计算模型与算法技术：9-Greedy Technique

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1、L o g oL o g o1Computational Model and Algorithmic TechniqueSouth China University of TechnologyDr.Han HuangL o g oL o g oCopyright 2007 Pearson Addison-Wesley.All rights reserved.L o g oL o g oGreedy TechniqueConstructs a solution to an optimization problem piece by piece through a sequence of choi

2、ces that are:vfeasiblevlocally optimalvirrevocableFor some problems,yields an optimal solution for every instance.For most,does not but can be useful for fast approximations.L o g oL o g oApplications of the Greedy StrategyvOptimal solutions:change making for“normal”coin denominations minimum spanni

3、ng tree(MST)single-source shortest paths simple scheduling problems Huffman codesvApproximations:traveling salesman problem(TSP)knapsack problem other combinatorial optimization problemsL o g oL o g oGreedy solution isvoptimal for any amount and“normal set of denominationsv may not be optimal for ar

4、bitrary coin denominationsGreedy TechniqueL o g oL o g oChange-Making ProblemGiven unlimited amounts of coins of denominations d1 dm,give change for amount n with the least number of coinsExample:d1=25c,d2=10c,d3=5c,d4=1c and n=48cL o g oL o g oMinimum Spanning Tree(MST)vSpanning tree of a connected

5、 graph G:a connected acyclic subgraph of G that includes all of Gs verticesvMinimum spanning tree of a weighted,connected graph G:a spanning tree of G of minimum total weightExample:cdba62431L o g oL o g oPrims MST algorithmvStart with tree T1 consisting of one(any)vertex and“grow”tree one vertex at

6、 a time to produce MST through a series of expanding subtrees T1,T2,TnvOn each iteration,construct Ti+1 from Ti by adding vertex not in Ti that is closest to those already in Ti(this is a“greedy”step!)vStop when all vertices are includedL o g oL o g oL o g oL o g oExample(graph ant its spanning tree

7、)bdca1523bdca123bdca153bdca152graphw(T1)=6w(T2)=9w(T3)=8T1 is the minimum spanning treeL o g oL o g oExample 1v1 C,D$700 2 C,E$800 v3 D,A$1200 4 A,B$900 vTotal:$3600 DCABE$900$1200$1300$1600$1000$700$800$2200$2000$1400L o g oL o g oExample 2v 1 b,f 1 v 2 a,b 2 v 3 f,j 2 v 4 a,e 3v 5 i,j 3v 6 f,g 3v

8、7 c,g 2v 8 c,d 1v 9 g,h 3v 10 h,l 3v 11 k,l 1abcefgkjidhl23143333112534243abcfgkjidhl21333112323eL o g oL o g oNotes about Prims algorithmvProof by induction that this construction actually yields MST vNeeds priority queue for locating closest fringe vertexvEfficiency O(n2)for weight matrix represen

9、tation of graph and array implementation of priority queue O(m log n)for adjacency list representation of graph with n vertices and m edges and min-heap implementation of priority queueL o g oL o g oAnother greedy algorithm for MST:KruskalsvSort the edges in nondecreasing order of lengthsv“Grow”tree

10、 one edge at a time to produce MST through a series of expanding forests F1,F2,Fn-1vOn each iteration,add the next edge on the sorted list unless this would create a cycle.(If it would,skip the edge.)L o g oL o g oL o g oL o g oExample 3v 1 c,d 1 v 2 k,l 1 v 3 b,f 1 v 4 c,g 2v 5 a,b 2v 6 f,j 2v 7 b,

11、c 3v 8 j,k 3v 9 g,h 3v 10 i,j 3v 11 a,e 4abcefgkjidhl231433331abcfgkjidhl2133311253424312323eL o g oL o g oExamplecdba42613cdba1cdba21cdba213L o g oL o g oNotes about Kruskals algorithmvAlgorithm looks easier than Prims but is harder to implement(checking for cycles!)vCycle checking:a cycle is creat

12、ed iff added edge connects vertices in the same connected componentvUnion-find algorithms see section 9.2L o g oL o g oShortest paths Dijkstras algorithmSingle Source Shortest Paths Problem:Given a weighted connected graph G,find shortest paths from source vertex sto each of the other verticesDijkst

