Minimum Spanning Trees My T. UF

Transcription

1 Introduction to Algorithms Minimum Spanning UF

2 Problem Find a low cost network connecting a set of locations Any pair of locations are connected There is no cycle Some applications: Communication networks Circuit design 2

3 Minimum Spanning Tree (MST) Problem Input: Undirected, connected graph G=(V, E), each edge (u, v) E has weight w(u, v) Output: acyclic subset T E that connects all of the vertices with minimum total weight w(t) = (u,v) T w(u,v) Bold edges form a Minimum Spanning Tree 3

4 Growing a minimum spanning tree Suppose A is a subset of some MST Iteratively add safe edge (u,v) s.t. A still a subset of some MST Generic algorithm: {(u,v)} is Key problem: How to find safe edges? Note: MST has V -1 edges 4

5 Some definitions A cut (S, V - S) is a partition of vertices into disjoint sets S and V - S An edge crosses the cut (S, V - S) if it has one end point in S, one end point in V - S A cut respects a set A of edges if and only if no edge in A crosses the cut, e.g. A is the set of bold edges 5

6 Some definitions An edge is a light edge crossing a cut if and only if its weight is minimum over all edges crossing the cut, e.g. edge (c, d) Observation: Any MST has at least one edge connect S and V S => one cross edge is safe for A 6

7 Find a safe edge Proof: Let T be a MST that includes A Case 1: (u, v) T => done. Case 2: (u, v) not in T: Exist edge (x, y) T cross the cut, (x, y) A Removing (x, y) breaks T into two components. Adding (u, v) reconnects 2 components T = T - {(x, y)} {(u, v)} is a spanning tree w(t ) = w(t) - w(x, y) + w(u, v) w(t) => T is a MST => done 7

8 Corollary In GENERIC-MST A is a forest containing connected components. Initially, each component is a single vertex. Any safe edge merges two of these components into one. Each component is a tree. 8

9 Kruskal s Algorithm Starts with each vertex in its own component Repeatedly merges two components into one by choosing a light edge that connects them (i.e., a light edge crossing the cut between them) Scans the set of edges in monotonically increasing order by weight. Uses a disjoint-set data structure to determine whether an edge connects vertices in different components 9

10 Disjoint-set data structure Maintain collection S = {S 1,..., S k } of disjoint dynamic (changing over time) sets Each set is identified by a representative, which is some member of the set Operations: MAKE-SET(x): make a new set S i = {x}, and add S i to S UNION(x, y): if x S x, y S y, then S S S x S y {S x S y } Representative of new set is any member of S x S y, often the representative of one of S x and S y. Destroys S x and S y (since sets must be disjoint). FIND-SET(x): return representative of set containing x In Kruskal s Algorithm, each set is a connected component 10

11 Pseudo code Running time: O(E lg V) ( First for loop: V MAKE-SETs Sort E: O(E lg E) - O(E lg V) is E is sorted) Second for loop: (o(e log V) (chapter 21) 11

12 Example 12

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14 Prim s Algorithm Builds one tree, so A always a tree Starts from an arbitrary root r At each step, find a light edge crossing cut (V A, V V A ), where V A = vertices that A is incident on. Add this edge to A. 14

15 Prim s Algorithm Uses a priority queue Q to find a light edge quickly Each object in Q is a vertex in V - V A Key of v is minimum weight of any edge (u, v), where u V A Then the vertex returned by Extract-Min is v such that there exists u V A and (u, v) is light edge crossing (V A, V V A ) Key of v is if v is not adjacent to any vertex in V A 15

16 Running time: O(E lgv) Using binary heaps to implement Q Initialization: O(V) Building initial queue : O(V) V Extract-Min s : O(V lgv) E Decrease-Key s : O(E lgv) Note: Using Fibonacci heaps can save time of Decrease-Key operations to constant (chapter 19) => running time: O(E + V lg V) 16

17 Example 17

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19 Summary MST T of connected undirect graph G = (V, E): Is a subgraph of G Connected Has V vertices, V -1 edges There is exactly 1 path between a pair of vertices Deleting any edge of T disconnects T Kruskal s algorithm connects disjoint sets of connects vertices until achieve a MST Run nearly linear time if E is sorted: 19

20 Summary Prim s algorithm starts from one vertex and iteratively add vertex one by one until achieve a MST Faster than Kruskal s algorithm if the graph is dense O(E + V lg V) vs O(E lg V) 20