Heap [堆]
# 介绍
堆(英语:Heap)是计算机科学中的一种特别的完全二叉树。若是满足以下特性,即可称为堆:“给定堆中任意节点 P 和 C,若 P 是 C 的母节点,那么 P 的值会小于等于(或大于等于)C 的值”。若母节点的值恒小于等于子节点的值,此堆称为最小堆(min heap);反之,若母节点的值恒大于等于子节点的值,此堆称为最大堆(max heap)。在堆中最顶端的那一个节点,称作根节点(root node),根节点本身没有母节点(parent node)。
# 堆树
堆树的定义如下:
- 堆树是一颗完全二叉树;
- 堆树中某个节点的值总是不大于或不小于其孩子节点的值;
- 堆树中每个节点的子树都是堆树。
当父节点的键值总是大于或等于任何一个子节点的键值时为最大堆。 当父节点的键值总是小于或等于任何一个子节点的键值时为最小堆。如下图所示,左边为最大堆,右边为最小堆。
# 性质
堆的实现通过构造二叉堆(binary heap),实为二叉树的一种;由于其应用的普遍性,当不加限定时,均指该数据结构的这种实现。这种数据结构具有以下性质。
- 任意节点小于(或大于)它的所有后裔,最小元(或最大元)在堆的根上(堆序性)。
- 堆总是一棵完全树。即除了最底层,其他层的节点都被元素填满,且最底层尽可能地从左到右填入。
- 将根节点最大的堆叫做最大堆或大根堆(大顶堆),根节点最小的堆叫做最小堆或小根堆(小顶堆)。
堆有许多种高级类型包含了适合制作双端队列的最大 — 最小堆及制作优先权队列的斐波那契堆等。
# 基本操作
某些堆实现还支持其他的一些操作,如斐波那契堆支持检查一个堆中是否存在某个元素。
# 实现
# JavaScript
class Heap {
/**
* @param {function} compare
* @param {array} [_values]
* @param {number|string|object} [_leaf]
*/
constructor(compare, _values, _leaf) {
if (typeof compare !== 'function') {
throw new Error('Heap constructor expects a compare function');
}
this._compare = compare;
this._nodes = Array.isArray(_values) ? _values : [];
this._leaf = _leaf || null;
}
/**
* Checks if a parent has a left child
* @private
*/
_hasLeftChild(parentIndex) {
const leftChildIndex = (parentIndex * 2) + 1;
return leftChildIndex < this.size();
}
/**
* Checks if a parent has a right child
* @private
*/
_hasRightChild(parentIndex) {
const rightChildIndex = (parentIndex * 2) + 2;
return rightChildIndex < this.size();
}
/**
* Compares two nodes
* @private
*/
_compareAt(i, j) {
return this._compare(this._nodes[i], this._nodes[j]);
}
/**
* Swaps two nodes in the heap
* @private
*/
_swap(i, j) {
const temp = this._nodes[i];
this._nodes[i] = this._nodes[j];
this._nodes[j] = temp;
}
/**
* Checks if parent and child should be swapped
* @private
*/
_shouldSwap(parentIndex, childIndex) {
if (parentIndex < 0 || parentIndex >= this.size()) {
return false;
}
if (childIndex < 0 || childIndex >= this.size()) {
return false;
}
return this._compareAt(parentIndex, childIndex) > 0;
}
/**
* Compares children of a parent
* @private
*/
_compareChildrenOf(parentIndex) {
if (!this._hasLeftChild(parentIndex) && !this._hasRightChild(parentIndex)) {
return -1;
}
const leftChildIndex = (parentIndex * 2) + 1;
const rightChildIndex = (parentIndex * 2) + 2;
if (!this._hasLeftChild(parentIndex)) {
return rightChildIndex;
}
if (!this._hasRightChild(parentIndex)) {
return leftChildIndex;
}
const compare = this._compareAt(leftChildIndex, rightChildIndex);
return compare > 0 ? rightChildIndex : leftChildIndex;
}
/**
* Compares two children before a position
* @private
*/
_compareChildrenBefore(index, leftChildIndex, rightChildIndex) {
const compare = this._compareAt(rightChildIndex, leftChildIndex);
if (compare <= 0 && rightChildIndex < index) {
return rightChildIndex;
}
return leftChildIndex;
}
/**
* Recursively bubbles up a node if it's in a wrong position
* @private
*/
_heapifyUp(startIndex) {
let childIndex = startIndex;
let parentIndex = Math.floor((childIndex - 1) / 2);
while (this._shouldSwap(parentIndex, childIndex)) {
this._swap(parentIndex, childIndex);
childIndex = parentIndex;
parentIndex = Math.floor((childIndex - 1) / 2);
}
}
/**
* Recursively bubbles down a node if it's in a wrong position
* @private
*/
_heapifyDown(startIndex) {
let parentIndex = startIndex;
let childIndex = this._compareChildrenOf(parentIndex);
while (this._shouldSwap(parentIndex, childIndex)) {
this._swap(parentIndex, childIndex);
parentIndex = childIndex;
childIndex = this._compareChildrenOf(parentIndex);
}
}
/**
* Recursively bubbles down a node before a given index
* @private
*/
