How to decrease key for a particular edge in a priority_queue<PI, vector<PI> ,greater<PI> >,trying to implement prim's algorithm?

Question

#include <bits/stdc++.h>
 using namespace std;

#define fast ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
#define LL long long int
#define pb push_back
#define mp make_pair
#define PI pair<int,int>
#define PL pair<LL,LL>
#define PIS pair< int,string>


#define test int t;cin>>t;while(t--)
#define ff first
#define ss second
#define INF 1000000000
#define input(a,n) for(i=1;i<=n;i++)cin>>a[i];
#define output(a,n) for(i=1;i<=n;i++)cout<<a[i]<<" ";
vector< vector<LL> >v(3002, vector<LL>(3002,-1));
priority_queue<PI, vector<PI> ,greater<PI> > pp;
LL w=0;
int vis[3002]={0};
/*void deck(int a ,int b, int *k)
{
    while(!pp.empty())
    {
        i=pp.top.ss;
        if(i==a)
          pp.top
    }
}*/
void prim(LL s, LL *k, LL *p,LL n)
{

    pp.push(mp(0,s));
    k[s]=0;
    LL i,x,a,b,c=0;
    vector<PI >::iterator it;
    while(true)
    {
        if(c==n)
          break;
        i=pp.top().ss;
        //cout<<i<<" ";
        if(vis[i]!=1)
        w=w+pp.top().ff;
        vis[i]=1;
        c++;
        pp.pop();
        for(x=1;x<=n;x++)
        {
            if(v[i][x]!=-1)
            {

            a=x;
            b=v[i][x];
            if(!vis[a] && b<k[a])
            {
                k[a]=b;
                p[a]=i;
                pp.push(mp(b,a));

            }
           }
       }
    }
}
int main()
{
    fast

    LL n,m,x,i,j,r,s;
    /*pp.push(mp(2,3));
    pp.push(mp(3,4));*/
    cin>>n>>m;
    LL k[n+1],p[n+1];
    v.resize(n+1);
    for(x=1;x<n+1;x++)
    {
        k[x]=INF;
        p[x]=-1;
    }
    for(x=0;x<m;x++)
    {
        cin>>i>>j>>r;
        /*v[i].pb(mp(j,r));
        v[j].pb(mp(i,r));*/
        if(v[i][j]!=-1)
        {
            if(v[i][j]>r)
            {
                v[i][j]=r;
                v[j][i]=r;
            }
       }
       else
       {
           v[i][j]=r;
           v[j][i]=r;
       }


    }
    cin>>s;
    prim(s,k,p,n);
    cout<<w;
    //cout<<pp.top().ss;    
}

I was not able to implement the function which searches a particular value ie the vertex and changes it's key value instead I pushed the changed pair, using

pp.push(mp(b,a));

I was able to get some test cases right by using the if statement

if(c==n)
break;

where 'c' represents the count of vertexes visited.

Answer 1

priority_queue in C++ does not provide the functionality of decrease key operation which means we cannot find a key in priority_queue and decrease its value. One way I know to achieve this is to implement a priority queue ourselves and then maintain another structure (a vector or a map ) which would store the indices of keys in the priority queue. Try to understand the code below which uses this idea.

// C++11 code
#include <iostream>
#include <vector>
#include <cstring>
#define SIZE 5          // number of vertices.
#define INF 100000000

/* This vector will contain a pair of integers where the first integer in
   the pair is value(cost) and the second is the vertex.
   This will be our priority queue.
*/
std::vector <std::pair <int, int> > pq (SIZE);
int size = SIZE;        // size of priority queue
int index_map[SIZE];

// Shift up the key with lower value.
void sift_up(int index) {
    int parent = (index-1)/2;
    while(index >= 0 && pq[index].first < pq[parent].first) {
        index_map[pq[index].second] = parent;
        index_map[pq[parent].second] = index;
        std::swap(pq[index], pq[parent]);
        index = parent;
        parent = (index - 1)/2;
    }
}

// Shift down the key with higher value.
void sift_down(int index) {
    int left = 2*index+1, right = 2*index+2;
    int min_index = index;
    if(left < size && pq[left].first < pq[min_index].first)
        min_index = left;
    if(right < size && pq[right].first < pq[min_index].first)
        min_index = right;
    if(min_index != index) {
        index_map[pq[index].second] = min_index;
        index_map[pq[min_index].second] = index;
        std::swap(pq[index], pq[min_index]);
        sift_down(min_index);
    }
}

// Extract the minimum element from priority queue.
std::pair <int, int> extract_min() {
    index_map[pq[0].second] = size-1;
    index_map[pq[size-1].second] = 0;
    std::swap(pq[0], pq[size-1]);
    size -= 1;
    sift_down(0);
    return pq[size];
}

// Change the value at index 'index' to 'value'.
void decrease_key(int index, int value) {
    int temp = pq[index].first;
    pq[index].first = value;
    if(value < temp)
        sift_up(index);
    else
        sift_down(index);
}

