Minimum spanning trees

Transcription

1 Minimum spanning trees [We re following the book very closely.] One of the most famous greedy algorithms (actually rather family of greedy algorithms). Given undirected graph G = (V, E), connected Weight function w : E IR For simplicity, all edge weights distinct Spanning tree: tree that connects all vertices, hence n = V vertices and n 1 edges MST T : w(t) = (u,v) T w(u, v) minimized What for? Chip design Communication infrastructure in networks Minimum Spanning Trees 1

2 1 Minimum Spanning Trees

3 Growing a minimum spanning tree First, generic algorithm. It manages set of edges A, maintains invariant: Prior to each iteration, A is subset of some MST. At each step, determine edge (u, v) that can be added to A, i.e. without violating invariant, i.e., A {(u, v)} is also subset of some MST. We then call (u, v) a safe edge. 1: A : while A does not form a spanning tree do 3: find an edge (u, v) that is safe for A 4: A A {(u, v)} : end while We use invariant as follows: Initialization. After line 1, A triv. satisfies invariant. Maintenance. Loop in lines maintains invariant by adding only safe edges. Termination. All edges added to A are in a MST, so A must be MST. Minimum Spanning Trees 3

4 But... how do we recognize safe edges? The following theorem provides a rule. Def. A cut (S, V S) of an undirected graph G = (V, E) is a partition of V. Def. An edge (u, v) crosses cut (S, V S) if one endpoint is in S, the other one in V S. Def. A cut respects a set A E if no edge in A crosses the cut. Def. An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut. Theorem 1. Let G = (V, E) be connected, undirected graph with real-valued weight fct. defined on E. Let A be a subset of E that is included in some MST 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, (u, v) is safe for A. Minimum Spanning Trees 4

5 Proof. let T be a MST that includes A (must be one) assume T does not include (u, v) (done otherwise) construct MST T that includes A {(u, v)}, showing that (u, v) is safe (by def.) (u, v) T, so there must be path p = (u = w 1 w w k = v) with (w i, w i+1 ) T for 1 i < k u and v are on opposite sides of cut (S, V S), so there must be at least one edge (x, y) of T crossing cut (x, y) is not in A because A respects cut (x, y) is on unique path from u to v, so removing (x, y) breaks T into two components adding (u, v) reconnects them to form new spanning tree T = T {(x, y)} {(u, v)} Minimum Spanning Trees

6 (u, v) is light edge crossing (S, V S), and (x, y) also crosses this cut, therefore w(u, v) w(x, y) and W(T ) = w(t) w(x, y) + w(u, v) W(T) But T is MST, i.e. w(t) w(t ), thus w(t ) = w(t) and T is MST also A T and (x, y) A (this was because (x, y) crosses cut but A respects cut), so A T also Since (u, v) T, we have A {(u, v)} T Since T is MST, (u, v) is safe for A q.e.d. Minimum Spanning Trees 6

7 We see: as algorithm proceeds, A is always acyclic (otherwise, MST including A would contain cycle) at any point, graph G A = (V, A) is a forest with components being trees some components may contain just one vertex (initially, A is empty, and forest contains V trees, one for each vertex) Any safe edge (u, v) for A connects distinct components of G A, since A {(u, v)} must be acyclic main loop is executed V 1 times (one iteration for every edge of the resulting MST) Minimum Spanning Trees 7

8 The following is going to be used later on. Corollary. Let G = (V, E) be connected, undirected graph with real-valued weight fct. defined on E. Let A be subset of E that is included in some MST for G, let C = (V C, E C ) be a connected component (tree) in forest G A = (V, A). If (u, v) is a light edge connecting C to some other component in G A, then (u, v) is safe for A. Proof. The cut (V C, V V C ) respects A (A defines the components of G A ), and (u, v) is a light edge for this cut. Therefore, (u, v) is safe for A. Minimum Spanning Trees 8

9 We ll see Kruskal s and Prim s algorithms, they differ in how they specify rules to determine safe edges. In Kruskal s, A is a forest; in Prim s, A is a single tree. Kruskal s algorithm Finds safe edge to add to growing forest by finding minimim-weight edge e that connects any two trees. If C 1, C denote the two trees that are connected by (u, v), then since (u, v) must be light edge connecting C 1 to some other tree, the corollary implies that (u, v) is safe for C 1. Kruskal s is greedy because at each step it adds an edge of least possible weight. Minimum Spanning Trees

10 This particular implementation uses Disjoint-Set data structure. Each set contains vertices in a tree of the current forest. Make-Set(u) initializes a new set containing just vertex u. Find-Set(u) returns representative element from set that contains u (so we can check whether two vertices u, v belong to same tree). Union(u, v) combines two trees (the one containing U with the one containing v). Time complexity depends on actual implementation. Minimum Spanning Trees

11 Given: graph G = (V, E), weight function w on E 1: A : for each vertex v V [G] do 3: Make-Set(u) 4: end for : sort edges of E into nondecr. order by weight w 6: for each edge (u, v) E, taken in nondecreasing order by weight w do 7: if Find-Set(u) Find-Set(v) then 8: A A {(u, v)} : Union(u, v) : end if 11: end for 1: return A Minimum Spanning Trees 11

12 Running time Depends on implementation of Disjoint-Set data structure. Using Disjoint-Set-Forest implementation with Union-By-Rank and Path-Compression heuristics (see book, Section 1.3), initializing A takes O(1) sorting edges takes O(E log E) main for loop performs O(E) Find-Set and Union operations; along with V Make-Set operation, this takes O((V +E)α(V )) time with α being a slow growing function (see Section 1.4 in book) since G is connected, E V 1, so Disjoint-Set operations take O(E α(v )) since α(v ) = O(log V ) = O(log E), total running time of Kruskal s is O(E log E) since E V, we have log E = O(log V ), and running time becomes O(E log V ) Minimum Spanning Trees 1

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