Introduction to Graphs
Blagovesta Stanoeva
26.11.2014
Software University
Table of contents
- The graph structure
- Types of graphs and their representation
- Problems of graph theory, Algorithms
(an example) - Applications of graph theory
What is a graph
?
Flashback
http://computationaltales.blogspot.com/
- Leonhard Euler - 1735
- The Seven Bridges of Königsberg
- Challenge - walk all 7 bridges without crossing a bridge twice
-
Negative resolution - that's when it all began
The Graph structure
-
Graph - ordered pair G = (V,E)
‣ V - finite set of vertices (nodes)
‣ E - finite set of edges (links) between nodes -
Captures pairwise relationship between objects
‣ Each edge is a pair (v,w), where v, w ∈ V
‣ Loops, parallel and adjacent edges
‣ Adjacent and isolated vertices
Types of Graphs
- Directed - the edges are ordered pairs (v, w) != (w, v)
- Undirected - the edges are unordered pairs (v, w) == (w, v)
- Mixed - combination of both
- Dense - number of edges is close to the maximum number
- Sparse - only few edges
-
Cyclic - directed graph with at least one cycle
‣ cycle - path along the edges from a vertex to itself - Weighted
- and many more ...
Graph Representations
-
Depends on:
‣ the graph structure
‣ the manipulation algorithm -
Two main ways
‣ Adjacency matrix - faster access, huge amounts of memory
‣ Adjacency list - smaller memory requirements, good for
sparse graphs
‣ Best way of representation: a combination of both
Graph Algorithms
-
Elementary: graph-searching algorithms
‣ Breadth-first search, Depth-first search -
Single-Source shortest paths:
‣ Algorithms of Bellman-Ford and Dijkstra -
All-Pairs shortest paths:
‣ Floyd-Warshall -
Minimum spanning trees:
‣ Algorithms of Kruskal and Prim -
Maximum flow:
‣ Ford-Fulkerson method
Prim's algorithm
..demo
Applications of Graph Theory
-
Routing problems
‣ Computer games - shortest path between two points
‣ Eulerian Path (Circuit) - postman, visiting each street (edge)
‣ Hamiltonian Path (Circuit) - postman, visiting each house (node)
‣ Map applications -
Minimum-spanning tree
‣ Modeling network topologies -
Maximum flow
‣ Transportation networks - traffic problems - and many more ...
Questions?
deck
By Beggie Angelova S
deck
- 1,062