新的根结点。 + +在根结点没有左子树或者没有右子树时,可以直接用那棵唯一的子树来替代根结点,从而完成了根结点的删除。但在根结点同时具有左右子树时,根据伸展树的局部性原理,仍可选取被删除结点的直接后继来作为新的根结点,这样数据的局部性仍然可以得到应用。 + +为了将根结点的直接后继提升为新的根结点,可以在$T_R$中再次查找根结点的关键码,尽管这次查找必然失败,却可以将$T_R$中的最小结点提升为根节点。并且由于它是最小结点,根据BST的局部有序性,新的根节点必然没有左子树,从而可以将$T_L$作为左子树与新的根结点进行连接。这样,就得到了一棵完整的二叉搜索树。整个过程如下图所示: + +![SplayTreeRemove](images/SplayTreeRemove.png) + +具体代码如下: + +```cpp +template +bool SplayTree::remove(T const &key){ + BinNodePosi(T) x = search(key); + if (!x || x->data.key != key) return false; + //else + if(!__root->leftChild){ + __root = __root->rightChild; + if(__root) __root->parent = nullptr; + search(key); + } + else if(!__root->rightChild){ + __root = __root->leftChild; + __root->parent = nullptr; + search(key); + } + else { + __root = x->rightChild; + __root->parent = nullptr; + search(key); //move x's succ to root, and __root has no left child(for succ has the smallest key + __root->leftChild = x->leftChild; + x->leftChild->parent = __root; + updateHeight(__root); + } + --__size; + delete x; + return true; +} +``` + +## 伸展树的评价 + +可以证明,伸展树单次操作的分摊时间复杂度为$O(logn)$,与AVL树相当,并且还不需要维护额外的平衡因子或者高度等信息,编程也比较简单(也不是很简单...)。关于证明看邓公的习题集。 + +并且,在一些场合下,伸展树还可以达到更高的性能。例如在局部性强的场合,缓存命中率将极高。假设树中一共有$n$个结点,其中$k$个结点是经常访问的结点,并且有$k << n$。这样,在经过多次访问后,这k个结点将被提升到树的根部,此时访问的时间复杂度甚至可以达到$O(logk)$。 + +但是伸展树并不能保证单次最坏情况的发生,所以不能适用于对效率非常敏感的场合,例如导弹卫星发射这种。 diff 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