How to update element priorities in a heap for Prim's Algorithm?

Question

I am studying Prim\'s Algorithm. There is a part within the code the next vertex across the cut will be coming to the set of the vertices belonging to the MST. Whi

Answer 1

Pointers enable efficient composite data structures

You have something like this (using pseudocode C++):

class Node
    bool visited
    double key
    Node* pi
    vector> adjacent //adjacent nodes and edge weights
    //and extra fields needed for PriorityQueue data structure
    // - a clean way to do this is to use CRTP for defining the base
    //   PriorityQueue node class, then inherit your graph node from that

class Graph
    vector vertices

CRTP: http://en.wikipedia.org/wiki/Curiously_recurring_template_pattern

The priority queue Q in the algorithm contains items of type Node*, where ExtractMin gets you the Node* with minimum key.

The reason you don't have to do any linear search is because, when you get u = ExtractMin(Q), you have a Node*. So u->adjacent gets you both the v's in G.Adj[u] and the w(u,v)'s in const time per adjacent node. Since you have a pointer v to the priority queue node (which is v), you can update it's position in the priority queue in logarithmic time per adjacent node (with most implementations of a priority queue).

To name some specific data structures, the DecreaseKey(Q, v) function used below has logarithmic complexity for Fibonnaci heaps and pairing heaps (amortized).

• Pairing heap: http://en.wikipedia.org/wiki/Pairing_heap
  • Not too hard to code yourself - Wikipedia has most of the source code
• Fibonacci heap: http://en.wikipedia.org/wiki/Fibonacci_heap

More-concrete pseudocode for the algorithm

MstPrim(Graph* G)
    for each u in G->vertices
        u->visited = false
        u->key = infinity
        u->pi = NULL
    Q = PriorityQueue(G->vertices)
    while Q not empty
        u = ExtractMin(Q)
        u->visited = true
        for each (v, w) in u->adjacent
            if not v->visited and w < v->key
                v->pi = u
                v->key = w
                DecreasedKey(Q, v) //O(log n)