Download presentation

Presentation is loading. Please wait.

Published byDasia Purington Modified about 1 year ago

1
Greedy Algorithms Pasi Fränti

2
Greedy algorithm 1.Coin problem 2.Minimum spanning tree 3.Generalized knapsack problem 4.Traveling salesman problem

3
Task: Given a coin set, pay the required amount (36 snt) using least number of coins. Coin problem

4
Coin problem Another coin set Amount to be paid: 30 Greedy: Optimal:

5
Blank space for notes

6
Minimum spanning tree When greedy works Needs problem definition! Tree = … (no cycles) Spanning Tree = … Minimum = … (couple of examples with simple graph)

7
Prim(V, E): RETURN T Select (u,v) E with min weight S S {u,v}; P P {(u,v)}; E E\{(u,v)}; REPEAT Select (u,v) with min weight (u,v) E, u S, v S S S {v}; P P {(u,v)}; E E\{(u,v)}; UNTIL S=V Return P; Minimum spanning tree Prim’s algorithm

8
Example of Prim

9

10
Proof of optimality General properties of spanning trees Spanning tree includes N-1 links There are no cycles Minimum spanning tree is the one with the smallest weights A B C A B C Remove Link BC Cycle No cycle

11
Proof of optimality Case: minimum link 2 A B C A B C 2 1 Link AB is minimum Suppose it is not in MST Path A → B must exist Add AB Adding AB we can remove another link (e.g. AC) Path A → C exists All nodes reached from C can now be reached from B

12
Proof of optimality Induction step A B C 4 3 E D Replace CD by CE MST solved for Subset S Suppose CE is minimum connecting S outside Path D → E must exist and is outside S

13
Proof of optimality Induction step A B C 4 3 E D MST solved for Subset S Path D → E still exist as before All nodes reachable via D can now be reached via C → D

14
Source of data just for fun

15
1. A // initially A is empty 2. for each vertex v V[G] // line 2-3 takes O(V) time 3. do Create-Set(v)// create set for each vertex 4. sort the edges of E by nondecreasing weight w 5. for each edge (u,v) E, in order by nondecreasing weight 6. do if Find-Set(u) Find-Set(v) // u&v on different trees 7. then A A {(u,v)} 8. Union(u,v) 9. return A Total running time is O(E lg E). Minimum spanning tree Kruskal’s algorithm Needs revisions: -Remove numbers -Change terminology

16
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

17
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

18
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

19
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

20
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

21
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

22
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

23
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

24
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

25
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

26
Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

27
Input:Weight of N items {w 1, w 2,..., w n } Cost of N items {c 1, c 2,..., c n } Knapsack limit S Output:Selection for knapsack: {x 1,x 2,…x n } where x i {0,1}. Sample input: w i ={1,1,2,4,12} c i = {1,2,2,10,4} S=15 Generalized Knapsack problem Problem definition

28
Will appear 2014… Generalized Knapsack problem

29
Semi-blank space

30
Traveling salesman problem GreedyTSP(V, E, home): RETURN T X[1] home; FOR i 1 TO N-1 DO Select (u,v) with min weight (u,v) E, u S, v S X[………….. S S {v}; …. UNTIL V≠{} Return something; Needs to be done

31
Greedy algorithms 1.Coin problem 2.Minimum spanning tree 3.Generalized knapsack problem 4.Traveling salesman problem Solved Not Solved Not

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google