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Graph Data Structures: Efficient Storage & Operations

Learn about graph data structures, storing vertices, edge storage methods, and operations like traversal, using adjacency matrix and lists.

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Graph Data Structures: Efficient Storage & Operations

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  1. sequence/linear (1 to 1) first ith last hierarchical (1 to many) graph (many to many) set abstract containers

  2. G = (V, E) a vertex may have: 0 or more predecessors 0 or more successors

  3. some problems that can be represented by a graph • computer networks • airline flights • road map • course prerequisite structure • tasks for completing a job • flow of control through a program • many more

  4. graph variations • undirected graph (graph) • edges do not have a direction • (V1, V2) and (V2, V1) are the same edge • directed graph (digraph) • edges have a direction • <V1, V2> and <V2, V1> are different edges • for either type, edges may be weighted or unweighted

  5. A E B C D a digraph V = [A, B, C, D, E] E = [<A,B>, <B,C>, <C,B>, <A,C>, <A,E>, <C,D>]

  6. graph data structures • storing the vertices • each vertex has a unique identifier and, maybe, other information • for efficiency, associate each vertex with a number that can be used as an index • storing the edges • adjacency matrix – represent all possible edges • adjacency lists – represent only the existing edges

  7. storing the vertices • when a vertex is added to the graph, assign it a number • vertices are numbered between 0 and n-1 • graph operations start by looking up the number associated with a vertex • many data structures to use • any of the associative data structures • for small graphs a vector can be used • search will be O(n)

  8. 0 ABCDE 0 1 2 3 4 A 4 1 E B C D 3 2 the vertex vector

  9. 0 ABCDE 0 1 2 3 4 A 4 1 E B 0 1 2 3 4 0 1 2 3 4 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 C D 3 2 adjacency matrix

  10. a V2 matrix is needed for a graph with V vertices maximum # edges?

  11. many graphs are “sparse” • degree of “sparseness” key factor in choosing a data structure for edges • adjacency matrix requires space for all possible edges • adjacency list requires space for existing edges only • affects amount of memory space needed • affects efficiency of graph operations

  12. 0 ABCDE 0 1 2 3 4 A 4 1 E B C D 3 2 2 3 adjacency lists 0 1 2 3 4 1 2 4 1

  13. Some graph operations adjacency matrixadjacency lists O(e) O(e) O(e) O(E) insertEdge isEdge #successors? #predecessors? O(1) O(1) O(V) O(V) Memory space used?

  14. Where to start? Will all vertices be visited? How to prevent multiple visits? traversing a graph ny bos dc la chi atl

  15. ny bos dc la chi atl breadth first traversal breadthFirstTraversal (v) put v in Q while Q not empty remove w from Q visit w mark w as visited for each neighbor (u) of w if not yet visited put u in Q

  16. depth first traversal ny bos dc la chi atl depthFirstTraversal (v) visit v mark v as visited for each neighbor (u) of v if not yet visited depthFirstTraversal (u)

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