Chapter 9. Greedy Technique. Copyright 2007 Pearson Addison-Wesley. All rights reserved.

Transcription

1 Chpter 9 Greey Tehnique Copyright 2007 Person Aison-Wesley. All rights reserve.

2 Greey Tehnique Construts solution to n optimiztion prolem piee y piee through sequene of hoies tht re: fesile lolly optiml irrevole For some prolems, yiels n optiml solution for every instne. For most, oes not ut n e useful for fst pproximtions. 9-2

3 Applitions of the Greey Strtegy Optiml solutions: hnge mking for norml oin enomintions minimum spnning tree (MST) single-soure shortest pths simple sheuling prolems Huffmn oes Approximtions: trveling slesmn prolem (TSP) knpsk prolem other omintoril optimiztion prolems 9-3

4 Chnge-Mking Prolem Given unlimite mounts of oins of enomintions 1 > > m, give hnge for mount n with the lest numer of oins Exmple: 1 = 25, 2 =10, 3 = 5, 4 = 1 n n = 48 Greey solution: Greey solution is optiml for ny mount n norml set of enomintions my not e optiml for ritrry oin enomintions 9-4

5 Minimum Spnning Tree (MST) Spnning tree of onnete grph G: : onnete yli sugrph of G tht inlues ll of G s verties Minimum spnning tree of weighte, onnete grph G: : spnning tree of G of minimum totl weight Exmple:

6 Prim s MST lgorithm Strt with tree T 1 onsisting of one (ny) vertex n grow tree one vertex t time to proue MST through series of expning sutrees T 1, T 2,, T n On eh itertion, onstrut T i+1 from T i y ing vertex not in T i tht is losest to those lrey in T i (this is greey step!) Stop when ll verties re inlue 9-6

7 Exmple

8 Notes out Prim s lgorithm Proof y inution tht this onstrution tully yiels MST Nees priority queue for loting losest fringe vertex Effiieny O(n 2 ) for weight mtrix representtion of grph n rry implementtion of priority queue O(m log n) ) for jeny list representtion of grph with n verties n m eges n min-hep implementtion of priority queue 9-8

9 Another greey lgorithm for MST: Kruskl s Sort the eges in noneresing orer of lengths Grow tree one ege t time to proue MST through series of expning forests F 1, F 2,, F n-1 On eh itertion, the next ege on the sorte list unless this woul rete yle. (If it woul, skip the ege.) 9-9

10 Exmple

11 Notes out Kruskl s lgorithm Algorithm looks esier thn Prim s ut is hrer to implement (heking for yles!) Cyle heking: yle is rete iff e ege onnets verties in the sme onnete omponent Union-fin lgorithms see setion

12 Minimum spnning tree vs. Steiner tree vs

13 Shortest pths Dijkstr s lgorithm Single Soure Shortest Pths Prolem: : Given weighte onnete grph G, fin shortest pths from soure vertex s to eh of the other verties Dijkstr s lgorithm: : Similr to Prim s MST lgorithm, with ifferent wy of omputing numeril lels: Among verties not lrey in the tree, it fins vertex u with the smllest sum v + w(v,u) where v is vertex for whih shortest pth hs een lrey foun on preeing itertions (suh verties form tree) v is the length of the shortest pth form soure to v w(v,u) ) is the length (weight) of ege from v to u 9-13

14 Exmple e Tree verties (-,0) Remining verties (,3) (-, ) (,7) e(-, ) e (,3) (,3+4) (,3+2) e(-, ) e (,5) (,7) e(,5+4) e (,7) e(,9) e(,9) e

15 Notes on Dijkstr s lgorithm Doesn t work for grphs with negtive weights Applile to oth unirete n irete grphs Effiieny O( V 2 ) for grphs represente y weight mtrix n rry implementtion of priority queue O( E log V ) for grphs represente y j. lists n min- hep implementtion of priority queue Don t mix up Dijkstr s lgorithm with Prim s lgorithm! 9-15

16 Coing Prolem Coing: : ssignment of it strings to lphet hrters Coewors: : it strings ssigne for hrters of lphet Two types of oes: fixe-length enoing (e.g., ASCII) vrile-length enoing (e,g., Morse oe) Prefix-free oes: : no oewor is prefix of nother oewor Prolem: If frequenies of the hrter ourrenes re known, wht is the est inry prefix-free oe? 9-16

17 Huffmn oes Any inry tree with eges lele with 0 s n 1 s yiels prefix-free oe of hrters ssigne to its leves Optiml inry tree minimizing the expete (weighte verge) length of oewor n e onstrute s follows Huffmn s lgorithm Initilize n one-noe trees with lphet hrters n the tree weights with their frequenies. Repet the following step n-1 times: join two inry trees with smllest weights into one (s left n right sutrees) n mke its weight equl the sum of the weights of the two trees. Mrk eges leing to left n right sutrees with 0 s n 1 s, respetively. 9-17

18 Exmple B _ C D B _ C D A A B _ A C D C 0.2 D A hrter A B C D _ frequeny B _ C D B _ A oewor verge its per hrter: 2.25 for fixe-length enoing: 3 ompression rtio: : (3-2.25)/3*100% = 25% 9-18