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Minimum Spanning Trees •Definition of MST •Generic MST algorithm •Kruskal's algorithm •Prim's algorithm 1 Definition of MST • Let G=(V,E) be a connected, undirected graph. • For each edge (u,v) in E, we have a weight w(u,v) specifying the cost (length of edge) to connect u and v. • We wish to find a (acyclic) subset T of E that connects all of the vertices in V and whose total weight is minimized. • Since the total weight is minimized, the subset T must be acyclic (no circuit). • Thus, T is a tree. We call it a spanning tree. • The problem of determining the tree T is called the minimum-spanning-tree problem. 2 Application of MST: an example • In the design of electronic circuitry, it is often necessary to make a set of pins electrically equivalent by wiring them together. • To interconnect n pins, we can use n-1 wires, each connecting two pins. • We want to minimize the total length of the wires. • Minimum Spanning Trees can be used to model this problem. 3 Here is an example of a connected graph and its minimum spanning tree: 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 Notice that the tree is not unique: replacing (b,c) with (a,h) yields another spanning tree with the same minimum weight. 4 Growing a MST(Generic Algorithm) GENERIC_MST(G,w) 1 A:={} 2 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)} 5 return A • Set A is always a subset of some minimum spanning tree. This property is called the invariant Property. • An edge (u,v) is a safe edge for A if adding the edge to A does not destroy the invariant. • A safe edge is just the CORRECT edge to choose to add to T. 5 How to find a safe edge We need some definitions and a theorem. • A cut (S,V-S) of an undirected graph G=(V,E) is a partition of V. • An edge crosses the cut (S,V-S) if one of its endpoints is in S and the other is in V-S. • An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut. 6 8 7 b c d 4 9 2 S↑ a 11 i 4 14 e ↑S 7 6 8 10 V-S↓ h g f ↓ V-S 1 2 • This figure shows a cut (S,V-S) of the graph. • The edge (d,c) is the unique light edge crossing the cut. 7 Theorem 25.1 • Let G=(V,E) be a connected, undirected graph with a real- valued weight 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 such that for any edge (u, v) in A, {u, v} S or {u, v} (V-S). • Let (u,v) be a light edge crossing (S,V-S). • Then, edge (u,v) is safe for A. • Proof: Let Topt be an minimum spanning tree.(Blue) • A --a subset of Topt and (u,v)-- a light edge crossing (S, V-S) • If (u,v) is NOT safe, then (u,v) is not in T. (See the red edge in Figure) • There MUST be another edge (u’, v’) in Topt crossing (S, V-S).(Green) • We can replace (u’,v’) in Topt with (u, v) and get another treeT’opt • Since (u, v) is light (the shortest edge connect crossing (S, V-S), T’opt is also optimal. 1 1 2 1 1 1 1 8 1 Corollary 25.2 • Let G=(V,E) be a connected, undirected graph with a real- valued weight function w defined on E. • Let A be a subset of E that is included in some minimum spanning tree for G. • Let C be a connected component (tree) in the forest GA=(V,A). • Let (u,v) be a light edge (shortest among all edges connecting C with others components) connecting C to some other component in GA. • Then, edge (u,v) is safe for A. (For Kruskal’s algorithm) 9 The algorithms of Kruskal and Prim • The two algorithms are elaborations of the generic algorithm. • They each use a specific rule to determine a safe edge in line 3 of GENERIC_MST. • In Kruskal's algorithm, – The set A is a forest. – The safe edge added to A is always a least-weight edge in the graph that connects two distinct components. • In Prim's algorithm, – The set A forms a single tree. – The safe edge added to A is always a least-weight edge connecting the tree to a vertex not in the tree. 10 Kruskal's algorithm(basic part) 1 (Sort the edges in an increasing order) 2 A:={} 3 while E is not empty do { 3 take an edge (u, v) that is shortest in E and delete it from E 4 if u and v are in different components then add (u, v) to A } Note: each time a shortest edge in E is considered. 11 Kruskal's algorithm (Fun Part, not required) MST_KRUSKAL(G,w) 1 A:={} 2 for each vertex v in V[G] 3 do MAKE_SET(v) 4 sort the edges of E by nondecreasing weight w 5 for each edge (u,v) in E, in order by nondecreasing weight 6 do if FIND_SET(u) != FIND_SET(v) 7 then A:=A∪{(u,v)} 8 UNION(u,v) 9 return A (Disjoint set is discussed in Chapter 21, Page 498) 12 Disjoint-Set (Chapter 21, Page 498) • Keep a collection of sets S1, S2, .., Sk, – Each Si is a set, e,g, S1={v1, v2, v8}. • Three operations – Make-Set(x)-creates a new set whose only member is x. – Union(x, y) –unites the sets that contain x and y, say, Sx and Sy, into a new set that is the union of the two sets. – Find-Set(x)-returns a pointer to the representative of the set containing x. 13 • Our implementation uses a disjoint-set data structure to maintain several disjoint sets of elements. • Each set contains the vertices in a tree of the current forest. • The operation FIND_SET(u) returns a representative element from the set that contains u. • Thus, we can determine whether two vertices u and v belong to the same tree by testing whether FIND_SET(u) equals FIND_SET(v). • The combining of trees is accomplished by the UNION procedure. • Running time O(|E| lg (|E|)). (The analysis is not required.) 