13、ras algorithm:Similar to Prims MST algorithm,with a different way of computing numerical labels:Among vertices not already in the tree,it finds vertex u with the smallest sum dv+w(v,u)where v is a vertex for which shortest path has been already found on preceding iterations(such vertices form a tree

14、)dv is the length of the shortest path form source to v w(v,u)is the length(weight)of edge from v to u.L o g oL o g oL o g oL o g oExample 4vfor i:=1 to nv L(vi):=vL(a):=0 vS:=be0zacd4105826321L o g oL o g oExample 4vu:=a vertex not in S with L(u)minimalvS:=S U ube0zacd41058263214(a)2(a)L o g oL o g

15、 oExample 4vif L(u)+w(u,v)L(v)thenv L(v):=L(u)+w(u,v)be0zacd41058263213(a,c)2(a)10(a,c)12(a,c)L o g oL o g oExample 4vif L(u)+w(u,v)L(v)thenv L(v):=L(u)+w(u,v)be0zacd41058263213(a,c)2(a)8(a,c)12(a,c)L o g oL o g oExample 4vif L(u)+w(u,v)L(v)thenv L(v):=L(u)+w(u,v)be0zacd41058263213(a,c)2(a)8(a,c,b)1

16、0(a,c,b,d)14(a,c,b,d)L o g oL o g oExample 4vif L(u)+w(u,v)L(v)thenv L(v):=L(u)+w(u,v)be0zacd41058263213(a,c)2(a)8(a,c,b)10(a,c,b,d)13(a,c,b,d,e)L o g oL o g oExample 4vThe shortest path is a,c,b,d,e,z with length 13.be0zacd41058263213(a,c)2(a)8(a,c,b)10(a,c,b,d)13(a,c,b,d,e)L o g oL o g oExample d4

17、 a(-,0)b(a,3)c(-,)d(a,7)e(-,)ab4e37625cabd4ce374625abd4ce374625abd4ce374625 b(a,3)c(b,3+4)d(b,3+2)e(-,)d(b,5)c(b,7)e(d,5+4)c(b,7)e(d,9)e(d,9)d4abdce374625L o g oL o g oNotes on Dijkstras algorithmvDoesnt work for graphs with negative weightsvApplicable to both undirected and directed graphsvEfficien

18、cy O(|V|2)for graphs represented by weight matrix and array implementation of priority queue O(|E|log|V|)for graphs represented by adj.lists and min-heap implementation of priority queuevDont mix up Dijkstras algorithm with Prims algorithm!L o g oL o g oCoding ProblemCoding:assignment of bit strings

19、 to alphabet charactersCodewords:bit strings assigned for characters of alphabetTwo types of codes:vfixed-length encoding(e.g.,ASCII)vvariable-length encoding(e,g.,Morse code)Prefix-free codes:no codeword is a prefix of another codewordProblem:If frequencies of the character occurrences are known,wh

20、at is the best binary prefix-free code?L o g oL o g oHuffman codesvAny binary tree with edges labeled with 0s and 1s yields a prefix-free code of characters assigned to its leavesvOptimal binary tree minimizing the expected(weighted average)length of a codeword can be constructed as followsHuffmans

21、algorithmInitialize n one-node trees with alphabet characters and the tree weights with their frequencies.Repeat the following step n-1 times:join two binary trees with smallest weights into one(as left and right subtrees)and make its weight equal the sum of the weights of the two trees.Mark edges l

22、eading to left and right subtrees with 0s and 1s,respectively.L o g oL o g oExamplecharacter AB C D _frequency 0.35 0.1 0.2 0.2 0.15codeword 11 100 00 01 101average bits per character:2.25for fixed-length encoding:3compression ratio:(3-2.25)/3*100%=25%L o g oL o g o0.1B0.15_0.2C0.2D0.35A0.250.1B0.15_0.2C0.2D0.35A1.2.L o g oL o g o0.250.1B0.15_0.35A3.0.2C0.2D0.44.0.250.1B0.15_0.35A0.60.2C0.2D0.4L o g oL o g o5.0.250.1B0.15_0.35A0.60.2C0.2D0.41.010011100L o g oL o g o36C l i c k t o e d i t c o m p a n y s l o g a n .