gr0 GRAPHS Hanan Samet

Transcription

1 g0 GRPHS Hanan Samet ompute Science epatment and ente fo utomation Reseach and Institute fo dvanced ompute Studies Univesity of Mayland ollege Pak, Mayland opyight 1997 Hanan Samet These notes may not be epoduced by any means (mechanical o electonic o any othe) without the expess witten pemission of Hanan Samet

2 GRPH (G) g1 Genealization of a tee 1. no longe a distinguished node called the oot implies no need to distinguish between leaf and nonleaf nodes. two nodes can be linked by moe than one sequence of edges Fomally: set of vetices (V) and edges (E) joining them, with at most one edge joining any pai of vetices (V 0, V 1,, V n ): path of length n fom V 0 to V n (chain) Simple Path: distinct vetices (elementay chain) onnected: path between any two vetices of G ycle: simple path of length 3 fom V 0 to V 0 (length in tems of edges) Plana: egee: Isomophic: cuves intesect only at points of gaph numbe of edges intesecting at the node if thee is a one-to-one coespondence between nodes and edges of two gaphs E

3 SMPLE GRPH PROLEM g Given n people at a paty who shake hands, show that at the paty s end, an even numbe of people have shaken hands with an odd numbe of people Theoem: Fo any gaph G an even numbe of nodes have an odd degee Poof: 1. each edge joins nodes. each edge contibutes to the sum of degees 3. sum of degees is even 4. thus an even numbe of nodes with odd degee

4 FREE TREES g3 onnected gaph with no cycles Given G as a fee tee with n vetices 1. onnected, but not so if any edge is emoved. One simple path fom V to V ( V V ) 3. No cycles and n 1 edges 4. G is connected with n 1 edges E F G iffeences fom egula tees: 1. No identification of oot. No distinction between teminal and banch nodes

5 FREE SUTREES 3 z 81 b g4 efinition: set of edges such that all the vetices of the gaph ae connected to fom a fee tee Ex: distibution of telephone netwoks London pl l ln Pais p Rio de Janeio n New Yok pb b bn uenos ies Fee subtee Given: connected gaph G n nodes (5) m edges (8) yclomatic Numbe = numbe of edges that must be deleted to yield a fee tee ( = m n + 1 )

6 IRETE GRPH g5 efinition: gaph with diection attached to the edges (V 0, V 1,, V n ): Elementay path: path of length n fom V 0 to V n all vetices ae distinct icuit: cycle (but can have length 1 o ) Elementay icuit: all vetices ae distinct Indegee: Outdegee: Stongly onnected: path fom any V to any V Rooted: at least one V with paths to all V V Note: stongly connected implies ooted but not vice vesa

7 T STRUTURES FOR GRPHS g6 Must decide what infomation is to be accessible and with what ease Most impotant infomation conveyed by a gaph is connectivity which is indicated by its edges Two choices 1. vetex-based keeps tack of nodes connected to each node can implement as aay of lists a. one enty fo each vetex p b. [p] is a list of all vetices P that ae connected to p by vitue of the existence of an edge between p and q whee q P (also known as an adjacency list). edge-based keeps tack of edges usually epesented as a list of pais of fom (p q) whee thee is an edge between vetices p and q dawback: need to seach entie set to detemine edges connected to a paticula vetex

8 JENY MTRIX g7 Hybid appoach Good fo epesenting a diected gaph ij = 1 if an edge exists fom i to j ij = 0 othewise = adjacency matix of distance Somewhat wasteful of space as thee is an enty fo evey possible edge even though the aay is usually spase 1. in such cases, a vetex-based epesentation such as an adjacency list is moe economical. adjacency matix is useful if want to detect if an edge exists between two vetices cumbesome when using a list as need to seach Useful if want to keep tack of all vetices eachable fom evey vetex Ex: Leftmost deivations b d c oolean matices = = = = 0 1 = 0 0 = = 0 n ycle of length n ii = 1 b c d b c d

9 ONNETE GRPH g8 ef: thee exists path between any two vetices of the gaph Ex: binay image image gaph image elements ae vetices hoizontal and vetical adjacencies between image elements ae edges. connected component labeling: detemine sepaate egions of binay image image gaph is stoed implicitly easy to access adjacent vetices given location of a vetex neithe a vetex-based o edge-based epesentation; instead algoithms ae based on them 1. vetex-based implies need to follow connectivity depth-fist o seed-filling appoach many page faults if disk-esident data. edge-based detemines edges by examining image ow-by-ow only need to access two ows simultaneously good fo disk-esident data 3. both take O(numbe of image elements) time 19 0

10 MINIMUM SPNNING TREE 7 z 6 5 v 4 g 3 z 81 b g9 ost ij associated with each edge fom i to j Find the fee subtee of G with minimum cost Solution: = connected nodes: initially { } U = unconnected nodes: initially { all nodes } 1. choose abitay node and place it in. select node in U that is closest to a node in and add edge; move node fom U to ; epeat until U is empty Ex: E 5 7 T Stat with node 0 is built by choosing: E T 0 5 T E

11 SHORTEST ELEMENTRY HIN 7 z 6 5 v 4 g 3 z 81 b g10 Given node X0 in G find the shotest (cheapest) chain joining X0 with all the nodes of G Solution: = connected nodes: initially X0 U = unconnected nodes: initially all but X0 E = set of edges: initially empty 1. find the closest node in U to X0 (say X1). move X1 fom U to 3. add (X0, X1) to E 4. fo each Xi in find Yi in U that is closest; choose Ym such that cost fom X0 to Ym is a minimum and add (Xm, Ym) to E; epeat until U is empty Ex: stat at node E 5 7 T Result: 0 (0,) (0,) o (,) (,) (,E) E (,E) o (,) (,T) 0 4 good fo designing flight schedules 5 T 3 1 E

12 EULERIN HINS N YLES 81 b g11 When is it possible to tace a plana gaph without tacing any edge moe than once so that the pencil is neve emoved fom the pape? I H E E G E F Euleian chain Euleian cycle Neithe Euleian cycle = edges ae all the edges of G (end up at point whee stated) Theoem: an Euleian cycle exists fo a connected gaph G wheneve all nodes have an even degee and vice vesa Poof: one diection: if an Euleian cycle exists, then each time we ente a node by one edge we leave by anothe edge othe diection: moe complex Euleian chain = joins nodes X and Y such that its edges ae all the edges of G (end up at point diffeent fom stating point) Theoem: an Euleian chain between nodes X and Y fo a connected gaph G exists if and only if nodes X and Y have odd degee and the emaining nodes have even degee

13 HMILTONIN HINS N YLES 81 b g1 When is it possible fo a salesman based in city X to cove his teitoy in such a way that he neve visits a city moe than once, whee not evey city is connected diectly to anothe city?? E F E F Hamiltonian cycle exists No Hamiltonian chain o cycle Hamiltonian chain exists (only one way fom E to F) Hamiltonian cycle = F E Hamiltonian cycle = cycle whee each vetex appeas once (salesman ends up at home!) Hamiltonian chain = chain whee each vetex appeas once (salesman need not end up at home!) Unlike Euleian chains and cycles, no necessay and sufficient conditions exist fo a gaph G to have a Hamiltonian chain o cycle Sufficient condition: Theoem: simple gaph with n 3 nodes such that fo any distinct nodes X and Y not joined by an edge and degee (X) + degee (Y) n, then G has a Hamiltonian cycle Ex: