Minimum Spanning Trees

Transcription

1 Miimum Spaig Trees

2 Miimum Spaig Trees Spaig subgraph Subgraph of a graph G cotaiig all the vertices of G Spaig tree Spaig subgraph that is itself a (free) tree Miimum spaig tree (MST) Spaig tree of a weighted graph with miimum total edge weight Applicatios ommuicatios etworks Trasportatio etworks A B 1 D 10 G E F 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

3 ycle Property ycle Property: Let T be a miimum spaig tree of a weighted graph G Let e be a edge of G that is ot i T ad let be the cycle formed by e with T f e For every edge f of, weight(f) weight(e) Proof: By cotradictio Replacig f with e yields a better spaig tree If weight(f) > weight(e) we ca get a spaig tree of smaller weight by replacig e with f f e 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

4 uts A cut i G is a partitio of the vertex set ito two parts U c Example: a w U- {a, e, b} w V {c, d, f} e Number of possible cuts w -1-1 Edges i the cut edges that coect vertices of oe partitio to vertices of the other partitio. b d V f Miimum Spaig Trees

5 Partitio Property Partitio Property: osider a partitio of the vertices of G ito subsets U ad V Let e be a edge of miimum weight across the partitio There is a miimum spaig tree of G cotaiig edge e Proof: Let T be a MST of G If T does ot cotai e, cosider the cycle formed by e with T ad let f be a edge of across the partitio By the cycle property, weight(f) weight(e) Thus, weight(f) = weight(e) We obtai aother MST by replacig f with e U U f e f e V Replacig f with e yields aother MST V 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

6 Prim-Jarik s Priciple Similar to Dijkstra s algorithm Utilizes the Partitio Property We pick a arbitrary vertex s ad we grow the MST as a cloud of vertices, startig from s A B D E F Miimum Spaig Trees

7 Prim-Jarik s Algorithm - Implemetatio Similar to Dijkstra s algorithm We pick a arbitrary vertex s ad we grow the MST as a cloud of vertices, startig from s We store with each vertex v label d(v) represetig the smallest weight of a edge coectig v to a vertex i the cloud At each step: We add to the cloud the vertex u outside the cloud with the smallest distace label We update the labels of the vertices adjacet to u 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

8 Example Miimum Spaig Trees B D A F E 0 B D A F E 0 B D A F E 0 B D A F E 0

9 Example (cotd.) A 0 B D F E A 0 B D F E Miimum Spaig Trees

10 Prim-Jarik Pseudo-code 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 10

11 Aalysis Graph operatios We cycle through the icidet edges oce for each vertex Label operatios We set/get the distace, paret ad locator labels of vertex z O(deg(z)) times Settig/gettig a label takes O(1) time Priority ueue operatios Each vertex is iserted oce ito ad removed oce from the priority ueue, where each isertio or removal takes O(log ) time The key of a vertex w i the priority ueue is modified at most deg(w) times, where each key chage takes O(log ) time Prim-Jarik s algorithm rus i O(( + m) log ) time provided the graph is represeted by the adjacecy list structure Recall that Σ v deg(v) = m The ruig time is O(m log ) sice the graph is coected 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 11

12 Kruskal s Priciple Greedy Algorithm greedily add the edge with miimum weight to the MST esure that the added edge does ot form a cycle 1 A B 11 E D G H F 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 1

13 Kruskal s Algorithm Maitai a partitio of the vertices ito clusters Iitially, sigle-vertex clusters Keep a MST for each cluster Merge closest clusters ad their MSTs A priority ueue stores the edges outside clusters Key: weight Elemet: edge At the ed of the algorithm Oe cluster ad oe MST 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 1

14 Example Miimum Spaig Trees 1 B G A F D 1 10 E H 11 B G A F D 1 10 E H 11 B G A F D 1 10 E H 11 B G A F D 1 10 E H 11

15 Example (cotd.) Miimum Spaig Trees 1 B G A F D 1 10 E H 11 B G A F D 1 10 E H 11 B G A F D 1 10 E H 11 four steps B G A F D 1 10 E H 11

16 Kruskal s Algorithm Miimum Spaig Trees 1

17 orrectess of Kruskal s Algorithm Let all the edges i the graph have distict weights Let the tree obtaied from Kruskal s algorithm cotai the edges g 1 <g <g <g -1 Let the Optimum MST have the edges f 1 <f <f...<f -1 Let i be the first edge that is differet betwee g s ad f s. w case1: g i <f i w case: f i <g i Merge Sort 1

