《算法导论》练习6.58—最小堆k路合并

　　《算法导论》第六章主要内容是关于堆和优先级队列，书中给出了一个练习题，非常有有意思，今天好好研究练习一下。题目如下：请给出一个时间为O(nlgk)、用来将k个已排序链表合并为一个排序链表的算法。此处n为所有输入链表中元素的总数。（提示：用一个最小堆来做k路合并）。

　　看到题目第个想到的是归并排序过程中的归并操作子过程，从头开始两两比较，找出最小的，然后接着往后比较，常用的是2路归并。而题目给的是k个已排好序的链表(k>=2)。如果没有提示，我半天不知道如何去实现，幸好提示说用最小堆来做k路合并，于是我想到可以这样做：创建一个大小为k的数组，将k个链表中的第一个元素依次存放到数组中，然后将数组调整为最小堆，这样保证数组的第一个元素是最小的，假设为min，将min从最小堆取出并存放到最终结果的链表中，此时将min所在链表的下一个元素到插入的最小堆中，继续上面的操作，直到堆中没有元素为止。举个例子如下图所示（只给出不部分操作）：

最终结果如下图所示：

现在采用C++语言，借助STL实现此过程，链表采用list，最小堆中存放的是list的迭代器，表示list中元素的位置。完整程序如下：

#include <iostream>
#include <vector>
#include <list>
#include <iterator>
#include <cstdlib>
using namespace std;

template<class T> class MinHeap
{
public:
    MinHeap();
    MinHeap(const size_t size);
    ~MinHeap();
    T get_min() const;
    void delete_min();
    void insert_element(const T& e);
    void adjust_min_heap(const size_t i);
    size_t get_heap_size() const;
    int compare(const T& t1,const T& t2);
private:
    T *heap;
    size_t heap_size;
};

template<class T>
MinHeap<T>::MinHeap():heap(NULL),heap_size(0){}

template<class T>
MinHeap<T>::MinHeap(const size_t size)
{
    if(!heap)
        delete [] heap;
    heap = new T[size+1];
    heap_size = 0;
}

template<class T>
MinHeap<T>::~MinHeap()
{
    if(!heap)
        delete [] heap;
    heap_size = 0;
}

template<class T>
T MinHeap<T>::get_min() const
{
    if(heap_size > 0)
        return heap[1];
    else
        return T();
}

template<class T>
void MinHeap<T>::delete_min()
{
    if(heap_size > 0)
    {
        heap[1] = heap[heap_size];
        heap_size = heap_size - 1;
        adjust_min_heap(1);
    }
    else
    {
        cout<<"Error: the min heap is empty"<<endl;
    }
}

template<class T>
void MinHeap<T>::insert_element(const T& e)
{
    size_t i,parent;
    T temp;
    heap_size = heap_size + 1;
    heap[heap_size] = e;
    i = heap_size;
    parent = i/2;
    while(i>1 && compare(heap[parent],heap[i]) > 0)
    {
        temp = heap[parent];
        heap[parent] = heap[i];
        heap[i] = temp;
        i = parent;
        parent = i/2;
    }
}

template<class T>
void MinHeap<T>::adjust_min_heap(const size_t i)
{
    size_t left,right,least;
    T temp;
    left = i*2;
    right = i*2+1;
    if(left <= heap_size && compare(heap[left],heap[i]) < 0)
        least = left;
    else
        least = i;
    if(right <= heap_size && compare(heap[right],heap[least]) < 0)
        least = right;
    if(least != i)
    {
        temp = heap[least];
        heap[least] = heap[i];
        heap[i] = temp;
        adjust_min_heap(least);
    }
}
template<class T>
size_t MinHeap<T>::get_heap_size() const
{
    return heap_size;
}

template<class T>
int MinHeap<T>::compare(const T& t1,const T& t2)
{
    return (*t1-*t2);
}

const static int k = 3;

int main()
{

    list<int> lists[k];
    list<int>::iterator iters[k];
    list<int> retlist;
    list<int>::iterator retiter;
    list<int>::iterator iter;
    MinHeap<list<int>::iterator> minheap(k);

    //first list <12,24,52>
    lists[0].push_back(12);
    lists[0].push_back(24);
    lists[0].push_back(52);
    cout<<"First list: ";
    for(iter=lists[0].begin();iter != lists[0].end();++iter)
          cout<<*iter<<"->";
    cout<<"NULL"<<endl;
    //second list <9,32>
    lists[1].push_back(9);
    lists[1].push_back(32);
    cout<<"Second list: ";
    for(iter=lists[1].begin();iter != lists[1].end();++iter)
          cout<<*iter<<"->";
    cout<<"NULL"<<endl;
    //third list <34,42,78>
    lists[2].push_back(34);
    lists[2].push_back(42);
    lists[2].push_back(78);
    cout<<"Third list: ";
    for(iter=lists[2].begin();iter != lists[2].end();++iter)
          cout<<*iter<<"->";
    cout<<"NULL"<<endl;
    iters[0] = lists[0].begin();
    iters[1] = lists[1].begin();
    iters[2] = lists[2].begin();

    minheap.insert_element(iters[0]);
    minheap.insert_element(iters[1]);
    minheap.insert_element(iters[2]);

    while(minheap.get_heap_size())
    {
        iter = minheap.get_min() ;
        retlist.push_back(*iter);
        minheap.delete_min();
        ++iter;
        if(iter != lists[0].end() && iter != lists[1].end()
           &&iter != lists[2].end())
            minheap.insert_element(iter);
    }
    cout<<"Merge the there list is: "<<endl;
    for(retiter = retlist.begin();retiter!= retlist.end();retiter++)
        cout<<*retiter<<"->";
    cout<<"NULL"<<endl;
    exit(0);
}

程序测试结果如下所示：