Algorithms Fourth Edition
Written By Robert Sedgewick & Kevin Wayne
Translated By 谢路云
Chapter 2 Section 4 优先队列
优先队列
优先队列API
N个数找到最大M个元素的时间成本
不同数据结构下的时间成本
堆
堆的定义
定义：当一棵二叉树的每个结点都大于等于它的两个子节点时，它称为堆有序
相应地，在堆有序的二叉树中，每个结点都小于等于它的父节点。从任意结点向上，我们都能得到一列非递减的元素；从任意结点向下，我们都能得到一列非递增的元素。特别的： 根结点是堆有序的二叉树中的最大结点。
二叉堆表示法
二叉堆：就是堆有序的完全二叉树，元素在数组中按照层级存储（一层一层的放入数组中，不用数组的第一个元素，因为0*2=0，递推关系不合适）。下面简称堆。
堆中：位置K的结点的父节点的位置为 ⌊k/2⌋ ，子节点的位置分别是 2k 和 2k+1
一个结论：一棵大小为N的完全二叉树的高度为 ⌊lgN⌋
用数组（堆）实现的完全二叉树是很严格的，但它的灵活性足以使我们高效地实现优先队列。
堆的算法
我们用数组pq[N+1]来表示大小为N的堆，我们不使用pq[0]。
上浮（由下至上的堆有序）
private void swim(int k) {
while (k > 1 && less(k / 2, k)) {
exch(k / 2, k);
k = k / 2;
}
}下沉（由上至下的堆有序）
private void sink(int k) {
while (2 * k <= N) {
int j = 2 * k;
if (j < N && less(j, j + 1)) j++; //找到子节点中更大的那个
if (!less(k, j)) break; //如果父结点比较大，则终止
exch(k, j);//如果父结点比较小，则把子节点中更大的那个jiaohuanshanglai
k = j;
}
}MaxPQ 代码
-
复杂度
插入：不超过lgN+1次比较
删除最大元素：不超过2lgN次比较
简易版
public class MaxPQ<Key extends Comparable<Key>> {
private Key[] pq; // heap-ordered complete binary tree
private int N = 0; // in pq[1..N] with pq[0] unused
public MaxPQ(int maxN) {
pq = (Key[]) new Comparable[maxN + 1];
}
public boolean isEmpty() {
return N == 0;
}
public int size() {
return N;
}
public void insert(Key v) {
pq[++N] = v; //添加到最后
swim(N); //上浮
}
public Key delMax() {
Key max = pq[1]; // Retrieve max key from top.最大的为根结点
exch(1, N--); // Exchange with last item.和最后一个结点交换，并减小N
pq[N + 1] = null; // Avoid loitering.删除原来的最后一位
sink(1); // Restore heap property.下沉
return max;
}
// See above
private boolean less(int i, int j)
private void exch(int i, int j)
private void swim(int k)
private void sink(int k)
}完整版
添加resize功能
public class MaxPQ<Key> implements Iterable<Key> {
private Key[] pq; // store items at indices 1 to N
private int N; // number of items on priority queue
private Comparator<Key> comparator; // optional Comparator
public MaxPQ(int initCapacity) {
pq = (Key[]) new Object[initCapacity + 1];
N = 0;
}
public MaxPQ() {
this(1);
}
public MaxPQ(int initCapacity, Comparator<Key> comparator) {
this.comparator = comparator;
pq = (Key[]) new Object[initCapacity + 1];
N = 0;
}
public MaxPQ(Comparator<Key> comparator) {
this(1, comparator);
}
public MaxPQ(Key[] keys) {
N = keys.length;
pq = (Key[]) new Object[keys.length + 1];
for (int i = 0; i < N; i++)
pq[i+1] = keys[i];
for (int k = N/2; k >= 1; k--)
sink(k);
assert isMaxHeap();
}
public boolean isEmpty() {
return N == 0;
}
public int size() {
return N;
}
public Key max() {
if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
return pq[1];
}
// helper function to double the size of the heap array
private void resize(int capacity) {
assert capacity > N;
Key[] temp = (Key[]) new Object[capacity];
for (int i = 1; i <= N; i++) {
temp[i] = pq[i];
}
pq = temp;
}
public void insert(Key x) {
// double size of array if necessary
if (N >= pq.length - 1) resize(2 * pq.length);
// add x, and percolate it up to maintain heap invariant
pq[++N] = x;
swim(N);
assert isMaxHeap();
}
public Key delMax() {
if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
Key max = pq[1];
exch(1, N--);
sink(1);
pq[N+1] = null; // to avoid loiterig and help with garbage collection
if ((N > 0) && (N == (pq.length - 1) / 4)) resize(pq.length / 2);
assert isMaxHeap();
return max;
}
private void swim(int k) {
while (k > 1 && less(k/2, k)) {
exch(k, k/2);
k = k/2;
}
}
private void sink(int k) {
while (2*k <= N) {
