Algorithms: Minimum Spanning Trees
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1 Algorithms: Minimum Spanning Trees Amotz Bar-Noy CUNY Spring 0 Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
2 Minimum Spanning Trees (MST) Input: An undirected and connected graph G = (V, E) with n vertices and m edges. A weight function w : E R on the edges. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
3 Minimum Spanning Trees (MST) Input: An undirected and connected graph G = (V, E) with n vertices and m edges. A weight function w : E R on the edges. A spanning tree: A connected sub-graph with n edges. A minimum spanning tree (MST): A spanning tree for which the sum of the weights of the n edges is minimum. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
4 Minimum Spanning Trees (MST) Input: An undirected and connected graph G = (V, E) with n vertices and m edges. A weight function w : E R on the edges. A spanning tree: A connected sub-graph with n edges. A minimum spanning tree (MST): A spanning tree for which the sum of the weights of the n edges is minimum. Two greedy algorithms: Kruskal - O(m log m)-time, a distributed algorithm. Prim - O(m + n log n)-time, a centralized algorithm. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
5 Example: A weighted Graph A D F 3 E 5 C 3 B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
6 Example: A BFS Spanning Tree A D F 3 E 5 C 3 W (T ) = B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
7 Example: A DFS Spanning Tree A D F 3 E 5 C 3 W (T ) = B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
8 Example: An MST A D F 3 E 5 C 3 W (T ) = 3 B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
9 Example: Another MST A D F 3 E 5 C 3 W (T ) = 3 B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 7 / 59
10 Cuts A cut S, (V S) in a graph is a partition of the set of vertices V into two disjoint sets: V = S (V S). An edge (u, v) crosses a cut S, (V S) if (u S & v (V S)) or (v S & u (V S)). An edge (u, v) is contained in a cut S, (V S) if (u, v) S or (u, v) (V S). A set A of edges is contained in a cut S, (V S) if all of the edges of A are contained in the cut. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 8 / 59
11 Example: A Cut A D F E C B An {ABDE}, {CF} cut Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 9 / 59
12 Example: Another Cut A D F E C An {AEF}, {BCD} cut B Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 0 / 59
13 Minimal Forests A set A of edges is a minimal forest if it is possible to add edges to A to become a minimal spanning tree. An empty set is in particular a minimal forest. A minimum spanning tree is in particular a minimal forest. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
14 A Minimal Forest A weighted graph: A B C D Minimal forests: A A B C B C D D Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
15 A Minimal Forest A weighted graph: A B C D Not a minimal forest: A B C D Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
16 The Greedy Lemma Input: A weighted connected graph G = (V, E). A minimal forest A E. A cut S, (V S) that contains A. An edge (u, v) crossing S, (V S) with minimum weight. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
17 The Greedy Lemma Input: A weighted connected graph G = (V, E). A minimal forest A E. A cut S, (V S) that contains A. An edge (u, v) crossing S, (V S) with minimum weight. Lemma: A {(u, v)} is also a minimal forest. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
18 Proof x u v y A vertex in S A vertex in V S An edge in A An edge in T A An edge not in T An MST T that contains A and (u, v) / T. (x, y) T crossing S, (V S) w(u, v) w(x, y). T = T {(x, y)} {(u, v)} T is also an MST. A {(u, v)} is a minimal forest. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
19 A Schematic Greedy Algorithm () A = () while A < n do (3) find (u, v) s.t. A {(u, v)} is a minimal forest () A = A {(u, v)} (5) return(a) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
20 A Schematic Greedy Algorithm () A = () while A < n do (3) find (u, v) s.t. A {(u, v)} is a minimal forest () A = A {(u, v)} (5) return(a) Correctness: Due to the lemma, step (3) is always possible. By definition, A is always a minimal forest. At the end A has n edges. A forest with n edges is a tree. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
21 A Schematic Greedy Algorithm () A = () while A < n do (3) find (u, v) s.t. A {(u, v)} is a minimal forest () A = A {(u, v)} (5) return(a) Correctness: Due to the lemma, step (3) is always possible. By definition, A is always a minimal forest. At the end A has n edges. A forest with n edges is a tree. Complexity: Depends on the implementation of step (3). Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
22 Kruskal Algorithm Sort the edges from the lightest to the heaviest. Consider the edges following this order. Start with an empty minimal forest. Add an edge to the minimal forest if it doesn t close a cycle. Terminate when there are n edges in the minimal forest. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 7 / 59
23 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 8 / 59
24 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 9 / 59
25 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 0 / 59
26 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
27 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
28 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
29 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
30 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
31 Example: Kruskal Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
