左偏树/左偏堆实现原理和代码实现指南 _二叉树

左偏树或左偏堆是使用二叉堆的变体实现的优先队列。每个节点都有一个s值(或等级或距离)到最近的叶子的距离。与二叉堆相反(始终是完整的二叉树), 左偏树可能非常不平衡。
以下是时间复杂度of左偏树/堆.

FunctionComplexityComparison 1) Get Min:O(1)[same as both Binary and Binomial] 2) Delete Min:O(Log n)[same as both Binary and Binomial] 3) Insert:O(Log n)[O(Log n) in Binary and O(1) in Binomial and O(Log n) for worst case] 4) Merge:O(Log n)[O(Log n) in Binomial]

左偏树是具有属性的二叉树：

普通最小堆性质：键(i)> =键(父(i))
左侧较重：dist(right(i))< = dist(left(i))。在这里, dist(i)是扩展的二叉树表示中从节点i到叶节点的最短路径上的边数(在此表示中, 空子级被视为外部或叶节点)。到达后代外部节点的最短路径是通过正确的子节点。每个子树也是左偏树, dist(i)= 1 + dist(right(i))。

例子：下面的左手树显示了通过上述过程为每个节点计算的距离。最右边的节点的等级为0, 因为此节点的右边子树为null, 并且其父节点的距离为1 x dist(i)= 1 + dist(right(i))。每个节点都遵循相同的规则, 并计算其s值(或等级)。

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从以上第二个属性, 我们可以得出两个结论：

从根到最右叶的路径是从根到叶的最短路径。
如果到最右边的叶子的路径有x个节点, 则左边的堆至少有2个X– 1个节点。这意味着对于具有n个节点的左偏堆, 最右叶的路径长度为O(log n)。

操作方式：

主要操作是merge()。
deleteMin()(或extractMin())可以通过删除根并为左右子树调用merge()来完成。
可以通过使用单个键(要插入的键)创建左偏树并为给定树和具有单个节点的树调用merge()来完成insert()。

合并背后的想法：
由于右子树较小, 因此其思想是将一棵树的右子树与其他树合并。以下是抽象步骤。

将值较小的根作为新根。
将其左子树挂在左侧。
递归合并其右子树和另一棵树。
从递归返回之前：
–更新合并根目录的dist()。
–如果需要, 在根目录下交换左右子树, 以保持合并的leftist属性
结果

资源：http://courses.cs.washington.edu/courses/cse326/08sp/lectures/05-leftist-heaps.pdf
合并的详细步骤：

比较两个堆的根。
将较小的键推入一个空堆栈, 然后移至较小键的右子项。
递归比较两个键, 然后继续将较小的键推入堆栈, 然后移至其正确的子级。
重复直到达到空节点。
以最后处理的节点为准, 使其成为堆栈顶部节点的右子节点, 如果违反了leftist堆的属性, 则将其转换为leftist堆。
递归地继续从堆栈中弹出元素, 并使它们成为新堆栈顶部的正确子元素。

例子：
考虑下面给出的两个左偏堆：

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将它们合并为一个左偏堆

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节点7的子树侵犯了左偏堆的属性, 因此我们将其与左子节点交换并保留了左偏堆的属性。

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转换为左偏堆。重复这个过程

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该算法在最坏情况下的时间复杂度在最坏情况下为O(log n), 其中n是左偏堆中的节点数。
合并两个左偏堆的另一个示例：

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【左偏树/左偏堆实现原理和代码实现指南】左偏树/左偏堆的实现：

