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03/11/04 Lecture 18
Vertex and Edge classes Class Edge { public Vertex dest; public double weight; public Edge (Vertex d,
double w) { dest = d; weight = w;
} }
Class Vertex { public String Name; public AnyType extraInfo; public List adj; public int dist; // double? public Vertex prev; public Vertex (String s) { Name = s; adj = new LinkedList(); reset(); } public reset () { dist=INFNT; path=null; } }
10/12/16 COP 3530: DATA STRUCTURES
Topological Sort void topSort () {
for( int j = 0; j < N; j++) { Vertex v = findVertexOfIndegZero(); if (v == null) return; // Cycle found v.topologicalNum = j; for each vertex w adjacent to v w.inDegree--; // use extraInfo field }
} 10/12/16 COP 3530: DATA STRUCTURES Time Complexity = O(n + m)
Unweighted SP algorithm Void BFS (Vertex s) { // same as unweighted SP Queue <Vertex> Q = new Queue <>; for each Vertex v except s { v.dist = INFNT;} s.dist = 0; s.prev = null; Q.enqueue(s); while ( !Q.isEmpty() ) {
v = Q.dequeue(); for each vertex w adjacent to v if (w.dist == INFNT) { w.dist = v.dist + 1; w.prev = v; Q.enqueue(w); }
} }
10/12/16 COP 3530: DATA STRUCTURES Time Complexity = O(n + m)
Dijkstra’s SP algorithm void Dijkstra (Vertex s) { // same as weighted SP PriorityQueue <Vertex> Q = new PriorityQueue <>; for each Vertex v except s { v.dist = INFNT; Q.insert(v); } s.dist = 0; s.prev= null; Q.insert(s); while ( !Q.isEmpty() ) {
v = Q.deleteMin(); for each vertex w adjacent to v if (w.dist > v.dist + weight of edge (v,w)) { w.dist = v.dist + weight of edge (v,w); w.prev= v; Q.updatePriority(w, v.dist + weight of edge (v,w)); }
} }
10/12/16 COP 3530: DATA STRUCTURES Time Complexity = O(n log n + m + m log n) = O(m log n)
Spanning Trees and MST u Graph G(V,E)
u Path: sequence of edges
u Cycle: closed path
u A subgraph G’ of G is given by G’(V’, E’)
u A tree is a graph with no cycles
u A connected graph has a path between all vertices
u A spanning tree is a tree that connects all vertices
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Minimum Spanning Trees u Input: Weighted Graph, G(V,E) with n vertices & m
edges, and edge weights w(e) for edge e
u Output: Find a spanning tree of minimum total weight
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Observations and Facts u Every spanning tree has n-1 edges.
u A spanning tree connects all vertices of a graph.
u A spanning tree has no cycles.
u If there is a cycle in the original graph, any spanning tree must miss at least one edge. q MST misses one heaviest edge on that cycle.
u Given any partition of vertices into (S, V-S), every spanning tree must include at least one edge from S to V-S. q MST includes one lightest edge from E(S,V-S).
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MST – Prim’s Algorithm 11/16/2016 Bookshelf Online: Data Structures and Algorithm Analysis in Java
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Figure 9.50 A graph G and its minimum spanning tree
This shows that greed works for the minimum spanning tree problem. The two algorithms we
present differ in how a minimum edge is selected.
10/12/16 COP 3530: DATA STRUCTURES
Prim’s Algorithm
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Figure 9.51 Prim’s algorithm after each stage
Figure 9.52 Initial configuration of table used in Prim’s algorithm
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10/12/16 COP 3530: DATA STRUCTURES
Reminder of Dijkstra’s Alg void Dijkstra (Vertex s) { // same as weighted SP PriorityQueue <Vertex> Q = new PriorityQueue <>; for each Vertex v except s { v.dist = INFNT; Q.insert(v); } s.dist = 0; s.prev= null; Q.insert(s); while ( !Q.isEmpty() ) {
v = Q.deleteMin(); for each vertex w adjacent to v if (w.dist > v.dist + weight of edge (v,w)) { w.dist = v.dist + weight of edge (v,w); w.prev= v; Q.updatePriority(w, v.dist + weight of edge (v,w)); }
} }
10/12/16 COP 3530: DATA STRUCTURES Time Complexity = O(n log n + m + m log n) = O(m log n)
Prim’s Algorithm ArrayList <Edges> Prim (Vertex s) { // computes MST PriorityQueue <Vertex> Q = new PriorityQueue <>; for each Vertex v except s { v.dist = INFNT; Q.insert(v); } s.dist = 0; s.prev= null; Q.insert(s); while ( !Q.isEmpty() ) {
v = Q.deleteMin(); for each vertex w adjacent to v if (w.dist > weight of edge (v,w)) { w.dist = weight of edge (v,w); w.prev= v; Q.updatePriority(w, weight of edge (v,w)); }
} for each Vertex v except s { compile List mst with edges (v.prev, v); } return mst; }
10/12/16 COP 3530: DATA STRUCTURES Time Complexity = O(n log n + m + m log n) = O(m log n)
Kruskal’s Algorithm
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Figure 9.58 Action of Kruskal’s algorithm on G
Figure 9.59 Kruskal’s algorithm after each stage
vertex is initially in its own set. If u and v are in the same set, the edge is rejected, because
since they are already connected, adding (u, v) would form a cycle. Otherwise, the edge is
accepted, and a union is performed on the two sets containing u and v. It is easy to see
that this maintains the set invariant, because once the edge (u, v) is added to the spanning
forest, if w was connected to u and x was connected to v, then x and w must now be
connected, and thus belong in the same set.
The edges could be sorted to facilitate the selection, but building a heap in linear time is a
much better idea. Then deleteMins give the edges to be tested in order. Typically, only a
small fraction of the edges need to be tested before the algorithm can terminate, although
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10/12/16 COP 3530: DATA STRUCTURES
Pseudocode: Kruskal’s ArrayList <Edges> Kruskal( List <Edge> edges, int N) {
PriorityQueue<Edge> Q = new PriorityQueue<>(edges); List<Edge> mst = new ArrayList<>(); while mst.size() != N-1) { Edge e = Q.deleteMin(); if (EdgeConnectsDifferentComponents(e)) { mst.add(e); UpdateComponents(e); } } return mst;
}
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Time Complexity = O(m log m) = O(m log n)