32 Kruskal Algorithm Data Structure A collection S of disjoint sets of vertices: {S, S,..., S k } S i S j = for all i j k. Initially, the collection is empty: S =. At the end S contains one set of all the vertices: S = {V }. Make-Set(x): Creates a new set containing only x and adds it to the collection S: S = S {{x}}. Find-Set(x): Finds the set in the collection S that contains x: S i S such that x S i. Union(S i, S j ): replaces in the collection S the two sets S i and S j with their union: S = S { S i, S j } { Si S j }. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 7 / 59
33 Kruskal Algorithm Code Variables: A: A minimal forest. Each tree in A is represented by a set of vertices. Algorithm: () A = () for all v V do Make-Set(v) (3) Sort(E) all edges from min to max () for each edge (u, v) in sorted order do (5) if Find-Set(u) Find-Set(v) then (6) A = A {(u, v)} (7) Union(Find-Set(u),Find-Set(v)) (8) return (A) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 8 / 59
34 Kruskal Algorithm Correctness Assume to the contrary that the output set A is not an MST. Let (u, v) be the first edge that was added to A such that A {(u, v)} is not a minimal forest. In particular, at that time, A is a minimal forest. Let C = S, (V S) be a cut for the set u S. (u, v) crosses C since v belongs to another set. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 9 / 59
35 Kruskal Algorithm Correctness Any lighter edge (x, y) is contained in C: If (x, y) A, then both x and y belong to the same set before (x, y) was examined. If (x, y) A, then both x and y belong to the same set after (x, y) was examined. Sorting (u, v) is a minimal crossing edge for the cut C. Greedy lemma A {(u, v)} is a minimal forest. A contradiction. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 30 / 59
36 Kruskal Algorithm Complexity Sorting complexity: O(m log m). Set operations complexity: There are n Make-Set operations. There are O(m) Find-Set operations. There are n Union operations. Possible to implement in time: O(m α(m, n)). α(m, n) is the inverse of the Ackerman s function. The α(m, n) function grows very slowly. For example, m, n 0 80 α(m, n). Overall complexity: O(m log m). Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
37 The Ackerman s Function n for k = 0 and n 0 A k (n) = A k () for k and n = A k (A k (n )) for k and n Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
38 The Ackerman s Function n for k = 0 and n 0 A k (n) = A k () for k and n = A k (A k (n )) for k and n A 0 (n) = + + = n The multiply-by- function. A (n) = = n The power-of- function. A (n) =... With n s, the tower-of- function. Each recursive level does the previous level s operation n times. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
39 The Ackerman s Function n for k = 0 and n 0 A k (n) = A k () for k and n = A k (A k (n )) for k and n A 0 (n) = + + = n The multiply-by- function. A (n) = = n The power-of- function. A (n) =... With n s, the tower-of- function. Each recursive level does the previous level s operation n times. A () = = 6 = Already A 3 () and A () must be extremely large! Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
40 Very Slow Growing Functions log n the inverse of the tower-of- function is the least x such that... x times is greater or equal to n. For example, log ( ) = 5. α(n) the inverse Ackerman s function is the least x such that A x (x) n. α(n) is much slower than log n. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 33 / 59
41 Kruskal Algorithm A Simple Implementation A set is represented by the following linked list: Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
42 Kruskal Algorithm A Simple Implementation A set is represented by the following linked list: Set Name Set Size Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
43 Kruskal Algorithm A Simple Implementation The head of the set contains two fields: the name and the size of the set. The head of the set has two pointers: to the head and the tail of a linked list of vertices. An array of n vertices each has two pointers: to the head of its set and to the next vertex in its linked list. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 35 / 59
44 Kruskal Algorithm A Simple Implementation Make-Set(x) O() complexity. S x Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 36 / 59
45 Kruskal Algorithm A Simple Implementation Find-Set(x) O() complexity. S s x Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 37 / 59
46 Kruskal Algorithm A Simple Implementation Union(R, S) O(s) complexity. R r a b S s c d RS r+s a b c d Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 38 / 59
47 The Union Operation Implementation Worst case complexity: Connect the larger set to the smaller set. Consider the n Union operations: Union(S, S ), Union(S 3, S ),..., Union(S n, S n ) The cost of Union(S i+, S i ) is Ω(i). Total cost for the Union operations: Ω() + Ω() + + Ω(n ) = Ω(n ). Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 39 / 59
48 The Union Operation Implementation Worst case complexity: Connect the larger set to the smaller set. Consider the n Union operations: Union(S, S ), Union(S 3, S ),..., Union(S n, S n ) The cost of Union(S i+, S i ) is Ω(i). Total cost for the Union operations: Ω() + Ω() + + Ω(n ) = Ω(n ). Modification: Connect the smaller set to the larger set. The pointer of each vertex is changed at most log n times, since after each Union operation the pointer points to a set whose size is at least twice the size of the previous set. All together, for the n Union operations, for all vertices, O(n log n) complexity. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 39 / 59
49 Implementation Complexity n Union operations: O(n log n). n Make-Set operations: n Θ() = Θ(n). O(m) Find-Set operations: O(m) Θ() = Θ(m). Sorting complexity: Θ(m log m) = Θ(m log n). Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 0 / 59
