【0】README
1) 本文部分内容转自 数据结构与算法分析,旨在理解 高级数据结构实现——自顶向下伸展树 的基础知识;
2) 源代码部分思想借鉴了数据结构与算法分析,有一点干货原创代码,for original source code, please visithttps://github.com/pacosonTang/dataStructure-algorithmAnalysis/tree/master/chapter12/p345_topdown_splay_tree
3) you can also refer to the link http://www.cnblogs.com/huangxincheng/archive/2012/08/04/2623455.html
4) for basic splay tree , please visit http://blog.csdn.net/PacosonSWJTU/article/details/50525435
【1】自顶向下伸展树相关
1)problem+solution
- 1.1)problem: 普通伸展树的展开操作需要从根沿树往下的一次遍历, 以及而后的从底向上的一次遍历。(详情,参见:http://blog.csdn.net/pacosonswjtu/article/details/50525435) 这可以通过保存一些父指针来完成, 或者通过将访问路径存储到一个栈中来完成。 但遗憾的 是, 这两种方法均需要大量的开销 ;
- 1.2)solution: 本节中, 我们指出如何在初始访问路径上施行一些旋转。结果得到在时间中更快的过程,只用到 O(1)的额外空间, 但却保持了 O(logN) 的摊还时间界;(干货——伸展树是基于AVL树的, 在AVL的基础上引入伸展树的目的是保持他的摊还时间界为 O(logN))
2)对伸展树的自顶向下的旋转操作:(单旋转+一字型旋转+之字型旋转)
- 2.1)这种伸展方式会把树切成三份,L树,M树,R树,考虑的情况有:单旋转,“一字型”旋转,“之字形”旋转。起初左树(L) 和 右树(R)均为空(NULL);
3)看个荔枝:
- 3.1)splay + deleting opeartions:
- 3.2)inserting opeartions:
4)source code at a glance
#include "topdown_splay_tree.h" // allocate memory for new node. Node makeNode(int value) { Node node; node = (Node)malloc(sizeof(struct Node)); if(!node) { Error("failed makeNode, for out of space !"); return NULL; } node->left = NULL; node->right = NULL; node->value = value; return node; } // left left single rotate TopDownSplayTree left_left_single_rotate(TopDownSplayTree root) { TopDownSplayTree temp; temp = root; // 1st step root = root->left; // 2nd step temp->left = root->right; // 3rd step root->right = temp; // 4th step return root; } // right_right_single_rotate TopDownSplayTree right_right_single_rotate(TopDownSplayTree root) { TopDownSplayTree temp; temp = root; // 1st step root = root->right; // 2nd step temp->right = root->left; // 3rd step root->left = temp; // 4th step return root; } // performing splay operations TopDownSplayTree topdown_splay(int value, TopDownSplayTree middle) { struct Node plusTree; Node leftTreeMax; Node rightTreeMin; leftTreeMax = &plusTree; rightTreeMin = &plusTree; while(value != middle->value) { if(middle->value < value) // the new node is greater. { if(middle->right == NULL) { break; } else if(middle->right->value < value && middle->right->right) { middle = right_right_single_rotate(middle); } leftTreeMax->right = middle; leftTreeMax = middle; middle = middle->right; leftTreeMax->right = NULL; } if(middle->value > value) // the new node is less. { if(middle->left == NULL) { break; } else if(middle->left->value > value && middle->left->left) { middle = left_left_single_rotate(middle); } rightTreeMin->left = middle; rightTreeMin = middle; middle = middle->left; rightTreeMin->left = NULL; } } leftTreeMax->right = middle->left; rightTreeMin->left = middle->right; middle->left = plusTree.right; middle->right = plusTree.left; return middle; } // delete the root of the TopDownSplayTree TopDownSplayTree deleteNode(int value, TopDownSplayTree root) { TopDownSplayTree newroot; if(root == NULL) { return root; } else // the splay tree is not null { root = topdown_splay(value, root); if(root->value == value) // find the node with given value. { if(root->left == NULL) { newroot = root->right; } else { newroot = root->left; // perform splay again with value towards the left subtree which is not null. newroot = topdown_splay(value, newroot); newroot->right = root->right; } free(root); root = newroot; } } return root; } // insert the node with value into the TopDownSplayTree TopDownSplayTree insert(int value, TopDownSplayTree root) { TopDownSplayTree node; node = makeNode(value); if(root == NULL) // the splay tree is null { return node; } else // the splay tree is not null { root = topdown_splay(value, root); if(root->value > value) { node->left = root->left; node->right = root; root->left = NULL; root = node; } else if(root->value < value) { node->right = root->right; root->right = NULL; node->left = root; root = node; } else { return root; } } return root; } // test for insert operation. int main1() { TopDownSplayTree root; int data[] = {5, 11, 23, 10, 17}; int size = 5; int i; printf(" === executing insert with {5, 11, 23, 10, 17} in turn.=== "); root = NULL; for(i=0; i<size; i++) { root = insert(data[i], root); printPreorder(1, root); } printf(" === executing insert with 8 in turn.=== "); root = insert(8, root); printPreorder(1, root); printf(" === executing insert with 18 in turn.=== "); root = insert(18, root); printPreorder(1, root); return 0; } // test for splay operation and deleting operation. int main() { TopDownSplayTree root; TopDownSplayTree temp; printf(" ====== test for splaying operation====== "); printf(" === original tree is as follows.=== "); root = makeNode(12); // root = 12 temp = root; temp->left = makeNode(5); temp->right = makeNode(25); temp = temp->right; // root = 25 temp->left = makeNode(20); temp->right = makeNode(30); temp = temp->left; // root = 20 temp->left = makeNode(15); temp->right = makeNode(24); temp = temp->left; // root = 15 temp->left = makeNode(13); temp->right = makeNode(18); temp = temp->right; // root = 18 temp->left = makeNode(16); printPreorder(1, root); printf(" === executing splay operation with finding value=19.=== "); root = topdown_splay(19, root); printPreorder(1, root); printf(" === executing deleting operation with value=15.=== "); root = deleteNode(15, root); printPreorder(1, root); return 0; } // analog print node values in the binominal tree, which involves preorder traversal. void printPreorder(int depth, TopDownSplayTree root) { int i; if(root) { for(i = 0; i < depth; i++) printf(" "); printf("%d ", root->value); printPreorder(depth + 1, root->left); printPreorder(depth + 1, root->right); } else { for(i = 0; i < depth; i++) printf(" "); printf("NULL "); } }