2IL65 Algorithms

1. 2IL65 Algorithms Fall 2013Lecture 7: Elementary Graph Algorithms

2. Elementary Graph Algorithms

3. computer network road network process 4 process 1 process 6 process 3 process 5 process 2 Networks and other graphs social network execution order for processes

4. Graphs: Basic definitions and terminology • A graph G is a pair G = (V, E) • V is the set of vertices or nodes of G • E⊂ VxV is the set of edges or arcs of G • If (u, v) ∈ E then vertex v is adjacent to vertex u undirected graph • (u, v) is an unordered pair ➨(u, v) = (v, u) • self-loops forbidden directed graph • (u, v) is an ordered pair ➨(u, v) ≠ (v, u) • self-loops possible

5. not a path cycle of length 3 a path path of length 4 Graphs: Basic definitions and terminology • Path in a graph: sequence ‹v0, v1, …, vk› of vertices, such that (vi-1, vi) ∈ E for 1 ≤ i ≤ k • Cycle: path with v0 = vk • Length of a path: number of edges in the path • Distance between vertex u and v: length of a shortest path between u and v (∞ if v is not reachable from u)

6. Graphs: Basic definitions and terminology • An undirected graph is connected if every pair of vertices is connected by a path. • A directed graph is strongly connected if every two vertices are reachable from each other.(For every pair of vertices u, v we have a directed path from u to v and a directed path from v to u.) connected components strongly connected components

7. Some special graphs Treeconnected, undirected, acyclic graph • Every tree with n vertices has exactly n-1 edges DAGdirected, acyclic graph Check appendix B.4 for more basic definitions

8. Some special graphs Planar graphs a graph that can be drawn in the plane without crossing edges Euler’s Formula for finite, connected, planar graphs: V − E + F = 2 Check appendix B.4 for more basic definitions

9. 2 1 5 4 3 1 2 3 4 3 3 3 2 2 1 1 4 5 Graph representation Graph G = (V, E) • Adjacency lists array Adj of |V| lists, one per vertex Adj[u] = linked list of all vertices v with (u, v) ∈ E (works for both directed and undirected graphs)

10. 2 1 5 4 3 1 2 3 1 2 2 3 4 5 Graph representation Graph G = (V, E) • Adjacency lists array Adj of |V| lists, one per vertex Adj[u] = linked list of all vertices v with (u, v) ∈ E (works for both directed and undirected graphs)

11. 2 1 5 4 3 1 2 3 4 5 1 1 1 2 1 1 • if (i, j) ∈ E • 0 otherwise aij = 3 1 1 1 4 1 5 Graph representation Graph G = (V, E) • Adjacency lists array Adj of |V| lists, one per vertex Adj[u] = linked list of all vertices v with (u, v) ∈ E • Adjacency matrix |V| x |V| matrix A = (aij) (also works for both directed and undirected graphs)

12. 2 1 5 4 3 1 2 3 4 5 1 1 2 • if (i, j) ∈ E • 0 otherwise aij = 3 1 1 4 1 5 Graph representation Graph G = (V, E) • Adjacency lists array Adj of |V| lists, one per vertex Adj[u] = linked list of all vertices v with (u, v) ∈ E • Adjacency matrix |V| x |V| matrix A = (aij) (also works for both directed and undirected graphs)

13. 1 1 2 3 4 5 2 1 1 1 3 4 3 3 2 2 3 1 1 2 1 1 3 4 1 1 1 4 1 5 5 Adjencency lists vs. adjacency matrix Θ(V + E) Θ(V2) Θ(degree(u)) Θ(V) better if the graph is sparse … use V for |V| and E for |E| Θ(degree(u)) Θ(1)

14. Searching a graph: BFS and DFS Basic principle: • start at source s • each vertex has a color: white = not yet visited (initial state) gray = visited, but not finished

15. Searching a graph: BFS and DFS Basic principle: • start at source s • each vertex has a color: white = not yet visited (initial state) gray = visited, but not finished • color[s] = gray; S = {s} • while S ≠ ∅ • do remove a vertex u from S • for each v ∈Adj[u] • doif color[v] == white • thencolor[v] = gray; S = S υ {v} • BFS and DFS choose u in different ways; BFS visits only the connected component that contains s. u s u u and so on …

16. Breadth-first search BFS uses a queue ➨ it first visits all vertices at distance 1 from s, then all vertices at distance 2, … BFS and DFS s

17. Breadth-first search BFS uses a queue (FIFO) ➨ it first visits all vertices at distance 1 from s, then all vertices at distance 2, … Depth-first search DFS uses a stack (LIFO) BFS and DFS s s

18. Breadth-first search (BFS) d(u) becomes distance from s to u BFS(G, s) • for each u ≠ s • do color[u] = white; d[u] =∞; π[u] = NIL • color[s] = gray; d[s] =0; π[s] = NIL • Q = ∅ • Enqueue(Q, s) • while Q ≠ ∅ • do u = Dequeue(Q) • for each v ∈ Adj[u] • doif color[v] == white • thencolor[v] = gray; d[v] = d[u]+1; π[v] = u; • Enqueue(Q, v) • color[u] = black π[u] becomes predecessor of u

