第四周上：Priority Queue

4.1 Priority Queues

1. Priority Queues

1. 区分：

   • Stack： 最后添加的item，最先被删（LIFO）
   • Queue：最早添加的item，最先被删（FIFO）
   • Randomized queue：随机删除一个item
   • Priority queue：删除最大（或最小）的item

2. 思路：

   • Plan A ：先不排序，insert后就放在序列尾；要remove时，找出最大的删除
   • Plan B：每insert一个，就按升序排一次序；要remove时，直接删除最后一个

3. Performance

   implementation	insert	delMax	max
   Plan A：Unordered array	1	N	N
   Plan B：ordered array	N	1	1

4. Java Implementation（Plan A）

   public class UnorderedMaxPQ>{
     private Key[] pq;  
     private int n;     // number of elements on pq
     
     // create an empty priority queue
     public UnorderedMaxPQ(int capacity){
       // no generic array creation
       pq = (Key[]) new Comparable[capacity];
     }
       
     // insert a key into the priority queue
     public void insert(Key v){
       pq[n++] = x;
     }
       
     // return and remove the largest key
     public Key delMax(){
       int max = 0;
       for(int i = 1; i < n; i++){
         if(less(pq, max, i)){
           max = i;
         }
       }
       exch(pq, max, n-1);
       return pq[--n];  // null out entry to prevent loitering
     }
     
     public Key max(){
       int max = 0;
       for(int i = 1; i < n; i++){
         if(less(pq, max, i)){
           max = i;
         }
       }
       return pq[max]; 
     }
     
     // is the priority queue empty?
     public boolean isEmpty(){
       return n == 0; 
     }
     
     private static boolean less(Comparable a, int i, int j){
        return a[i].CompareTo(a[j]) < 0;
    }
   
    private static void exch(Comparable[] a, int i, int j){
        Comparable swap = a[i];
        a[i] = a[j];
        a[j] = swap;
    }
   }
   

2. Binary heaps

1. complete tree

   • 除了最底层，其他都完全对称

   • N个节点，高度：[lgN]

2. 思路

   • Array代表了一个heap-ordered complete binary tree
     • heap-ordered complete binary tree
       • Keys在各个Nodes
       • Parent‘s key不能小于children’s key
     • Array
       • 让indices从1开始
       • Nodes按层排序
   • array[1]就是最大的key，即为binary tree的root
   • 对于array[k]
     • Parent: array[k/2]
     • Children: array[2k]，array[2k+1]

3. Performance

   implementation	insert	delMax	max
   Plan A：Unordered array	1	N	N
   Plan B：ordered array	N	1	1
   Binary heap	lgN	lgN	1

4. Java implementation

   public class MaxPQ>{
    private Key[] pq;
    private int n;
   
    public MaxPQ(int capacity){
        pq = (Key[]) new Comparable[capacity+1]; // 因为我们假设从1开始，所以capacity需要＋1
    }
   
    public boolean isEmpty(){
        return n == 0;
    }
   
    public void insert(Key key){ // at most 1+lgN compares
        pq[++n] = x;
        swim(n);
    }
   
    public Key delMax(){ // at most 2lgN compares
        Key max = pq[1];
        exch(1, n);
        sink(1);
        pq[n] = null;
        n--;
        return max;
    }
   
    // Child's key变得比它的parent‘s key大 
    private void swim(int k){
        while(k>1 && less(k/2, k)){
            exch(k/2, k);
            k = k/2;
        }
    }
   
    // Parent's key 变得比它的其中一个child或者比两个children都小
    private void sink(int k){
        while(2*k <= n){
            int j = 2*k;
            if(j < n && less(j, j+1)){ //找出两个children中较大的那个
                j++;
            }
            if(!less(k, j)){// parent和较大的那个child比较
                break;
            }
            exch(k, j);
            k = j;
        }
    }
   
    private static boolean less(int i, int j){
        return pq[i].CompareTo(pq[j]) < 0;
    }
   
    private static void exch(int i, int j){
        Key swap = pq[i];
        pq[i] = pq[j];
        pq[j] = swap;
    }
   }
   
   

3. Immutability

1. immutable data type: 一旦创建，data type value不能改变

   • immutable：String， Integer，Double，Vector

     // can't override instance methods
     public final class Vector{
       // all instance variable private and final
       private final int n;
       private final double[] data;
       
       public Vector(double[] data){
         this.n = data.length;
         this.data = new double[n];
         for(int i = 0; i < n; i++){
           this.data[i] = data[i];
         }
       }
       
       // instance methods don't change instance variable
       ...
     }
     

• mutable：StringBuilder，Stack，Counter，Java array

4. Heapsort

1. 思路：

   • 开始有一个无序的array

   • 建一个有n个keys的max-heap（依然假设index是从1到n）

   • 不断地取出最大的key

2. Performance

   • Heap construction : compares and exchanges
   • Heapsort: compares and exchanges

3. Java implementation

   public class Heap{
    public static void sort(Comparable[] a){ // 假设 array 0 到 1
        int n = a.length;
        for(int k = k/2; k >=1; k--){
            sink(a, k, n);
        }
        while(n > 1){
            exch(a, 1, n);
            sink(a, 1, --n);
        }
    }
   
    private static void sink(Comparable[] a, int k, int n){
        while(2*k <= n){
            int j = 2*k;
            if(j < n && less(a, j, j+1)){ //找出两个children中较大的那个
                j++;
            }
            if(!less(a, k, j)){// parent和较大的那个child比较
                break;
            }
            exch(a, k, j);
            k = j;
        }
    }
   
    private static boolean less(Comparable[] a, int i, int j){ // 注意：是1-based indexing
        return a[i-1].CompareTo(a[j-1]) < 0; 
    }
   
    private static void exch(Comparable[] a, int i, int j){ // // 注意：是1-based indexing
        Key swap = pq[i-1];
        pq[i-1] = pq[j-1];
        pq[j-1] = swap;
    }
   }
   

4. 总结（Sorting algorithms）

   	inplace?	stable?	worst	average	best	remarks
   selection	√					N exchanges
   insertion	√	√				适用small N或partially ordered
   shell	√		?	?		code少
   quick	√					probabilistic guarantee（用shuffle），fastes in practice
   3-way quick	√					对quicksort的改良，适用于有很多duplicate keys
   merge		√				guarantee，stable
   heap	√					guarantee，inplace