_heapifyDownUntil(index) {
let parentIndex = 0;
let leftChildIndex = 1;
let rightChildIndex = 2;
let childIndex;
while (leftChildIndex < index) {
childIndex = this._compareChildrenBefore(
index,
leftChildIndex,
rightChildIndex
);
if (this._shouldSwap(parentIndex, childIndex)) {
this._swap(parentIndex, childIndex);
}
parentIndex = childIndex;
leftChildIndex = (parentIndex * 2) + 1;
rightChildIndex = (parentIndex * 2) + 2;
}
}
/**
* Inserts a new value into the heap
* @public
* @param {number|string|object} value
* @returns {Heap}
*/
insert(value) {
this._nodes.push(value);
this._heapifyUp(this.size() - 1);
if (this._leaf === null || this._compare(value, this._leaf) > 0) {
this._leaf = value;
}
return this;
}
/**
* Removes and returns the root node in the heap
* @public
* @returns {number|string|object}
*/
extractRoot() {
if (this.isEmpty()) {
return null;
}
const root = this.root();
this._nodes[0] = this._nodes[this.size() - 1];
this._nodes.pop();
this._heapifyDown(0);
if (root === this._leaf) {
this._leaf = this.root();
}
return root;
}
/**
* Applies heap sort and return the values sorted by priority
* @public
* @returns {array}
*/
sort() {
for (let i = this.size() - 1; i > 0; i -= 1) {
this._swap(0, i);
this._heapifyDownUntil(i);
}
return this._nodes;
}
/**
* Fixes node positions in the heap
* @public
* @returns {Heap}
*/
fix() {
for (let i = 0; i < this.size(); i += 1) {
this._heapifyUp(i);
}
return this;
}
/**
* Verifies that all heap nodes are in the right position
* @public
* @returns {boolean}
*/
isValid() {
const isValidRecursive = (parentIndex) => {
let isValidLeft = true;
let isValidRight = true;
if (this._hasLeftChild(parentIndex)) {
const leftChildIndex = (parentIndex * 2) + 1;
if (this._compareAt(parentIndex, leftChildIndex) > 0) {
return false;
}
isValidLeft = isValidRecursive(leftChildIndex);
}
if (this._hasRightChild(parentIndex)) {
const rightChildIndex = (parentIndex * 2) + 2;
if (this._compareAt(parentIndex, rightChildIndex) > 0) {
return false;
}
isValidRight = isValidRecursive(rightChildIndex);
}
return isValidLeft && isValidRight;
};
return isValidRecursive(0);
}
/**
* Returns a shallow copy of the heap
* @public
* @returns {Heap}
*/
clone() {
return new Heap(this._compare, this._nodes.slice(), this._leaf);
}
/**
* Returns the root node in the heap
* @public
* @returns {number|string|object}
*/
root() {
if (this.isEmpty()) {
return null;
}
return this._nodes[0];
}
/**
* Returns a leaf node in the heap
* @public
* @returns {number|string|object}
*/
leaf() {
return this._leaf;
}
/**
* Returns the number of nodes in the heap
* @public
* @returns {number}
*/
size() {
return this._nodes.length;
}
/**
* Checks if the heap is empty
* @public
* @returns {boolean}
*/
isEmpty() {
return this.size() === 0;
}
/**
* Clears the heap
* @public
*/
clear() {
this._nodes = [];
this._leaf = null;
}
/**
* Builds a heap from a array of values
* @public
* @static
* @param {array} values
* @param {function} compare
* @returns {Heap}
*/
static heapify(values, compare) {
if (!Array.isArray(values)) {
throw new Error('Heap.heapify expects an array of values');
}
if (typeof compare !== 'function') {
throw new Error('Heap.heapify expects a compare function');
}
return new Heap(compare, values).fix();
}
/**
* Checks if a list of values is a valid heap
* @public
* @static
* @param {array} values
* @param {function} compare
* @returns {boolean}
*/
static isHeapified(values, compare) {
return new Heap(compare, values).isValid();
}
}
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# 参考
编辑 (opens new window)
上次更新: 2022/10/11, 17:42:20