int j = 2*k;
if (j < N && less(j, j+1)) j++;
if (!less(k, j)) break;
exch(k, j);
k = j;
}
}
private boolean less(int i, int j) {
if (comparator == null) {
return ((Comparable<Key>) pq[i]).compareTo(pq[j]) < 0;
}
else {
return comparator.compare(pq[i], pq[j]) < 0;
}
}
private void exch(int i, int j) {
Key swap = pq[i];
pq[i] = pq[j];
pq[j] = swap;
}
// is pq[1..N] a max heap?
private boolean isMaxHeap() {
return isMaxHeap(1);
}
// is subtree of pq[1..N] rooted at k a max heap?
private boolean isMaxHeap(int k) {
if (k > N) return true;
int left = 2*k, right = 2*k + 1;
if (left <= N && less(k, left)) return false;
if (right <= N && less(k, right)) return false;
return isMaxHeap(left) && isMaxHeap(right);
}
public Iterator<Key> iterator() {
return new HeapIterator();
}
private class HeapIterator implements Iterator<Key> {
// create a new pq
private MaxPQ<Key> copy;
// add all items to copy of heap
// takes linear time since already in heap order so no keys move
public HeapIterator() {
if (comparator == null) copy = new MaxPQ<Key>(size());
else copy = new MaxPQ<Key>(size(), comparator);
for (int i = 1; i <= N; i++)
copy.insert(pq[i]);
}
public boolean hasNext() { return !copy.isEmpty(); }
public void remove() { throw new UnsupportedOperationException(); }
public Key next() {
if (!hasNext()) throw new NoSuchElementException();
return copy.delMax();
}
}
public static void main(String[] args) {
MaxPQ<String> pq = new MaxPQ<String>();
while (!StdIn.isEmpty()) {
String item = StdIn.readString();
if (!item.equals("-")) pq.insert(item);
else if (!pq.isEmpty()) StdOut.print(pq.delMax() + " ");
}
StdOut.println("(" + pq.size() + " left on pq)");
}
}索引优先队列
增加索引
增加change, contains, delete方法
索引优先队列API
各方法的时间成本
IndexMinPQ 代码
简易版
public class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
private int maxN; // maximum number of elements on PQ
private int N; // number of elements on PQ
private int[] pq; // binary heap using 1-based indexing
private int[] qp; // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
private Key[] keys; // keys[i] = priority of i
public IndexMinPQ(int maxN) {
this.maxN = maxN;
keys = (Key[]) new Comparable[maxN + 1]; // 存一发原来的数组
pq = new int[maxN + 1]; // 这是二叉树，比如1位置放的是想要记录的是keys[3]，但是记录了3，即pq[1]=3
qp = new int[maxN + 1]; // 反过来，keys[3]放在哪里了呢？放在了树的1位置, qp[3]=1
for (int i = 0; i <= maxN; i++)
qp[i] = -1;
}
public void insert(int i, Key key) {
if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
N++;
qp[i] = N; // i放到了树最后的位置N，通过原数组i找到树中的位置N
pq[N] = i; // 树的最后位置N放了i，通过树中的位置N找到原数组i
keys[i] = key; //具体是什么
swim(N); //上浮
}
private void swim(int k) {
while (k > 1 && greater(k/2, k)) {
exch(k, k/2); //在这里pq,qp都换好了
k = k/2;
}
}
private void exch(int i, int j) {
int swap = pq[i];
pq[i] = pq[j];
pq[j] = swap;
qp[pq[i]] = i; //因为是逆运算
qp[pq[j]] = j;
}
public int delMin() {
if (N == 0) throw new NoSuchElementException("Priority queue underflow");
int min = pq[1];
exch(1, N--);
sink(1);
qp[min] = -1; // delete
keys[min] = null; // to help with garbage collection
pq[N+1] = -1; // not needed
return min;
}
public void changeKey(int i, Key key) {//改的是原来的数组
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
keys[i] = key;
swim(qp[i]); //可能往上
sink(qp[i]); //可能往下
}
} 完整版
public class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
private int maxN; // maximum number of elements on PQ
private int N; // number of elements on PQ
