Graphs

1. Graphs • 1 Definition • 2 Terminology • 3 Properties • 4 Internal representation • Adjacency list • Adjacency matrix • 5 Exploration algorithms • 6 Other algorithms

2. What is a graph? • Definition • A graphG is a finite set V of verticesand a finite set E of edgesconnecting pairs of vertices: • G = (V,E) • Directed vs. undirected • G is undirected if its edges are undirected (top fig.), • G is directed if its edges are • directed (bottom fig.).

3. Applications • Network representation: traffic, aerial routing, internet… • Automata: languages, discrete state systems • Dynamic system modeling • Probabilistic model: Bayesian network, neural network… Algorithms • Shortest path • Optimal flow • Optimal tour: traveling salesman • Clustering: k-neighboring • Complexity of a network

4. Terminology • The end-verticesof an edge are the vertices connected by that edge. • Adjacent vertices: directly linked by an edge • Adjacent edges: share a common end-vertex • An edge is incidentto a vertex if it connects that vertex to another vertex. • The degreeof a vertex is the number of edges that are incident to that vertex.

5. Properties • Let G be a graph with n vertices and m edges. • Property 1 • Proof: each edge is counted twice. • Property 2 • If G is undirected with no self-loops and no multiple edges: • m ≤n(n − 1)/2 • Proof: the maximum number of edges is obtained when each vertex is connected to all the other vertices of the graph. We have, • (n − 1) + (n − 2) + (n − 3) + . . . + 2 + 1 = n(n − 1)/2

6. Terminology b e c a c d a • Path: sequence of vertices v1 ,v2 ,. . .vk such that all consecutive vertices are adjacent. • Simple path: no repeated vertex • Cycle: simple path, excepted that the last vertex is the same as the first one.

7. Terminology • Connex graph: each pair of vertices is linked by a path • Sub-graph: sub-set of vertices and edges forming a graph • Connex component: sub-graph connex • example: le graph bellow has 3 connex components

8. Terminology • Tree – connex graph without cycle • Forest - collection of trees

9. Connectivity Let n = # vertices m = # edges • Complete graph (clique)– each pair of vertices are adjacent • Each of the n vertices are incident to n-1 edges, but each edge is summed two time! Therefore, m = n(n-1)/2. • So iff a graph is notcomplete then m < n(n-1)/2 n = 5 m= (5 * 4)/2 = 10

10. Clique: A subgraph in which each pair of vertices are adjacent: a complete subgraph. • Search for the maximum clique: a naïve algorithm. • Examine each set of k vertices to determine if it is a clique. • But the number of possible cliques of size k in a graph of size V • Lot of research on heuristic algorithms to find good non-exact solutions

11. Connectivity • In case of a tree m = n - 1 • if m < n - 1, G is not connex n = 5 m = 4 n = 5 m = 3

12. Spanning tree • A spanning treeof G is a sub-graph which is a tree and which contains all vertices of G G Spanning tree ofG

13. Curiosity • Euler and the bridges of Koenigsberg: the first problem of graph theory? • Is it possible to make a walk crossing each bridge one and only one time and to come back to the starting point?

14. Curiosity • The graph model • Eulerian circuit: path which use each edge exactly once and come back to the initial vertex. • Euler’s theorem: a connected graph has an Eulerian circuit iff it has no vertex of odd degree •  No, it is not possible!

15. More definitions • Oriented graph: each edge go only in one direction • Acyclic oriented graph Without cycle With cycle

16. Accessibility A tree rooted in v contains all accessible vertices from v using oriented path strongly connex each vertex is accessible from each other using an oriented path

17. Strongly connex component • Transitive closure • It is the graph G* obtained from the graph G after applying the following rule: • If there exists an oriented path from a to b in G then add an oriented edge from a to b in G*. { a , c , g } { f , d , e , b }

18. Graph representation • Adjacency list • Definition • The adjacency list of a graph with n vertices is an array of n lists of vertices. • The list i contains vertex j if there is an edge from vertex i to • vertex j.

19. Example 2 in an oriented graph

20. Adjacency matrix • Definition • Let adjacency matrix of a graph with n vertices is a n × n matrix A where: • Remark • The adjacency matrix of an undirected graph must be symmetric.

21. Example 1

22. Example 2

23. Exploration algorithms • They explore the vertices that are reachable starting from a source vertex. • Depth-First Search • Breadth-First Search (level-order search)

24. Breath-First Search • Algorithm that explores the vertices that are reachable starting from a source vertex s and constructs a breadth-first search tree(spanning tree). • Compute a distance from each vertices to the source vertex. • Color terminology • WHITE vertices are unexplored. • BLACK vertices are fully explored vertices. • GRAY vertices are being explored: these vertices define the ”frontier” between explored and unexplored vertices.

