Chapter 23: Minimum Spanning Trees: A graph optimization problem Given undirected graph G(V,E) and a weight function w(u,v) defined on all edges (u,v)  E, find an acyclic subset T  E that connects all the vertices and for which the total weight is a minimum w(T) =

Any acyclic subset T  E that connects all the vertices is a spanning tree. Any T with minimum total weight is a minimum-weight spanning tree. Most often called “minimum spanning tree

Chapter 23 covers 2 approaches for solving the minimum spanning tree problem; Kruskal’s algorithm and Prim’s algorithm. Both are examples of greedy algorithms (no global search strategy) Generic greedy MST(G,w) A = null set while A is not a minimum spanning tree add the lightest edge unless it generates a cycle (trees are acyclic graphs)

Growing a minimum spanning tree (MST): G(V,E) is an undirected graph. A is subset of V that defines part of a MST. edge (u,v) is a safe edge if A  (u,v) is also part of a MST. Kruskal and Prim algorithms differ in how they search for safe edges. Safe edges are defined by “cuts”.

Theorem 23.1(next slide) provides a rule for finding safe edges A cut (S,V-S) is a partition of vertices in G(V,E) Edge (u,v) crosses cut (S,V-S) if one end is in S and the other is V-S A cut respects set A if no edge in A crosses the cut An edge that crosses a cut is a light edge if it has the minimum weight of edges crossing the cut Edges in A are shaded. White and black V’s are in different parts of partition. (c,d) is a light edge and a safe edge for growing MST.

Let (S,V-S) be a cut in G that respects A, a growing MST. Theorem 23.1 Let (S,V-S) be a cut in G that respects A, a growing MST. Let (u,v) be a light edge crossing (S,V-S). Then (u,v) is a safe edge for A. Proof: Let T be a MST that contains A (shaded) and (x,y) but not (u,v). Both (u,v) and (x,y) cross (S, V-S) (connect B&W vertices). Define T = {T – (x,y)}{(u,v)} w(u,v) < w(x,y) because (u,v) is a light edge w(T) = w(T) – w(x,y) + w(u,v) < w(T) but T is a MST  w(T) < w(T)   w(T) must also be a MST  (u,v) is a safe edge for A

Corollary 23.2: G(V,E) a is connected, undirected, weighted graph. A is a subset of E that is included in some MST of G. GA (V,A) is a forest of connected components (VC,EC), each of which is a tree. If (u,v) is a light edge connecting C to some other component of A, then (u,v) is a safe edge of A Proof: cut (VC, V-VC) respects A and crosses (u,v)  (u,v) is a safe edge by Theorem 23.1

Corrllary 23.2 is basis for Krustal algorithm Look for a cut that respects the growing MST, A, and crosses the lightest edge not already in A. If such a cut exists, include lightest edge in A; otherwise repeat with next lightest edge. Repeat until all vertices are in the MST Resolve ambiguity by including the edge that connects to the vertex with lowest alphabetical order.

Example from text

Pseudo-code for Krustal’s algorithm: p 569 Uses operations on disjoint-set data structures that are described in Chapter 21, p499.   Make-Set(x) creates a new distinct set whose only member, x, is the representative of that set. Find-Set(x) returns a pointer to the representative of the unique set that contains element x. Union-Set(x,y) unites the sets Sx and Sy that contain elements x and y, respectively. Chooses a representative for Sx  Sy and destroys Sx and Sy

MST-Kruskal(G,w) (initialization) A  0 for each v  V[G] do Make-Set(v) sort edges E[G] in non-decreasing order by weight (process edges in sorted order) for all edges (u,v) do if Find-Set(u)  Find-Set(v) (vertices u and v are on opposite sides of a cut) (u,v,) is a light edge then A  A  {(u,v)} Union-Set(u,v) return A Running time: Sorting edges by weight takes O(E lgE) (most time consuming step) Disjoint-set operations (21.3 p505-509) O((E+V)lgV) Total O(ElgE + (E+V)lgV) -> O(ElgE) if E >> V

Prim’s MST algorithm Rationale of Prim’s algorithm: Starts from an arbitrary root.   At each step, consider all edges that connect the growing MST, A, to a vertex not already in the tree. Add the lightest of these if it is a safe edge (i.e. does not create a cycle). In hand calculation, resolve any ambiguity by alphabetical order of vertices that could be added to A (i.e, add to A the edge that connects A to the vertex with lowest alphabetical order). As MST is built, record p[v] if ask to draw the final MST.