//C++ program for leftist heap / leftist tree #include < bits/stdc++.h> using namespace std; // Node Class Declaration class LeftistNode { public : int element; LeftistNode *left; LeftistNode *right; int dist; LeftistNode( int & element, LeftistNode *lt = NULL, LeftistNode *rt = NULL, int np = 0) { this -> element = element; right = rt; left = lt, dist = np; } }; //Class Declaration class LeftistHeap { public : LeftistHeap(); LeftistHeap(LeftistHeap & rhs); ~LeftistHeap(); bool isEmpty(); bool isFull(); int & findMin(); void Insert( int & x); void deleteMin(); void deleteMin( int & minItem); void makeEmpty(); void Merge(LeftistHeap & rhs); LeftistHeap & operator =(LeftistHeap & rhs); private : LeftistNode *root; LeftistNode *Merge(LeftistNode *h1, LeftistNode *h2); LeftistNode *Merge1(LeftistNode *h1, LeftistNode *h2); void swapChildren(LeftistNode * t); void reclaimMemory(LeftistNode * t); LeftistNode *clone(LeftistNode *t); }; // Construct the leftist heap LeftistHeap::LeftistHeap() { root = NULL; }// Copy constructor. LeftistHeap::LeftistHeap(LeftistHeap & rhs) { root = NULL; * this = rhs; }// Destruct the leftist heap LeftistHeap::~LeftistHeap() { makeEmpty( ); }/* Merge rhs into the priority queue. rhs becomes empty. rhs must be different from this.*/ void LeftistHeap::Merge(LeftistHeap & rhs) { if ( this == & rhs) return ; root = Merge(root, rhs.root); rhs.root = NULL; }/* Internal method to merge two roots. Deals with deviant cases and calls recursive Merge1.*/ LeftistNode *LeftistHeap::Merge(LeftistNode * h1, LeftistNode * h2) { if (h1 == NULL) return h2; if (h2 == NULL) return h1; if (h1-> element < h2-> element) return Merge1(h1, h2); else return Merge1(h2, h1); }/* Internal method to merge two roots. Assumes trees are not empty, and h1's root contains smallest item.*/ LeftistNode *LeftistHeap::Merge1(LeftistNode * h1, LeftistNode * h2) { if (h1-> left == NULL) h1-> left = h2; else { h1-> right = Merge(h1-> right, h2); if (h1-> left-> dist < h1-> right-> dist) swapChildren(h1); h1-> dist = h1-> right-> dist + 1; } return h1; }// Swaps t's two children. void LeftistHeap::swapChildren(LeftistNode * t) { LeftistNode *tmp = t-> left; t-> left = t-> right; t-> right = tmp; }/* Insert item x into the priority queue, maintaining heap order.*/ void LeftistHeap::Insert( int & x) { root = Merge( new LeftistNode(x), root); }/* Find the smallest item in the priority queue. Return the smallest item, or throw Underflow if empty.*/ int & LeftistHeap::findMin() { return root-> element; }/* Remove the smallest item from the priority queue. Throws Underflow if empty.*/ void LeftistHeap::deleteMin() { LeftistNode *oldRoot = root; root = Merge(root-> left, root-> right); delete oldRoot; }/* Remove the smallest item from the priority queue. Pass back the smallest item, or throw Underflow if empty.*/ void LeftistHeap::deleteMin( int & minItem) { if (isEmpty()) { cout< < "Heap is Empty" < < endl; return ; } minItem = findMin(); deleteMin(); }/* Test if the priority queue is logically empty. Returns true if empty, false otherwise*/ bool LeftistHeap::isEmpty() { return root == NULL; }/* Test if the priority queue is logically full. Returns false in this implementation.*/ bool LeftistHeap::isFull() { return false ; }// Make the priority queue logically empty void LeftistHeap::makeEmpty() { reclaimMemory(root); root = NULL; }// Deep copy LeftistHeap & LeftistHeap::operator =(LeftistHeap & rhs) { if ( this != & rhs) { makeEmpty(); root = clone(rhs.root); } return * this ; }// Internal method to make the tree empty. void LeftistHeap::reclaimMemory(LeftistNode * t) { if (t != NULL) { reclaimMemory(t-> left); reclaimMemory(t-> right); delete t; } }// Internal method to clone subtree. LeftistNode *LeftistHeap::clone(LeftistNode * t) { if (t == NULL) return NULL; else return new LeftistNode(t-> element, clone(t-> left), clone(t-> right), t-> dist); }//Driver program int main() { LeftistHeap h; LeftistHeap h1; LeftistHeap h2; int x; int arr[]= {1, 5, 7, 10, 15}; int arr1[]= {22, 75}; h.Insert(arr[0]); h.Insert(arr[1]); h.Insert(arr[2]); h.Insert(arr[3]); h.Insert(arr[4]); h1.Insert(arr1[0]); h1.Insert(arr1[1]); h.deleteMin(x); cout< < x < < endl; h1.deleteMin(x); cout< < x < < endl; h.Merge(h1); h2 = h; h2.deleteMin(x); cout< < x < < endl; return 0; }

输出如下：