// Initialise and heapify the priority queue.
void make_heap(int start_index) {
    for(int i = 0; i < SIZE; i++) {
        pq[i].first = INF;
        pq[i].second = i;
    }
    pq[0].first = 0;
    pq[start_index].second = start_index;

    for(int i = SIZE-1; i >= 0; i--)
        sift_down(i);
}

int main() {

    /* Graph is represent using adjacency list. It takes the following form:
       graph[u] = {{v_1, c_1}, {v_2, c_2}, ..., {v_n, c_n}};
       The above line means that there is an undirected edge 
       between vertices 'u' and 'v_1' with cost 'c_1'. 
       Similarly for (u, v_2, c_2), ..., (u, v_n, c_n).
    */ 
    std::vector <std::vector <std::pair <int, int> > > graph (SIZE);

    graph[0] = {{1, 20}, {2, 50}, {3, 70}, {4, 90}};
    graph[1] = {{0, 20}, {2, 30}};
    graph[2] = {{0, 50}, {1, 30}, {3, 40}};
    graph[3] = {{0, 70}, {2, 40}, {4, 60}};
    graph[4] = {{0, 90}, {3, 60}};

    int visited[SIZE];
    memset(visited, 0, sizeof(visited));
    visited[0] = 1;
    make_heap(0);           // Assuming algorithm to start from vertex 0.

    for(int i = 0; i < SIZE; i++)
        index_map[pq[i].second] = i;

    int answer = 0;
    while(size != 0) {

        std::pair <int, int> p = extract_min();
        /* p.first will contain the cost of the next edge to be added in our
           answer and p.second will give the vertex number.
        */
        visited[p.second] = 1;
        answer += p.first;

        for(int i = 0; i < graph[p.second].size(); i++) {
            if(!visited[graph[p.second][i].first]) {
                if(graph[p.second][i].second < pq[index_map[graph[p.second][i].first]].first) {
                    decrease_key(index_map[graph[p.second][i].first], graph[p.second][i].second);
                }
            }
        }

    }
    std::cout << answer << "\n";
}

Answer 2

The standard std::priority_queue does not allow to peak inside of it so changing priority keys is not possible. It is a classic queue implementation with pushing elements at one side and popping them back at the other. So probably you should look for a more general implementation of the heap data structure.

If you insist on using std::priority_queue you might need to do something as ugly as popping the full queue content to a vector, updating elements and restoring the queue.

Answer 3

There are three ways I know of.

1. If you insist on using the standard library's priority_queue , you can insert each vertex multiple times, but ignore it every time you see it except the first. In your code, you can change if(vis[i]!=1) w=w+pp.top().ff; into if(vis[i]==1) continue; (perhaps; didn't test). The downside is that your priority_queue can grow to O(|E|) memory.

2. You can also use the standard library's set instead of priority_queue . Then, whenever you want to insert a pair (distance, vertex) , first you have to find and remove the pair (old_distance, vertex) if it is in the set. To know old_distance to each vertex at all times, you have to maintain an array (or vector) with the current distances. This way, you keep memory to O(|V|), but set has a greater constant factor than priority_queue .

3. Finally, you can implement a priority queue which allows deletion. Let's say you implement the priority queue as a binary heap. Then, you will have to maintain an inverse permutation of elements in the priority queue: for each element v , store and track what is the current heap index of v . One of the other answers looks like it implements this approach, the inverse permutation being index_map .

Answer 4

I found a way to implement the algorithm using

std::priority_queue

By changing approach, would like to share my code

    #include<iostream>
    #include<algorithm>
    #include<vector>
    #include<set>
    #include<limits.h>
    #include<map>
    #include<stack>
    #include<stdio.h>
    #include<queue>
    #include<cmath>
    #include<string.h>

    using namespace std;
    #define pb push_back
    #define mp make_pair
    #define ff first
    #define ss second
    #define PII pair<int,int>
    vector<PII> v[3001];
    bool vis[3001];
    int sum=0;
    void prim(int s)
    {
        int y,x,i;
        PII p;
        priority_queue<PII, vector<PII> , greater<PII> > q;
        q.push(mp(0,s));
        while(!q.empty())
        {
            p = q.top();
            q.pop();
            x = p.ss;
            if(vis[x]==true)
                continue;
            sum+=p.ff;
            vis[x]=true;
            for(i=0;i<v[x].size();i++)
            {
                y = v[x][i].ss;
                if(vis[y]==false)
                    q.push(v[x][i]);
            }
        }
    }
        int main() {

        fast
        int f1=0,max=0,y,a1,x,j,w=0,f=0,l,m,b,c1=0,r=0;
        char t,a[4][4],c;
        int n,i=0,d=1,k;
        cin>>n>>k;
        for(x=0;x<k;x++)
        {
            cin>>i>>j>>r;
            v[i].pb(mp(r,j));
            v[j].pb(mp(r,i));
        }
        cin>>i;
        prim(i);
        cout<<sum;
        return 0;

}