MST-Prim(G,w,r) for each u  V[G] do key[u]  , p[u]  NIL key[r]  0 Q  V[G] (initialization completed) while Q  0 do u  Extract-Min(Q) (get the external vertex with the lightest connecting edge) for each v  Adj[u] do if v  Q and w(u,v) < key[v] (update the fields of all vertices that are adjacent to u but not in the tree) then p[v]  u, key[v]  w(u,v) Performance of Prim: (assuming Q is a binary min-heap) Build-Min-Heap requires O(V) time while Q  0 loop executes |V| times  Extract-Min operations take total O(V lgV) time body of the for loop over adjacency lists executes |E| times Q membership test can be implemented in constant time Decrease-Key to update key[v] requires O(lgV) time   Total time O((V + E)lgV ) same as disjoint-set operations in Krustal but Krustal require ElgE in addition to sort edges by weight.

Implementation of Prim’s algorithm: Start from an arbitrary root. At each step, add a light edge that connects growing MST to a vertex not already in the tree. (a safe edges by Corollary 23.2) To facilitate selection of new edges, maintain a minimum priority queue of vertices not in the growing MST. key[v], which defines the priority of v, is the minimum weight among all the edges that connect v to any vertex in the current MST. key[v] =  if no such edge exist. Algorithm terminates when queue is empty. {(v, p[v]): v  V – {r} –Q} implicitly defines the growing MST = A

Executing Prim’s MST algorithm by hand In hand calculation, resolve any ambiguity by alphabetical order of vertices in A (i.e, if vertices have equal priority, add to A the edge that connects A to the vertex with lowest alphabetical order). As MST is built, record p[v] if ask to draw the final MST.

Example from text

Krustal Prim

Completes solution by Prim’s algorithm

c d e b f g a h Assignment 22: Due 4/12/19 Use the rationale of Prim’s algorithm to find a minimum-weight spanning tree in the connected graph below. Use vertex h as the root. Record the predecessor of vertices as they are added to the MST. Resolve any ambiguity about vertex discovery by the alphabetical order of vertex labels. Use the predecessor of vertices to draw the MST. c d e b f g a h 5 8 18 7 14 3 1 11 6 2 4 10 9

To be discussed in class Ex 23.1-1, 23.1-3, and 23.1-4: text p629 Ex 23.1-6 and 23.1-10: text p630

Ex 23. 1-1: (u,v) is minimum-weight edge in a connected graph G Ex 23.1-1: (u,v) is minimum-weight edge in a connected graph G. Show that (u,v) is in some MST of G Proof: Let A be null set edges. Let (s, v-s) be any cut that crosses (u,v) By Theorem 23.1 (u,v) is a safe edge of A Theorem 23.1 Let (S,V-S) be a cut in G that respects A. Let (u,v) be a light edge crossing (S,V-S). Then (u,v) is a safe edge for A

Ex 23. 1-3: (u,v) is contained in some MST of G Ex 23.1-3: (u,v) is contained in some MST of G. Show that (u,v) is a light edge crossing some cut of G Proof: Removal (u,v) from MST breaks it into 2 parts. This allows for a cut that respects the 2 parts and crosses (u,v). (u,v) is a light edge because any other edge crossed by cut was not part of the MST.

Not an MST because cyclic Ex 23.1-4: Every edge (u,v) of a connected graph has the property that there exist a cut (s, v-s) such that (u,v) is a light edge of (s, v-s). Give a simple example of such a graph that is not an MST B A C w(A,B) = w(B,C) = w(C,A) Not an MST because cyclic

Ex 23. 1-6: For every cut of G there exist a unique light edge Ex 23.1-6: For every cut of G there exist a unique light edge. Show that the MST of G is unique. Proof: Given MSTs T and T’ of G, show that T = T’ Remove (u,v) from T then there exist a cut (s,v-s) such that (u,v) is a light edge (ex 12.1-3) Suppose (x,y) of T’ also crosses (s,v-s), then it is also a light edge. Otherwise it would not be part of T’ (ex 12.1-3) But for every cut of G the light edge is unique. Therefore (u,v) = (x,y) and T = T’

Ex 23. 1-6: For every cut of G there exist a unique light edge Ex 23.1-6: For every cut of G there exist a unique light edge. Show that the MST of G is unique. Show by an example that the converse, “If MST is unique then for every cut there exist a unique light edge” is not true

Ex 23. 1-10: T is an MST of G that includes (x,y) Ex 23.1-10: T is an MST of G that includes (x,y). w(x,y) is reduced by k. Show that T remains an MST of G. Proof: Let w(t) be the sum of weights of any spanning tree t Let w’(t) be the sum of weights after w(x,y) is reduced by k Let T’ by any spanning tree that is different from MST T. Then w(T’) > w(T) If (x,y) is not part of T’ then w’(T’) = w(T’) > w(T) > w’(T) If (x,y) is part of T’ then w’(T’) = w(T’) – k > w(T) – k = w’(T) In either case, w’(T’) > w’(T); hence T remains an MST