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GRAPHS & COMBINATORICS

GRAPHS & COMBINATORICS. Antonio Hervás Jorge ahervas@mat.upv.es. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II. Basic components . Connectivity and conected components Connectivity in non-directed graphs Graph representations Adjacency matrix Incidence matrix

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GRAPHS & COMBINATORICS

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  1. GRAPHS & COMBINATORICS Antonio Hervás Jorge ahervas@mat.upv.es

  2. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II • Basic components • . Connectivity and conected components • Connectivity in non-directed graphs • Graph representations • Adjacency matrix • Incidence matrix • Accesibility matrix • Adjacency lists • Search methods in graphs • Breadth First Search (BFS) • Depth First Search (DFS) • The problem of the shortest path with just one origin • Minimum paths joining all the nodes of a graph The problem of the shortest paths A.H.J.-2.003

  3. Definition.- Let G =(V,E) be a non-directed graph. We shall call chain from node v1 to node vk , the series of nodes and edges: P = v1, a1, v2 , a2 , ..., vk , ak , vk+1 so that For all j, 1 ≤ j ≤ k , aj = ( vj, vj+1 ) Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II A.H.J.-2.003

  4. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II All edges being different Simple chain Whenever v1 = v k+1Closed chain All nodes being different P path Whenever a path, v1 = v k+1P cycle Number of edges of a path lenght A.H.J.-2.003

  5. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Definition.- Let G = (V,E) be a directed graph . We shall call directed semi-path from node v1 to node vk , the series of nodes and edges: P = v1, a1, v2 , a2 , ..., vk , ak , vk+1 so that For all j, 1 ≤ j ≤ k ,aj = < vj , vj+1 > o aj = < v j+1 , vj > A.H.J.-2.003

  6. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II A.H.J.-2.003

  7. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II A.H.J.-2.003

  8. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II SEMI-CYCLE SEMI DIRECTED PATH A.H.J.-2.003

  9. Connectivity and connected components

  10. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Connectivity in non-directed graphs • Properties of the connectivity relation: • A node is connected to itself by means of a path whose length is zero. • Whenever there is a path from u to v, there will also exist a path from v to u. • Whenever there is a path from uto v and another from v to w, then u and w nodes are connected A.H.J.-2.003

  11. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II CONNECTIVITY relation defined on a set of nodes of a non-directed graph VERIFIES THE PROPERTIES:  Reflexive  Symetric  Transitive →Binary Relation of Equivalence A.H.J.-2.003

  12. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II • [1]= {1,3,5} =[3]=[5] • [2]={2,4,6} A.H.J.-2.003

  13. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Example.- 1   3  2 Graph 2-connected 5   4 made up of: 2 connected components A.H.J.-2.003

  14. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II How can we define the best directed connectivity? How can we define the conectivuty in directed graphs? A.H.J.-2.003

  15. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Theorem.- Let G be a non-directed and connected. G bypartite G has no odd length cycles A.H.J.-2.003

  16. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II THEOREM G non-directed graphG has: connected non-triviala zero degree node  a cycle  or both A.H.J.-2.003

  17. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II THEOREM Let G =( V, E) be a non-directed connected graph and C a cycle in G. e edge from CG - {e } connected A.H.J.-2.003

  18. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II THEOREM G non-directed connected graph with |V| 2 Nº edges  |V| - 1 from G A.H.J.-2.003

  19. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II THEOREM G directed strongly connected graph with |V| 2 nº edges  |V| from G A.H.J.-2.003

  20. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II THEOREM V set of nodes  non-directed connected graph with |V | - 1 edges A.H.J.-2.003

  21. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II TEOREMA V set of nodes  directed strongly connected graph with  |V| edges A.H.J.-2.003

  22. Graph representation.

  23. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II The existence of some edges between the nodes enables for a double representation of the graph The connectivity among nodes results in  Adjacence Matrix.  Incidence Matrix.. Accesibility Matrix.  Accesibility Matrix. A.H.J.-2.003

