1. Algorithms and data structures
Graph representation
Selected problems

2. Graph 11 Simple Directed Multiple edges 10 Loops 7 Weights
In more formal way:
graph is an ordered pair G = (V, E) comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V

3. Graph representation Adjacency matrix Adjacency listsB D A B C D
Adjacency matrix A B C D 1 C
Adjacency lists A B C D B A C D B D B C
If matrix is symetrical (graph is undirected), we can store only a half of matrix.

4. Graph representation Incidence matrix Incidence lists m p m n o p n 11 o
Incidence lists m n o p p n m o p p n m n o

5. Weights representation2 7 B D A B C D
Adjacency matrix 3 A B C D 5 7 2 3 5 C 2
Adjacency lists A B C D
A,2
B,3
D,2
B,7
C,5

6. Connected graph, connected component
DFS

7. Minimal Spanning Tree 1 3 6 2 4 8 5
Application: road building

8. Minimal Spanning Tree Kruskal’s algorithm:
Pick the edge with the smallest possible weights and avoid cycles.
Pot. problem: eficient cicle detection.
Prim-Dijkstra’s algorithm:
CurrentTree = edges with the smallest weight
CurrentEdge = Pick the smallest edge incident with CurrentTree
Add CurrentEdge to CurrentTree.
Pot. problem: disconnected graph.

9. Shortest path 1 3 6 7 4 9 2
Applications: - shortest (fastest) road; - the least expensive (optimal) technology process.

10. Dijkstra’s Algorithm A.: Weights are not less than 0
Starting vertex s: di = , Visited = s, ds = 0;
For every vertex i adjacent to Visited di = e(s,i)
Repeat until reach destination vertex (or there are unreached vertexes in general case):
From V–Visited pick vertex j with smallest dj
Visited = Visited + j
For any neighbar i of j vertex from V–Visited update:
di = min{ dj, di+e(j,i) }

11. Euler’s Cycle, Path
Road – (open or not), that conatains all the edges from graph G
Application:
Chinese Postman Problem Drawing/cutting with plotter aid.

12. Euler’s Cycle - Fleury’s alg.
Start from a vertex current, where deg(current) is odd (if such vertex exists)
while G contains any edge pick any edge d incident to current but avoid bridges
traverse the edge d and update current
remove d from G

13. Euler’s Cycle – stack algorithm
A: G is an Euler’s graph
current = Pick any vertex from G
while G contains edges and stack is not empty
if there exist any edge d incident with current
push current to the stack
traverse edge d and update current
remove edge d from G
else
move current to the solution pop current from the stack

14. Graph coloring Coloring
Colors are assigned while avoiding conflicts with neighbars
Coloring
vertices
edges

15. Coloring Some interesting applications could be pointed i.e.
Assigning frequencies
Scheduling of lessons
Codes ressist for trasmission errors
Placing elements on curcuit board
Many more or less sophisticated models are studied
In general finding the optimal solution is very time-consuming so not optimal (but fast) algorithms are used

16. LF – heuristic (Largest First)
Pick an unpainted vertex with a max degree;
Assign minimal possible color.
LF quality is linear i.e for any n there exist graph that will require n colors more than optimal solution. 1 2 2 3 1

17. SL – heuristic (Smallest Last)
Pick a vertex with smallest degree in remaining subgraph
Push vertex to the stack
Remove vertex (and incident edges) from graph
Color vertexes from the stack. ?

18. Similar problems could be diffienent
Euler cycle (visit all edges) is easy
Hamilton cycle (visit all vertices) is hard
Shortest path is easy
Longest path is hard