MST. GENERIC-MST.G; w/ 1 A D; 2 while A does not form a spanning tree 3 findanedge.u; / that is safe for A 4 A D A [ f.

Transcription

1 Greedy Algorithm

2 Greedy Algorithm A greedy algorithm always makes the choice that looks best at the moment. That is, it makes a locally op:mal choice in the hope that this choice will lead to a globally op:mal solu:on.

3 MST GENERIC-MST.G; w/ 1 A D; 2 while A does not form a spanning tree 3 findanedge.u; / that is safe for A 4 A D A [ f.u; /g 5 return A We use the loop invariant as follows: 4 a 8 b 8 7 c 2 11 i h g f 1 2 d 9 10 e

4 Matroid A matroid is an ordered pair M D.S; / satisfying the following conditions. 1. S is a finite set. 2. is a nonempty family of subsets of S, calledtheindependent subsets of S, such that if B 2 and A B, thena 2.Wesaythat is hereditary if it satisfies this property. Note that the empty set ; is necessarily a member of. 3. If A 2, B 2,andjAj < jbj, thenthereexistssomeelementx 2 B A such that A [ fxg 2.WesaythatM satisfies the exchange property. The word matroid is due to Hassler Whitney. He was studying matric ma-

5 Graphic Matroid asks you to show, this structure defines a matroid. As another example of matroids, consider the graphic matroid M G D.S G ; G / defined in terms of a given undirected graph G D.V; E/ as follows: The set S G is defined to be E,thesetofedgesofG. If A is a subset of E, thena 2 G if and only if A is acyclic. That is, a set of edges A is independent if and only if the subgraph G A D.V; A/ forms a forest. Thegraphic A matroid matroid is an ordered is closely pair M related D.S; to / the satisfying minimum-spanning-tree the following conditions. problem, 1. S is a finite set. 2. is a nonempty family of subsets of S, calledtheindependent subsets of S, such that if B 2 and A B, thena 2.Wesaythat is hereditary if it satisfies this property. Note that the empty set ; is necessarily a member of. 3. If A 2, B 2,andjAj < jbj, thenthereexistssomeelementx 2 B A such that A [ fxg 2.WesaythatM satisfies the exchange property. The word matroid is due to Hassler Whitney. He was studying matric ma-

6 Matroid Homework: Prove the exchange property for the graphical matroid

7 Matroid Homework: Prove or Read (Pag. 438) the exchange property for the graphical matroid

8 Weighted Matroid j j is called a spanning tree of. We say that a matroid M D.S; / is weighted if it is associated with a weight function w that assigns a strictly positive weight w.x/ to each element x 2 S. The weight function w extends to subsets of S by summation: w.a/ D X x2a w.x/ for any A S. Forexample,ifweletw.e/ denote the weight of an edge e in a graphic matroid M G,thenw.A/ is the total weight of the edges in edge set A. Greedy algorithms on a weighted matroid

9 [ f g Greedy GREEDY.M; w/ 1 A D; 2 sort M:S into monotonically decreasing order by weight w 3 for each x 2 M:S, takeninmonotonicallydecreasingorderbyweightw.x/ 4 if A [ fxg 2 M: 5 A D A [ fxg 6 return A Line 4 checks whether adding each element to would maintain as an inde- A matroid is an ordered pair M D.S; / satisfying the following conditions. 1. S is a finite set. 2. is a nonempty family of subsets of S, calledtheindependent subsets of S, such that if B 2 and A B, thena 2.Wesaythat is hereditary if it satisfies this property. Note that the empty set ; is necessarily a member of. 3. If A 2, B 2,andjAj < jbj, thenthereexistssomeelementx 2 B A such that A [ fxg 2.WesaythatM satisfies the exchange property. The word matroid is due to Hassler Whitney. He was studying matric ma-

10 Task Scheduling Problem (1 cpu) the second task begins at time and finishes at time,andsoon. The problem of scheduling unit-time tasks with deadlines and penalties for a single processor has the following inputs: asets D fa 1 ;a 2 ;:::;a n g of n unit-time tasks; asetofn integer deadlines d 1 ;d 2 ;:::;d n,suchthateachd i satisfies 1 d i n and task a i is supposed to finish by time d i ;and asetofn nonnegative weights or penalties w 1 ;w 2 ;:::;w n,suchthatweincur apenaltyofw i if task a i is not finished by time d i,andweincurnopenaltyif ataskfinishesbyitsdeadline. We wish to find a schedule for that minimizes the total penalty incurred for Find a permuta:on of S (a schedule) minimizing the penal:es Task a i d i w i

11 Matroid Problems Union of Matroids (polynomial) Intersec:on of Matroids (2 polynomial, 3 NPhard) Matroid Par::oning, par::on the elements of a matroid into as few independent sets as possible (Polynomial)

12 Huffman Code HUFFMAN.C / 1 n D jc j 2 Q D C 3 for i D 1 to n 1 4 allocate anew node 5 :left D x D EXTRACT-MIN.Q/ 6 :right D y D EXTRACT-MIN.Q/ 7 :freq D x:freq C y:freq 8 INSERT.Q; / 9 return EXTRACT-MIN.Q/ // return the root of the tree For our example, Huffman s algorithm proceeds as shown in Figure Example f:5 e:9 c:12 b:13 d:16 a:45

13 Huffman Code (b) c:12 b:13 14 d:16 a:45 f:5 e:9 (c) 14 f:5 e:9 d:16 25 a:45 c:12 b:13 (d) 25 c:12 b: d:16 f:5 e:9 a:45 (e) a: c:12 b:13 14 d:16 f:5 e:9 (f) 100 a: c:12 b:13 14 d:16 f:5 e:9

14 Card Deck Card Deck

15 Frac:onal Knapsack