private int[] pq; // binary heap using 1-based indexing
private int[] qp; // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
private Key[] keys; // keys[i] = priority of i
public IndexMinPQ(int maxN) {
if (maxN < 0) throw new IllegalArgumentException();
this.maxN = maxN;
keys = (Key[]) new Comparable[maxN + 1];
pq = new int[maxN + 1];
qp = new int[maxN + 1];
for (int i = 0; i <= maxN; i++)
qp[i] = -1;
}
public boolean isEmpty() {
return N == 0;
}
public boolean contains(int i) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
return qp[i] != -1;
}
public int size() {
return N;
}
public void insert(int i, Key key) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
N++;
qp[i] = N;
pq[N] = i;
keys[i] = key;
swim(N);
}
public int minIndex() {
if (N == 0) throw new NoSuchElementException("Priority queue underflow");
return pq[1];
}
public Key minKey() {
if (N == 0) throw new NoSuchElementException("Priority queue underflow");
return keys[pq[1]];
}
public int delMin() {
if (N == 0) throw new NoSuchElementException("Priority queue underflow");
int min = pq[1];
exch(1, N--);
sink(1);
assert min == pq[N+1];
qp[min] = -1; // delete
keys[min] = null; // to help with garbage collection
pq[N+1] = -1; // not needed
return min;
}
public Key keyOf(int i) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
else return keys[i];
}
public void changeKey(int i, Key key) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
keys[i] = key;
swim(qp[i]);
sink(qp[i]);
}
public void change(int i, Key key) {
changeKey(i, key);
}
public void decreaseKey(int i, Key key) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
if (keys[i].compareTo(key) <= 0)
throw new IllegalArgumentException("Calling decreaseKey() with given argument would not strictly decrease the key");
keys[i] = key;
swim(qp[i]);
}
public void increaseKey(int i, Key key) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
if (keys[i].compareTo(key) >= 0)
throw new IllegalArgumentException("Calling increaseKey() with given argument would not strictly increase the key");
keys[i] = key;
sink(qp[i]);
}
public void delete(int i) {
if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
int index = qp[i];
exch(index, N--);
swim(index);
sink(index);
keys[i] = null;
qp[i] = -1;
}
private boolean greater(int i, int j) {
return keys[pq[i]].compareTo(keys[pq[j]]) > 0;
}
private void exch(int i, int j) {
int swap = pq[i];
pq[i] = pq[j];
pq[j] = swap;
qp[pq[i]] = i;
qp[pq[j]] = j;
}
private void swim(int k) {
while (k > 1 && greater(k/2, k)) {
exch(k, k/2);
k = k/2;
}
}
private void sink(int k) {
while (2*k <= N) {
int j = 2*k;
if (j < N && greater(j, j+1)) j++;
if (!greater(k, j)) break;
exch(k, j);
k = j;
}
}
public Iterator<Integer> iterator() { return new HeapIterator(); }
private class HeapIterator implements Iterator<Integer> {
// create a new pq
private IndexMinPQ<Key> copy;
// add all elements to copy of heap
// takes linear time since already in heap order so no keys move
public HeapIterator() {
copy = new IndexMinPQ<Key>(pq.length - 1);
for (int i = 1; i <= N; i++)
copy.insert(pq[i], keys[pq[i]]);
}
public boolean hasNext() { return !copy.isEmpty(); }
public void remove() { throw new UnsupportedOperationException(); }
public Integer next() {
if (!hasNext()) throw new NoSuchElementException();
return copy.delMin();
}
}
}
java
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