CSC 1700 Analysis of Algorithms: Minimum Spanning Tree

Transcription

1 CSC 1700 Analysis of Algorithms: Minimum Spanning Tree Professor Henry Carter Fall 2016

2 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic programming achieves this by saving the result of overlapping subproblems Can be executed bottom-up or top-down (using memory functions) Practical applications include transitive closure and allpairs shortest-path 2

3 Greedy Algorithms Global optimization that makes a series of locally optimal choices Three requirements at each iteration: Feasible Locally optimal Irrevocable Recall: change making Other examples? 3

4 Proofs of Optimality Induction Other iterative approaches cannot do better Show the output is always optimal 4

5 Minimum Spanning Tree Given: a weighted, undirected, connected graph Find: a subgraph that connects all vertices with the cheapest edge weights Applications: network infrastructure, data set clustering, approximation algorithms 5

6 Greedy Approaches How could we greedily add vertices to the MST? How could we greedily add edges to the MST? 6

7 Prim s Algorithm Greedy approach to adding vertices Maintains three sets: vertices in the MST, edges in the MST, vertices outside the MST Iterate: add the vertex outside the MST with the cheapest connection 7

8 Example b 1 d a 5 c 5 e f 8

9 Proof of Optimality 9

10 Prim s Algorithm Prim(G) input : A weighted connected graph G = hv,ei. output: E T, the set of edges composing a minimum spanning tree of G. V T {v 0 } E T ; for i 1 to V 1 do Find a minimum-weight edge e =(v,u ) among all the edges (v, u) such that v is in V T and u is in V V T V T V T [ {u } E T E T [ {e } end return E T 10

11 Analysis Main loop: V Unordered list priority queue: V 2 Min-heap: log V Total: O( E log V ) 11

12 Kruskal s Algorithm Greedy approach to edges Maintains one set: edges in the MST (and counters for number of edges considered and number of edges in the MST) Add the minimum cost edge that does NOT form a cycle 12

13 Example b 1 d a 5 c 5 e f 13

14 Kruskal s Algorithm Kruskal(G) input : A weighted connected graph G = hv,ei. output: E T, the set of edges composing a minimum spanning tree of G. Sort E in nondecreasing order of the edge weights w(e i1 ) apple apple w(e i E ) E T ;; ecounter 0 k 0 while ecounter < V 1 do k k +1 if E T [ {e ik } is acyclic then E T E T [ {e ik }; ecounter ecounter +1 end end return E T 14

15 Analysis Worst case: E iterations Testing for cycles (i.e., searching for disconnected components): E log V Union Find Sorting edges by weight: O( E log E ) 15

16 Recap Greedy algorithms iterate locally optimal choices to construct a globally optimal solution Minimum spanning trees are subgraphs that connect all vertices with the cheapest edge weights Prim s and Kruskal s algorithms illustrate greedy design on adding vertices or edges 16

17 Practice Run Prim s algorithm on the following graph:

18 Next Time... Levitin Chapter 9.3 Remember, you need to read it BEFORE you come to class! Homework: 9.1: 3, 5, 9 9.2: 1, 2, 5 18