In the last post, I defined a spanning tree and gave an algorithm called the ‘Kruskal’s Algorithm’. In this post, I am going to describe another algorithm that also helps in computing the minimum spanning tree of a graph. This algorithm is called the ‘Prim’s Algorithm’ . The basic intuition behind the algorithm goes as follows: Firstly, every vertex should be reachable from every other vertex for it to be a tree. So we will try to build the tree by adding one vertex after another into the connected component. Since we want a “minimum” such tree, we will use the edge between the new vertex and the old component, that is of the minimum weight. This intuition is formalized below as an algorithm:
Set ConnectedSet = Pick a random vertex v of vertex set Set ToBeAddedSet = set of vertices except vertex v Set ListOfBridgeEdges = Set of Edges while ToBeAddedSet not empty: - Select the minimum edge e from the ListOfBridgeEdges such that it has exactly one end in ConnectedSet - Add the other end of e to the ConnectedSet and remove it from the ToBeAddedSet
Let us now analyze the complexity of the above algorithm:
1) The outer while loop runs for O(|V|) times.
2) Inside each loop iteration, it takes O(|E|) to select the minimum weighted edge such that it has exactly one end in ConnectedSet.
Therefore the overall running time of this algorithm is O(|V| * |E|). And in the worst case |E| is O(|V|^2). Hence, the worst case running time will be O(|V|^3). We can now improve the above algorithm by using a different data structure to store the bridge edges. We will store the edges in a min priority queue(a heap structure).
- Initialise an empty min priority queue Q. - Store a <key,value> pair in the queue,where key is the comparator for the queue. - for all the vertices v in V: Initialise the key as infinity Initialise the value as NULL - Choose a random vertex v from V. Initialise the key as 0. - Push v into Q - while Q is not NULL: Assign u as the extract minimum from priority queue Q for all v neighbour of u: if v is present in Q and edgeweight(u,w)< key value of v: Assign v->value as u Assign v->key as edgeweight(u,w)
Now using the predecessors information present in the v->value, we can build the MST.
Let us now analyse the complexity of this modified algorithm
1) The outermost while loop runs for O(|V|) iterations.
2) For each iteration, the extract minimum from the priority queue takes O(log |V|). Hence the complexity is O(|V| log |V| ) for L13.
3) To analyse the inner loop is slightly tricky. Notice that the total number of times the outer loop + the inner loop executes is exactly 2*|E|. Because each edge is counted twice. Once for each of its end vertices.
4) Each time inside the inner loop, the value of the key is potentially changed. And to insert this back in the priority queue it takes a complexity of O(log |V|). Hence the inner loop overall takes O(|E| log |V|).
Hence, with this modified algorithm, the overall complexity is O(|V| log |V| + |E| log |V|). And for |E| = O(|V|^2), the algorithm runs in O(|V|^2 log |V|), which is a improvement over O(|V|^3).
A JAVA implementation of the Prim’s Algorithm can be found here.