Minimum Spanning Trees

Minimum Spanning Trees Kruskal's and Prim's algorithms

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Definition • A minimum spanning tree of an undirected, connected graph is a spanning tree of minimum weight. • A spanning tree is a tree composed of all the vertices of a graph, and some or all of its edges. • Note that there may be multiple minimum spanning trees (but only one if edge weights are unique).

Optimality conditions • Cycle property: Any edge in the graph not in the MST forms a cycle with the MST, and this edge must be the maximum edge on the cycle. • Cut property: Any edge in the MST crossing a cut C of the graph will be the minimum edge crossing the cut.

Prim's algorithm • Start the tree with a single source node. • Consider all edges adjacent to one tree node and one non-tree node. • Pick the minimum edge and add it to the tree. • Repeat until the tree covers all vertices. • Use a simple array for O(n^2) time or a priority queue for O(mlogn) time. • Similar to Dijkstra's algorithm.

Prim's algorithm pseudocode typedef pair<int, int> pii; priority_queue<pii> pq; pq.push(pii(0, start)); while (!pq.empty()) { pii p = pq.top(); pq.pop(); int cur = p.second; if (vis[cur]) continue; //add edge to MST vis[cur]=true; for pii p2 in adj[cur]: pq.push(p2); }

Kruskal's algorithm • Start with each vertex as a separate tree. • Sort edges by nondecreasing weight. • For each edge, if the two vertices are part of the same tree, ignore it. Otherwise, use this edge to combine the two trees. • Repeat until there is only a single connected tree. • Runs in O(mlogn) time, or O(m a(n)) time if edges are pre-sorted. • Use Union-Find data structure.

Kruskal's algorithm pseudocode for(int i=0;i<n;i++) MakeSet(i); sort(edges); for(int i=0;i<length(edges);i++) { int a=edges[i].src, b=edges[i].dest; if(find(a) != find(b)) { union(find(a), find(b)); //add edge to MST } }

Potw • Farmer John is irrigating his fields and wants to bring water to his N pastures. • He needs all of his pastures to be connected by some sequence of pipes to pasture 1, the water tank. • FJ has M possible pipes he can use, each connecting two pastures, and pipe has a cost c_i. • Pipes connected to pastures disturb the cows there, so there is an additional cost of p_i * d for each pasture, where d is the number of pipes going through/connected to that pasture. • Help FJ find the minimum possible cost.

Potw Input format: Line 1: N M, # pastures and # pipes Line 2...M+1: a_i b_i c_i, the pastures the pipe is connected to and the cost of the pipe Line M+2...M+N+1: p_i, additional cost for this pasture for each pipe going through it Output format: Line 1: C, the minimum total cost (might not fit within a 32 bit integer)

Potw Sample input: 4 5 1 2 2 1 3 4 1 4 1 2 3 6 3 4 3 10 2 1 5 Sample output: 32 Constraints: 1<= c_i, p_i <= 500,000,000 10 points: 1<=N, M<=100 15 points: 1<=N, M<=1,000 25 points: 1<=N, M<=100,000

Minimum Spanning Trees