Dijkstra with a heap. How to update the heap after relaxation?

问题

I am trying to implement Dijkstra algorithm.

 foreach distance d
   d = INFINITY
 d[source] = 0

 create_heap_based_on_Distances();

 while(true)
   bestedge = heap[0]

   remove_minimum_from_heap //it will be heap[0]

   foreach adjacency of bestedge
      if (weight + bestedge_distance < current_distance)
      {
          current_distance = weight + bestedge_distance

          // Now I have to update heap, how can I do that?
      }
   if (heap_empty) break

So, in the relaxation, how can I update the heap, so it would have the correct order? I don't have the heap's index for that node in that step. Does that mean I have to create a new array like nodes[edge] = heapIndex, so I could get a heap's index for that node? But it seems very inefficient as I need then to update insert_to_heap, remove_minimum functions.

C or JAVA code is okay.

回答1:

Does that mean I have to create a new array like nodes[edge] = heapIndex, so I could get a heap's index for that node?

Yes.

But it seems very inefficient as I need then to update insert_to_heap, remove_minimum functions.

Array updates are very cheap, both in theory and in practice, and you only have to do a few of those per heap operation, so this is not inefficient at all. Also, the memory usage of such an array is very cheap compared to the storage cost of a graph data structure.

回答2:

Does that mean I have to create a new array like nodes[edge] = heapIndex, 
so I could get a heap's index for that node?

I don't what the exactly mean of node[edge]. In my opinion, it should be a Map(a array indeed) f that f[node]=HeapIndex(It gives the index of that node in Heap). Storage of the node[edge] is not efficient.

Then how to implement the MapHeap? I have implemented a efficient MapHeap, but not so much note in the code:

template<class DT>
struct MapHeap
{
    DT f[HEAP_SIZE+5];//store the distance
    int mp1[HEAP_SIZE+5];//val -> index
    // I assume the val is unique.
    // In the dijk, the val is the nodeId,so it must be unique. 
    // mp1[nodeId] gives the index of that node in my heap
    int mp2[HEAP_SIZE+5];//index -> val
    int nv;// number of node in my heap now
    MapHeap():nv(0)
    {
        memset(mp1,-1,sizeof(mp1));
        memset(mp2,-1,sizeof(mp2));
    }
    void print(int n)
    {
        for(int i=1;i<=n;i++) printf("%d ",f[i]);
        puts("");
        for(int i=1;i<=n;i++) printf("%d ",mp1[i]);
        puts("");
        for(int i=1;i<=n;i++) printf("%d ",mp2[i]);
        puts("");
    }
    void print(){print(nv);}
    bool resize(int n)
    {
        if (nv<0||nv>HEAP_SIZE) return 0;
        for(int i=n+1;i<=nv;i++)
        {
            mp1[mp2[i]]=-1;
            mp2[i]=-1;
        }
        nv=n;
        return 1;
    }
    DT top()//get the smallest element
    {
        if (nv<1) return DT(-1);
        return f[1];
    }
    DT get(int idx)
    {
        if (idx<1||idx>nv) return DT(-1);
        return f[idx];
    }
    // it's for unpdating the heap. It should be pravite method. 
    // Because I write this code for competition, so I just ignore the accsee controling
    void movedown(int now,int val,const DT &x)//this node is larger than son
    {
        for(;now*2<=nv;)
        {
            int a=now*2;
            int b=now*2+1;
            if (b<=nv&&f[b]<f[a]) a=b;
            if (f[a]>=x) break;
            f[now]=f[a];
            mp1[mp2[a]]=now;
            mp2[now]=mp2[a];
            now=a;
        }
        f[now]=x;
        mp1[val]=now;
        mp2[now]=val;
    }
    void moveup(int now,int val,const DT &x)//this node is smaller than father
    {
        for(;now>1;now>>=1)
        {
            int par=now>>1;
            if (f[par]<=x) break;
            f[now]=f[par];
            mp1[mp2[par]]=now;
            mp2[now]=mp2[par];
        }
        f[now]=x;
        mp1[val]=now;
        mp2[now]=val;
    }
    bool pop(int idx=1)//pop a element, pop the smallest element by default
    {
        if (idx<1||idx>nv) return 0;
        DT &x=f[nv];
        int v1=mp2[nv];
        int v2=mp2[idx];
        mp1[v1]=idx;
        mp2[idx]=v1;
        mp1[v2]=-1;
        mp2[nv]=-1;
        nv--;
        if (idx!=nv+1) movedown(idx,v1,x);
        x=0;
        return 1;
    }
    bool push(const DT &x,int val)//push a node, and with the value of val(in dijk, the val is the nodeId of that node)
    {
        int now=++nv;
        if (now>HEAP_SIZE) return 0;
        moveup(now,val,x);
        return 1;
    }
};

来源：https://stackoverflow.com/questions/16217261/dijkstra-with-a-heap-how-to-update-the-heap-after-relaxation