Minimum Spanning Tree

Minimum Spanning Tree CS 583 Analysis of Algorithms CS583 Fall'06: Minimum Spanning Tree

Outline • Breadth-First Tree • Predecessor subgraph • Printing paths • Minimum Spanning Trees • Definitions • Generic algorithm • Correctness • Self-test • 23.1-1, 23.1-4 CS583 Fall'06: Minimum Spanning Tree

Breadth-First Trees • BFS builds a breadth-first tree as it searches the graph. • It is represented by the  field in each vertex. • Define the predecessor subgraph of G=(V, E) as G = (V, E): • V = {v  V : [v]  NIL}  {s} • E = {([v],v) : v  V - {s}} • The predecessor graph is a breadth-first tree if: • V consists of the vertices reachable from s, and • For all v  V there is a unique path from s to v in G that is also a shortest path in G. • The breadth-first tree is a tree since (see Theorem B.2): • It is connected (there’s a path between any two vertices), and • | E | = | V | - 1 CS583 Fall'06: Minimum Spanning Tree

Breadth-First Trees in BFS Lemma 22.6 When applied to a directed or undirected graph G=(V,E), BFS constructs  so that the predecessor subgraph G = (V, E) is a breadth-first tree. Proof. Line 16 of BFS sets [v] = u if and only if (u,v)  E and (s,v) < . Hence V consists of the vertices in V reachable from s. Since G forms a tree, it contains a unique path from s to v. Using Theorem 22.5 we can show by induction that every such path is a shortest path.  CS583 Fall'06: Minimum Spanning Tree

Printing Shortest Paths The following procedure prints out the vertices on a shortest path from s to v, assuming that BFS has already run. Print-Path(G,s,v) 1 if v = s 2 print s 3 else 4 if [v] = NIL 5 print “No path from s to v exists!” 6 else 7 Print-Path(G,s,[v]) 8 print v The procedure runs in O(|V|) time since each recursive call is for a path one vertex shorter. CS583 Fall'06: Minimum Spanning Tree

The Networking Problem • Design a network that connects n sites at a minimum total cost. • A weighted graph represents the cost (an edge’s weight) of connecting any two sites. • An optimal network will be a subgraph of such a graph containing the minimum number of edges (n-1) at a total minimum sum of the weights of these edges. • Such a subgraph forms a spanning tree. CS583 Fall'06: Minimum Spanning Tree

The Minimum Spanning Tree Problem • Assume we have a connected, undirected graph G = (V,E) with the weight function w : E  R. • For each edge (u,v)  E, we have weight w(u,v). • We wish to find an acyclic subset T  E that connects all of the vertices, and whose total weight is minimized. • w(T) = (u,v)  T w(u,v) CS583 Fall'06: Minimum Spanning Tree

Generic MST • A “generic” algorithm uses a greedy approach in that it grows the minimum spanning tree one edge at a time. • The algorithm manages a set of edges A and maintains the following loop invariant: • Prior to each iteration, A is a subset of some minimum spanning tree. • At each step we determine an edge that can be added to A without violating the invariant. • Such edge is called a safe edge. CS583 Fall'06: Minimum Spanning Tree

Generic MST: Pseudocode The following procedure sketches an algorithm to find a subset of edges of a weighted graph that forms a minimum spanning tree. Generic-MST(G,w) 1 A =  2 while <A does not form a spanning tree> 3 <find an edge (u,v) that is safe for A> 4 A = A  {(u,v)} 5 return A CS583 Fall'06: Minimum Spanning Tree

Generic MST: Loop Invariant • Initialization: • After line 1, the set A is empty, which is a subset of any spanning tree, • Maintenance: • The while loop 2-4 maintains the invariant by adding only safe edges. • Termination: • All edges added to A are in a minimum spanning tree. • Once the spanning tree is formed it is the minimum spanning tree. CS583 Fall'06: Minimum Spanning Tree

Finding Safe Edges • The key is to find safe edges (line 3) • Such edge always exists according to the loop invariant. • A cut(S, V-S) of an undirected graph G = (V,E) is a partition of V. • An edge (u,v)crosses the cut, if one of its endpoints is in S, and the other is in V-S. • A cut respects a set A of edges if no edge in A crosses the cut. • An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut. CS583 Fall'06: Minimum Spanning Tree

Recognizing Safe Edges Theorem 23.1 • Let G = (V,E) be a connected, undirected graph with a real valued function w defined on E. • Let A be a subset of E that is included in some minimum spanning tree for G. • Let (S, V – S) be any cut of G that respects A. • Let (u,v) be a light edge crossing (S, V – S). • Then edge (u, v) is safe for A. Proof Let T be a minimum spanning tree that includes A. If it contains (u,v), then it can be safely added to A. If it does not, we construct another minimum spanning tree T1 that includes A  (u,v). CS583 Fall'06: Minimum Spanning Tree

Recognizing Safe Edges (cont.) • The edge (u,v) forms a cycle with edges on the path from u to v in T. • Since u and v are on the opposite sides of the cut (S, V – S), there’s at least one edge in T on this path that also crosses the cut. • Let this edge be (x,y). • Since (x,y) is on unique path from u to v in T, removing (x,y) breaks T. • Adding (u,v) reconnects it to form a new spanning tree: T1 = T – {(x,y)}  {(u,v)} CS583 Fall'06: Minimum Spanning Tree

Recognizing Safe Edges (cont.) • Show that T1 is a minimum spanning tree: • Since (u,v) is a light edge and (x,y) also crosses the cut, we have: • w(u,v)  w(x,y) • w(T1) = w(T) – w(x,y) + w(u,v)  w(T) • Since T is a minimum spanning tree, T1 is also a minimum spanning tree. • Show that (u,v) is a safe edge for A: • A  T • (x,y)  A • A  {(u,v)}  T1 • Since T1 is a minimum spanning tree, (u,v) is safe for A. CS583 Fall'06: Minimum Spanning Tree

Generic-MST: Framework • As the algorithm proceeds, the set A is always acyclic. • MST cannot contain a cycle. • At any step of the algorithm the graph GA = (V,A) is a forest. • Each of the connected components is a tree. • Any safe edge for A connects distinct components of GA. • A  {(u,v)} must be acyclic. • The loop 2-4 is executed |V| - 1 times to add each edge of an MST. CS583 Fall'06: Minimum Spanning Tree

Generic-MST: Framework (cont.) Corollary 23.2 • Let G = (V,E) be a connected, undirected graph with a real valued function w defined on E. • Let A be a subset of E that is included in some minimum spanning tree for G. • Let C = (VC ,EC) be a connected component (tree) in the forest GA. • Let (u,v) be a light edge connecting C to some other component in GA. • Then edge (u, v) is safe for A. Proof The cut (VC , V - VC) respects A, and edge (u,v) is a light edge for this cut. Therefore, (u,v) is safe for A.  CS583 Fall'06: Minimum Spanning Tree

Minimum Spanning Tree