Recommended
More Related Content
What's hot
What's hot (20)
Similar to Learn Graph Data Structures and Algorithms
Similar to Learn Graph Data Structures and Algorithms (20)
More from fika sweety
More from fika sweety (20)
Recently uploaded
Recently uploaded (20)
Learn Graph Data Structures and Algorithms
- 2. GRAPHS Graphs = formalism for representing relationships between objects • A graph G = (V, E) • V is a set of vertices • E V V is a set of edges 2 Han Leia Luke V = {Han, Leia, Luke} E = {(Luke, Leia), (Han, Leia), (Leia, Han)}
- 3. DIRECTED VS. UNDIRECTED GRAPHS • In directed graphs, edges have a specific direction: • In undirected graphs, they don’t (edges are two-way): • Vertices u and v are adjacent if (u, v) E 3 Han Leia Luke Han Leia Luke
- 4. WEIGHTED GRAPHS 4 20 30 35 60 Mukilteo Edmonds Seattle Bremerton Bainbridge Kingston Clinton There may be more information in the graph as well. Each edge has an associated weight or cost.
- 5. PATHS AND CYCLES A path is a list of vertices {v1, v2, …, vn} such that (vi, vi+1) E for all 0 i < n. A cycle is a path that begins and ends at the same node. 5 Seattle San Francisco Dallas Chicago Salt Lake City p = {Seattle, Salt Lake City, Chicago, Dallas, San Francisco,
- 6. PATH LENGTH AND COST Path length: the number of edges in the path Path cost: the sum of the costs of each edge 6 Seattle San Francisco Dallas Chicago Salt Lake City 3.5 2 2 2.5 3 2 2.5 2.5 length(p) = 5 cost(p) = 11.5
- 7. CONNECTIVITY Undirected graphs are connected if there is a path between any two vertices Directed graphs are strongly connected if there is a path from any one vertex to any other 7
- 8. CONNECTIVITY Directed graphs are weakly connected if there is a path between any two vertices, ignoring direction A complete graph has an edge between every pair of vertices 8
- 9. CONNECTIVITY A (strongly) connected component is a subgraph which is (strongly) connected CC in an undirected graph: SCC in a directed graph: 9
- 10. DISTANCE AND DIAMETER • The distance between two nodes, d(u,v), is the length of the shortest paths, or if there is no path • The diameter of a graph is the largest distance between any two nodes • Graph is strongly connected iff diameter < 10
- 11. OTHER GRAPHS • Geography: • Cities and roads • Airports and flights (diameter 20 !!) • Publications: • The co-authorship graph • E.g. the Erdos distance • The reference graph • Phone calls: who calls whom • Almost everything can be modeled as a graph ! 11
- 12. GRAPH REPRESENTATION 1: ADJACENCY MATRIX A |V| x |V| array in which an element (u, v) is true if and only if there is an edge from u to v 12 Han Leia Luke Han Luke Leia Han Luke LeiaRuntime: iterate over vertices iterate ever edges iterate edges adj. to vertex edge exists? Space requirements:
- 13. GRAPH REPRESENTATION 2: ADJACENCY LIST A |V|-ary list (array) in which each entry stores a list (linked list) of all adjacent vertices 13 Han Leia Luke Han Luke Leia space requirements: Runtime: iterate over vertices iterate ever edges iterate edges adj. to vertex edge exists?
- 14. TREES AS GRAPHS • Every tree is a graph with some restrictions: • the tree is directed • there are no cycles (directed or undirected) • there is a directed path from the root to every node 14 A B D E C F HG JI BAD!
- 15. DIRECTED ACYCLIC GRAPHS (DAGS) DAGs are directed graphs with no cycles. 15 main() add() access() mult() read() Trees DAGs Graphs if program call graph is a DAG, then all procedure calls can be in-lined
- 16. APPLICATION OF DAGS: REPRESENTING PARTIAL ORDERS 16 check in airport call taxi taxi to airport reserve flight pack bagstake flight locate gate
- 17. TOPOLOGICAL SORT Given a graph, G = (V, E), output all the vertices in V such that no vertex is output before any other vertex with an edge to it. 17 check in airport call taxi taxi to airport reserve flight pack bags take flight locate gate
- 18. TOPOLOGICAL SORT 18 check in airport call taxi taxi to airport reserve flight pack bags take flight locate gate