50 Implementation Complexity n Union operations: O(n log n). n Make-Set operations: n Θ() = Θ(n). O(m) Find-Set operations: O(m) Θ() = Θ(m). Sorting complexity: Θ(m log m) = Θ(m log n). Kruskal algorithm complexity: Θ(m log m). Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 0 / 59
51 Prim Algorithm Start with an arbitrary vertex as a singleton tree. Maintain a minimal forest initially set to be this tree. Find the closest vertex to the minimal forest. Add this vertex and its closest edge to the minimal forest. Continue until all vertices are in the minimal forest. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
52 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
53 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 3 / 59
54 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 / 59
55 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
56 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 6 / 59
57 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 7 / 59
58 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 8 / 59
59 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 9 / 59
60 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 50 / 59
61 Example: Prim Algorithm B 8 7 C D A I 9 E H G F Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
62 Prim Algorithm Implementing the Greedy Idea Data structure: A minimal tree (forest) A that will be the MST. A starting vertex r that is the root of the MST. A priority queue Q for vertices not yet in A. A distance key(v) from A for vertices in Q. A candidate edge (v, Π(v)) for any vertex v in Q. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
63 Prim Algorithm Implementing the Greedy Idea Data structure: A minimal tree (forest) A that will be the MST. A starting vertex r that is the root of the MST. A priority queue Q for vertices not yet in A. A distance key(v) from A for vertices in Q. A candidate edge (v, Π(v)) for any vertex v in Q. The greedy idea: Repeat adding to A the edge (u, Π(u)) for the vertex u that has the minimum value for key( ) in Q. Update, if necessary, the distance key(v) and the candidate edge (v, Π(v)) for each neighbor v of u that is still in Q. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
64 Prim Algorithm Code () initialize Π(r) = nil; A = ; Q = V {r} () for all u Q do key(u) = (3) u = r () Repeat (5) for each neighbor v of u do (6) if (v Q) and w(u, v) < key(v) then (7) key(v) = w(u, v); Π(v) = u (8) u =Extract-Min(Q) (9) A = A {(u, Π(u))} (0) until Q = () return (A) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 53 / 59
65 Prim Algorithm Correctness Assume to the contrary that the output set A is not an MST. Let (u, v) be the first edge that was added to A such that A {(u, v)} is not a minimal forest. In particular, at that time, A is a minimal forest. Let C = Q, (V Q) be a cut. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
66 Prim Algorithm Correctness Assume to the contrary that the output set A is not an MST. Let (u, v) be the first edge that was added to A such that A {(u, v)} is not a minimal forest. In particular, at that time, A is a minimal forest. Let C = Q, (V Q) be a cut. The algorithm guarantees that C contains A. The priority queue implies that (u, v) is a minimal crossing edge for the cut C. Greedy lemma A {(u, v)} is a minimal forest. A contradiction. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 5 / 59
67 Prim Algorithm Complexity I Queue operations: One time building a priority queue. n times the operation Extract-Min. At most m times updating a value of a key. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 55 / 59
68 Prim Algorithm Complexity I Queue operations: One time building a priority queue. n times the operation Extract-Min. At most m times updating a value of a key. An implementation with an unsorted array: Θ(n) to build a queue. Θ(n) for the Extract-Min operation. Θ() for the Update-Queue operation. Unsorted array total complexity: Θ(n) + (n ) Θ(n) + Θ(m) Θ() = Θ(n ) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 55 / 59
69 Prim Algorithm Complexity II Queue operations: One time building a priority queue. n times the operation Extract-Min. At most m times updating a value of a key. An implementation with a sorted array: Θ(n) to build a queue. Θ() for the Extract-Min operation. Θ(n) for the Update-Queue operation. Sorted array total complexity: Θ(n) + (n ) Θ() + Θ(m) Θ(n) = Θ(nm) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 56 / 59
70 Prim Algorithm Complexity III Queue operations: One time building a priority queue. n times the operation Extract-Min. At most m times updating a value of a key. An implementation with a heap: Θ(n) to build a queue. Θ(log n) for the Extract-Min operation. Θ(log n) for the Update-Queue operation. Heap total complexity: Θ(n) + (n ) Θ(log n) + Θ(m) Θ(log n) = Θ(m log n) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 57 / 59
71 Prim Algorithm Complexity IV Queue operations: One time building a priority queue. n times the operation Extract-Min. At most m times updating a value of a key. An implementation with a Fibonacci heap: Θ(n) to build a queue. Θ(log n) for the Extract-Min operation. Θ() (amortized) for the Update-Queue operation. Fibonacci heap total complexity: Θ(n) + (n ) Θ(log n) + Θ(m) Θ() = Θ(m + n log n) Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 58 / 59
72 Kruskal vs. Prim Complexity: Kruskal is an O(m log m) algorithm. Prim is an O(m + n log n) algorithm. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 59 / 59
73 Kruskal vs. Prim Complexity: Kruskal is an O(m log m) algorithm. Prim is an O(m + n log n) algorithm. Implementation: Kruskal is a distributed algorithm. Prim is a centralized algorithm. Amotz Bar-Noy (CUNY) Minimum Spanning Trees Spring 0 59 / 59
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