  24. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Definition.- Let G = (V,E) be a graph. We shall say that vj is whithin reach from vi , if there is a path (either directed or not) fromvito vj . A.H.J.-2.003

  25. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Definition.- We define the Accesibility matrix of a graph with n nodes and denote it by R = [r(i,j)] n  n as r(i,j) = { 1, whenever vj is whithin reach from vi 0, otherwise. A.H.J.-2.003

  26. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II We shall call R(vi) the nodes within reach from vi. We obtain these nodes from using the function  That is: R(vi) = whithin reach from (vi) = = {vi} (vi)  . . . n (vi) A.H.J.-2.003

  27. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Definition.- We shall define the Access Matrix as the transposed to the accesibility matrix. It is denoted by Q = [q(i,j)] n  n and defined as: q(i,j) = { 1, whenever vi it is within reach from vj 0,otherwise. If the graph The accesibility matrix is a non-directed one and the access matrix coincide ( it is symetric) A.H.J.-2.003

  28. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Graph Search Methods • Breadth First Search (BFS) • Depth First Search (DFS) A.H.J.-2.003

  29. Algorithm (BFS) BFS (v) procedure /* se aplica sobre un grafo G de n vértices */ global G, n, ALCANZADO (1:n); cola COLA; x  v; ALCANZADO (v)  1; inicializar la cola a vacío; bucle para todos los vértices w no adyacentes desde x hacer si ALCANZADO (w) = 0 entonces ALCANZADO (w)  1 añadir w a COLA si COLA está vacia entonces return borrar el vértice x de COLA fin del bucle fin BFS A.H.J.-2.003 Graphs & Combinatorics.

  30. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Results from applying the BFS Algorithm on any G graph: A.H.J.-2.003

  31. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II We shall use the algorithm to find the connected components of an unconnected non-directed graph. Results:  If the node i is whithin reach from the first node of the path, then: ALCANZADO (i) = 1  In non-directed graphs, if the node ALCANZADO has only got 1 numbers , then: CONNECTED Graph A.H.J.-2.003

  32. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Algorithm (DFS) DFS (v) procedure /* se aplica sobre un grafo G de n vértices */ global G, n, ALCANZADO (1:n); integer v, w; ALCANZADO (v)  1; para todos los vértices w (no alcanzados) adyacentes desde x hacer call DFS (w) fin del para fin BFS A.H.J.-2.003

  33. The problem of the shortest paths.

  34. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II The problem of finding the shortest path/s from a x node to another node and/orto the rest of the graph nodes among each and every one of the graph nodes. A.H.J.-2.003

  35. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II The problem of the shortest paths with just one origin G = (V,E) a directed graph. C = [ C(p,q) ]n  n the weight matrix of the graph G. s Vthe origin node. Find the cost of the shortest path from the origin node sto the rest of the nodes inV. (The cost of the path is the result of adding up the weights of the path arcs) A.H.J.-2.003

  36. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Wheights of the graph edges POSITIVE NEGATIVE BELLMAN-FORD Algorithm. DIJKSTRA Algorithm. A.H.J.-2.003

  37. Departamento de Matemática Aplicada DIJKSTRA Algorithm. •  The weights C(p,q) from all edges must be positive • Whenever an edges is not on the graph we labell it with weight +  • Every node will be labelled as l(xi). • - It will represent a higher level of the length of the • shortest path from the initial node to the node xi .. • - It will be variable in principle, but in every iteration • one will be fixed. •  In every iteration the node labells disminish. • ( As the nodes are reached from the origin node) •  The algorith finishes when the labell of the searched node • is fixed( or all labells are fixed) A.H.J.-2.003

  38. Departamento de Matemática Aplicada DIJKSTRA Algorithm. [Step1] Let s be the origin node, we labell it as l(s) = 0. l(xi) = + xi  V /* variable */ Let P = s be. [Step 2] For all xi(P) with variable labell, update the labells: l(xi) = min [ l(xi) , l(P) + C(P, xi) ] [Step 3] Let xi = min [ l(xj) ] , xj with variable labell. [Step 4] Mark the labell from xi as fix and take P = xi . * * * A.H.J.-2.003