18 Is the MST uiue? yes, if the edge weights are distict. o, otherwise. 01 Goodrich, Tamassia, Goldwasser Merge Sort 1

19 Data Structure for Kruskal s Algorithm The algorithm maitais a forest of trees A priority ueue extracts the edges by icreasig weight A edge is accepted it if coects distict trees We eed a data structure that maitais a partitio, i.e., a collectio of disjoit sets, with operatios: makeset(u): create a set cosistig of u fid(u): retur the set storig u uio(a, B): replace sets A ad B with their uio Total time O(m log + time for Uio* + time for Fid* m) 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 1

20 Kruskal s Algorithm Miimum Spaig Trees 0

21 Partitios with Uio-Fid Operatios makeset(x): reate a sigleto set cotaiig the elemet x ad retur the positio storig x i this set uio(a,b ): Retur the set A U B, destroyig the old A ad B fid(p): Retur the set cotaiig the elemet at positio p 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 1

22 List-based Implemetatio Each set is stored i a seuece represeted with a liked-list Each ode should store a object cotaiig the elemet ad a referece to the set ame 01 Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

23 Aalysis of List-based Represetatio Uio mergig of two liked lists costat time Fid search through the list O() Total time O(m log )+O()+O(m). Miimum Spaig Trees

24 Tree-based Implemetatio Each elemet is stored i a ode, which cotais a poiter to a set ame A ode v whose set poiter poits back to v is also a set ame Each set is a tree, rooted at a ode with a selfreferecig set poiter For example: The sets 1,, ad : Goodrich, Tamassia, Goldwasser Miimum Spaig Trees 1

25 Uio-Fid Operatios To do a uio, simply make the root of oe tree poit to the root of the other To do a fid, follow set-ame poiters from the startig ode util reachig a ode whose set-ame poiter refers back to itself Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

26 Uio-Fid Heuristic 1 Uio by size/rak: Whe performig a uio, make the root of smaller tree poit to the root of the larger Implies O( log ) time for performig uio-fid operatios: Each time we follow a poiter, we are goig to a subtree of size at least double the size of the previous subtree Thus, we will follow at most O(log ) poiters for ay fid Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

27 Uio-Fid Heuristic Path compressio: After performig a fid, compress all the poiters o the path just traversed so that they all poit to the root Goodrich, Tamassia, Goldwasser Miimum Spaig Trees

28 Ed Semester Exam Thursday, November, losed book, closed otes exam Format Similar to the mid-semester exam - uestios w Defiitios/Examples w Algorithm Desig w Data Structure Desig w Data Structure Aalysis Write the solutio i the space provided i the aswer sheet. additioal sheets will be provided durig the exam. 01 Goodrich, Tamassia, Goldwasser Merge Sort

29 Searchig ad Search Tree Structures (chapter 11) Biary Search Trees isertio ad deletio operatios Balaced Search Trees AVL Trees operatios Red-Black Trees operatios (,) Trees multi-way search trees Splay Trees Splayig operatio

30 Strig Processig ad Dyamic Programmig (hapter 1) Patter Matchig Algorithms The Boyer-Moore Algorithm w Bad character ad good suffix rules The Kuth-Morris-Pratt algorithm w Failure fuctios Tries Stadard Tries ompressed Tries Suffix Tries w Prefix Matchig Huffma odig (covered before mid-semester) Dyamic Programmig Methodology 0

31 Sortig (hapter 1) Divide ad ouer Paradigm Merge Sort algorithm complexity aalysis Quick Sort/Radomized Quick Sort algorithm complexity aalysis I-Place partitioig Radix Sort Radix Exchage Sort Straight Radix Sort w Stable way Bucket sort 1

32 Graphs (hapter 1) Graphs Termiology Data Structures for represetig graphs edge list, adjacecy list, adjacecy map, adjacecy matrix Graph Traversals BFS, DFS Directed Graphs Trasitive losure Directed Acyclic Graphs topological order Shortest Paths weighted graphs, Dijkstra s algorithm, Bellma-ford algorithm Miimum Spaig Trees Prim-Jarik Algorithm, Kruskal s Algorithm, Data structure for Kruskal s algorithm.

33 Labs ad Quizzes - Lab - hopefully before this weeked - Lab - before the exam for sure. - Quiz - hopefully before this weeked - Quiz (tomorrow) - before this weeked

34 Good Luck