25. BSF algorithm

26. Example

27. The resulting BSF spanning tree

28. Analysis of BFS • Let n be the number of vertices and m the number of edges in a graph. • Each vertex is enqueued once in the queue: to enqueue all vertices it takes O(n). • Each edge is visited at most once: visiting all edges takes O(m). • Complexity of BFS • As a result, BFS takes O(n + m) time.

29. Depth-First Search • Start from a vertex s. • Set s as the current vertex u. Mark u as ‘visited’. • Select arbitrarily one adjacent vertex v of u. • If v is ‘visited’ go back to u • Else mark v ‘visited’. V become the current vertex. Repeat the previous steps • When all vertices adjacent to the current vertex are ‘visited’ backtrack to a previous ‘visited’ vertex. Repeat the previous steps. • When backtrack leads to vertex s and if all the adjacent vertices of s are ‘visited’, the algorithm stop.

30. Algorithme DFS(u); Input: a vertex u of G Output: a graph with all vertices labeled ‘visited’ for each edge e incident to u do let v be the other extremity of e if vertex vis not ‘visited’ then mark v ‘visited’ recursively call DFS(v)

31. Example 2) 1) 3) 4)

32. 5) 6)

33. Definition of a weighted graph • A graph, in which each edge has an associated numerical value, is called • a weighted graph. • The numerical value associated to an edge is the weight of the edge. • The weight of an edge can represent a distance, a cost. . . etc. • Applications • Weighted graphs find their application in various problems such as communication or transportation networks.

34. Definition of the MST (Minimum Spanning Tree) The MST is a spanning tree of a connex, weighted and undirectedgraph with minimum total weight. Example The weight of the MST: W(MST) = 8 + 2 + 4 + 7 + 4 + 2 + 9 + 1 = 37 is minimal.

35. Formal definition of a MST Given a connex, weighted and undirected graph G = (V,E), find an acyclic subset T E connecting all vertices in V such that: weight(u, v) is the weight the edge (u, v).

36. Safe edges • Definition • Let A be a subset of edges of a MST of a graph G. • An edge (u, v) of G is safe for A if A  {(u, v)} is also a subset of a MST. • We can deduce from the above definition that finding a MST can be done by greedily grow a set of safe edges:

37. More definitions... • The cut of a graph • A cut of a graph G = (V,E) is a partition of the vertices of the graph into 2 sets: S and V − S. • The cut is denoted: (S,V − S).

38. An edge crossing the cut • An edge crosses the cut (S,V − S) if one of its end-vertices is in S and the other one in V − S. • The edges (b, c), (c, d), (d, f ), (a, h), (e, f) and (b, h) cross the cut (S,V-S).

39. A cut respects a set... • A cut respects a set A of edges if no edge in A crosses the cut. • Light edges • An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut. • Characterization of a safe edge • Theorem • Let G = (V,E) and A E included in some MST of G. • Let (S,V − S) be a cut of G that respects A. • Let e = (u, v) be a light edge crossing (S,V − S). • Then, edge e is safe for A, which mean that e is in the MST of G

40. Proof by contradiction • Suppose you have a MST T not containing e; then we want to show that T is not the MST. • Let e=(u,v), with u in S and v in V - S. • Then because T is a spanning tree it contains a unique path from u to v, which together with e forms a cycle in G. • This path has to include another edge f connecting S to V - S. • T+e-f is another spanning tree (it has the same number of edges, and remains connected since you can replace any path containing f by one going the other way around the cycle). • It has smaller weight than T since e has smaller weight than f. • So T was not minimum, which is what we wanted to prove.

41. Prim’s algorithm for finding a MST • Minimum-Spanning-Tree-by-Prim(G, weight-function, source) • for each vertex uin graph G • set key ofuto ∞ • set parent ofutonil • set key of source vertex tozero • enqueue to minimum-heap Q all vertices in graph G. • whileQ is not empty • extract vertex u from Q// u is the vertex with the lowest key that is in Q • for each adjacent vertex vofudo • if (v is still in Q) and (weight-function(u, v) < key of v) then • setuto be parent ofv// in minimum-spanning-tree • update v's key to equal weight-function(u, v) • Complexity: O(nlogn + mlogn) using a heap

42. Prim’s algorithm on an example...