  39. Departamento de Matemática Aplicada [Step 5] (1) if we only wish to know the shortest path from s to t. if P = t then l(P) is the length of the shortest path searched STOP. otherwise go to STEP2. (2) if we only wish to know the shortest paths from s to the rest of the nodes. If all nodes have a fix labell then tese labells indicate the length of the shortest paths STOP. Otherwise go to step 2. A.H.J.-2.003

  40. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Theorem. The DIJKSTRA Algorithm gives the The shortest paths from a v node to the rest of the nodes in a connected graph with a positive weight matrix. A.H.J.-2.003

  41. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Algoritmo de BELLMAN-FORD.  It gives the shortest path among two or more nodes in a connected graphwheighted with the general weight matrix (either positive or negative). There must not exist any cycles with negative total weight  Nodes will be labelled as lk(x). - It represents the shortest path from node s to node x having k or less edges. - They remain variable until the last iteration. - At the end of k iteration we will calculate the labell k + 1  The algorithm will finish as soon as it calculates paths of n - 1 length ( or the largest if they are shorter ). A.H.J.-2.003

  42. Departamento de Matemática Aplicada BELLMAN-FORD Algorithm. [Step 1] Inicialización. S =  (s); k = 1; and labells: l1 (s) = 0; for all xi  (s) : lk (xi) = C(s, xi) for the rest of the nodes lk (xi) =  [Step 2] For all xi (S) update the labells: lk+1 (xi) = min [ lk (xi) , minx j Ti { lk (xj) + C(xj,xi) } ] Ti = -1 (xi)  S lk+1 (xi) = lk (xi) for xi(S) 1 A.H.J.-2.003

  43. Departamento de Matemática Aplicada [Step 3]Ending test. a) If k  n-1 and xi, lk+1(xi) = lk(xi)  STOP. We have obtained the length of the shortest paths given by the current labells. b) if k < n-1 and there is any xi , lk+1(xi)  lk(xi) STEP4. c) If k = n-1 and there is any xi , lk+1 lk(xi)  there is NO SOLUTION. STOP. [Step 4] S = { xi / lk+1 (xi)  lk (xi) } S has the nodes whose shortest paths is of k+1 cardinality. [Step 5] k = k+1; go to STEP 2. A.H.J.-2.003

  44. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Minimum path among all the nodes of a graph. FLOYD-WARSHALL Algorithm.  It acts directly on the weight matrix, but does not labell the nodes.  It calculates the length of the shortest paths amongall the nodes of a graph. It finds the cycles with negative weight (as far as they exist). A.H.J.-2.003

  45. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II Weight matrix of the graph.  it is updated in every iteration. C = (cij) Weight matrix of a graph where: - the nodes are those of the original graph - The edges are the shortest paths with k or less edges in the initial graph. Ck A.H.J.-2.003

  46. Graphs & Combinatorics. ESTRUCTURAS MATEMÁTICAS PARA LA INFORMÁTICA II At the begining, in the diagonal of the C matrix there are only zeros.  If any value of the diagonal turns out to be negative  Cycle of total weight is negative.  There is no solution.  Otherwiseafter n iterations we will obtain the matrix with the length of the shortest paths. A.H.J.-2.003

  47. Departamento de Matemática Aplicada FLOYD-WARSHALL Algorithm. [Step 1]k = 0 [Step 2]k = k+1 [Step 3]i  k / cik j  k / ckj [Step 4] a) If cii < 0 STOP,negative weights circuit. b)If i, cii0 ^ k = nSTOP. [cij] nn represents the length of the shortest paths of xi a xj c) cii > 0 , i, k < n  go to STEP 2. } cij = min { cij, cik + ckj } A.H.J.-2.003

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