14 The execution of Kruskal's algorithm (Moderate part) •The edges are considered by the algorithm in sorted order by weight. •The edge under consideration at each step is shown with a red weight number. 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 15 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 16 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 17 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 18 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 19 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 20 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 21 Prim's algorithm(basic part) MST_PRIM(G,w,r) 1. A={} 2. S:={r} (r is an arbitrary node in V) 3. Q=V-{r}; 4. while Q is not empty do { 5 take an edge (u, v) such that (1) u S and v Q (v S ) and (u, v) is the shortest edge satisfying (1) 6 add (u, v) to A, add v to S and delete v from Q } 22 Prim's algorithm(Fun part, not required) MST_PRIM(G,w,r) 1 for each u in Q do 2 key[u]:=∞ 3 parent[u]:=NIL 4 key[r]:=0 5 QV[Q] 6 while Q!={} do 7 u:=EXTRACT_MIN(Q) 8 for each v in Adj[u] do 9 if v in Q and w(u,v)<key[v] 10 then parent[v]:=u 11 key[v]:=w(u,v) 23 • Grow the minimum spanning tree from the root vertex r. • Q is a priority queue, holding all vertices that are not in the tree now. • key[v] is the minimum weight of any edge connecting v to a vertex in the tree. • parent[v] names the parent of v in the tree. • When the algorithm terminates, Q is empty; the minimum spanning tree A for G is thus A={(v,parent[v]):v∈V-{r}}. • Running time: O(|E|+|V|lg |V|). (Analysis is not required) 24 The execution of Prim's algorithm(moderate part) 8 7 the root b c d 4 9 vertex 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 25 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 26 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 27 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 28 8 7 b c d 4 9 2 a 11 i 4 14 e 7 6 8 10 h g f 1 2 Bottleneck spanning tree: A spanning tree of G whose largest edge weight is minimum over all spanning trees of G. The value of the bottleneck spanning tree is the weight of the maximum-weight edge in T. Theorem: A minimum spanning tree is also a bottleneck spanning tree. 29 2-optimal Euler path problem 2-optimal Euler path problem 30 Euler circuit in a directed graph: • An Euler circuit in a directed graph is a directed circuit that visits every edge in G exactly once. b v4 v1 v3 d e a g c v2 v6 v5 f Figure 1 31 Theorem 2: • If all the vertices of a connected graph have equal in-degrees and out-degrees, then the graph has an Euler circuit. • in-degree-- number of edges pointing to the node • out-degree-- number of edges pointing out from the node. 32 2-path: • A 2-path is a directed subgraph consisting of two consecutive edges. For example: u v w • v is called the midpoint. • Every 2-path is given a cost (positive number) • A Euler circuit of m edges contains m 2-paths,and the cost of the Euler circuit is the total weight of the m 2-paths. 33 An Example: There are four 2-paths: b (a, b), (b, c), (c, d) a c and (d, a). If c(a,b)=1, c(b, c)=2, c(c,d)=3, and c(d, a)=3, d then the total cost of the four 2-paths is 1+2+3+3=9. 34 2-optimal Euler circuit problem: • Instance: A directed graph, each of its vertices has in- degree at most 2 and out-degree at most 2, and 2-path has a cost. • Question: Find an Euler circuit with the smallest cost among all possible Euler circuits. • Back to the example of the last slide. – If c(a,b)=1, c(b, c)=2, c(c,d)=3, and c(d, a)=3, the total cost of the four 2-paths is 1+2+3+3=9. 35 An Euler circuit: • An Euler circuit in the directed graph is: a, b, g, f, e, d, c, a. b v4 v1 v3 d e a g c v2 v6 v5 f Figure 1 The 2-paths are: (a, b), (b, g), (g, f), (f, e), (e,d), (d, c) and (c, a) 36 Line graph: • If we view each edge in the above graph G(Figure 1) as a vertex, and each 2-path in the above graph as an edge, we get another graph L(G), called line graph. b d e a c g f 37 Continue: • Observation: An Euler circuit in graph G corresponds to a Hamilton circuit in graph L(G). • 2-optimal Euler path problem becomes a TSP problem in line graph. • Theorem: TSP in line graph L(G), where G has in-degree at most 2 and out-degree at most 2, can be solved in polynomial time. (TSP in general is NP-complete) 38 Line graph: • In the line graph, the corresponding Hamilton circuit is: • a, b, g, f, e, d, c, a. Some edges in the line graph are not used, e.g., (b, c), and (d, g). b d e a c g f 39

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