19. ∞ ∞ BFS on an undirected graph Adjacency lists 1: 3 2: 6, 7, 5 3: 1, 6, 4 4: 5, 3 5: 2, 4, 8 6: 3, 2 7: 2, 9 8: 9, 5 9: 7, 8 10: 11 11: 10 Source s = 6 2 7 ∞ 2 ∞ 1 6 ∞ 3 9 ∞ 1 2 0 ∞ 1 3 ∞ 3 8 11 ∞ 2 ∞ 2 4 5 10 Queue Q: 6 3 2 1 4 7 5 9 8

20. ∞ ∞ BFS on an undirected graph Adjacency lists 1: 3 2: 6, 7, 5 3: 1, 6, 4 4: 5, 3 5: 2, 4, 8 6: 3, 2 7: 2, 9 8: 9, 5 9: 7, 8 10: 11 11: 10 Source s = 6 2 7 2 tree edges ∞ 1 6 ∞ 3 9 1 2 0 1 3 3 8 11 2 2 4 5 10 Queue Q: 6 3 2 1 4 7 5 9 8 Note: BFS only visits the nodes that are reachable from s

21. Dequeue Enqueue u d[∙] = d[u] 0 or more nodes with d[∙] = d[u]+1 BFS: Properties Invariants • Q contains only gray vertices • gray and black vertices never become white again • the queue has the following form: • the d[∙] fields of all gray vertices are correct • for each white vertex we have: (distance to s) > d[u]

22. BFS: Analysis Invariants • Q contains only gray vertices • gray and black vertices never become white again This implies • every vertex is enqueued at most once ➨ every vertex is dequeued at most once • processing a vertex u takes Θ(1+|Adj[u]|) time ➨ running time at most ∑uΘ(1+|Adj[u]|) = O(V + E)

23. BFS: Properties After BFS has been run from a source s on a graph G we have • each vertex u that is reachable from s has been visited • for each vertex u we have d[u] = distance to s • if d[u] < ∞, then there is a shortest path from s to u that is a shortest path from s to π[u] followed by the edge (π[u], u) Proof: follows from the invariants (details see book) BFS computes a shortest path tree (breadth-first tree) on the connected component of G containing s, consisting of edges (u, v) such that u is gray and v is white when (u, v) is explored.

24. Depth-first search (DFS) DFS(G) • for each u ∈ V • do color[u] = white; π[u] = NIL • time = 0 • for each u ∈ V • doif color[u] == white then DFS-Visit(u) DFS-Visit(u) • color[u] = gray; d[u] =time; time = time + 1 • for each v ∈ Adj[u] • doif color[v] == white • thenπ[v] = u; DFS-Visit(v) • color[u] = black; f[u] =time; time = time + 1 time = global timestamp for discovering and finishing vertices d[u] = discovery time f[u] = finishing time

25. DFS on a directed graph Adjacency lists 1: 2, 4 2: 5 3: 5, 6 4: 2 5: 4 6: -- f=7 f=6 f=11 f=10 d=1 d=0 d=8 d=9 2 1 3 6 4 5 d=2 d=3 f=5 f=4

26. DFS on a directed graph Adjacency lists 1: 2, 4 2: 5 3: 5, 6 4: 2 5: 4 6: -- f=7 f=6 f=11 f=10 d=1 d=0 d=8 d=9 2 1 3 6 4 5 tree edges d=2 d=3 f=5 f=4 Note: DFS always visits all vertices

27. DFS: Properties • DFS visits all vertices and edges of G • Running time: • DFS forms a depth-first forest comprised of ≥ 1 depth-first trees.Each tree is made of edges (u, v) such that u is gray and v is white when (u, v) is explored. Θ(V + E)

28. DFS: Edge classification Tree edgesedge (u, v) is a tree edge if v was first discovered by exploring edge (u, v); the tree edges form a forest, the DF-forest Back edgesedges (u, v) connecting a vertex u to an ancestor v in a depth-first tree Forward edgesnon-tree edges (u, v) connecting a vertex u to a descendant v Cross edgesall other edges f=7 f=6 f=11 f=10 d=1 d=0 d=8 d=9 2 1 3 6 4 5 d=2 d=3 f=5 f=4

29. DFS: Edge classification Tree edgesedge (u, v) is a tree edge if v was first discovered by exploring edge (u, v); the tree edges form a forest, the DF-forest Back edgesedges (u, v) connecting a vertex u to an ancestor v in a depth-first tree Forward edgesnon-tree edges (u, v) connecting a vertex u to a descendant v Cross edgesall other edges Undirected graph ➨ (u, v) and (v, u) are the same edge; • classify by first type that matches.

30. DFS: Properties • DFS visits all vertices and edges of G • Running time: • DFS forms a depth-first forest comprised of ≥ 1 depth-first trees.Each tree is made of edges (u, v) such that u is gray and v is white when (u, v) is explored. • DFS of an undirected graph yields only tree and back edges. No forward or cross edges. • Discovery and finishing times have parenthesis structure. Θ(V + E) [ ]{ }[{ }]{[ ]}{[}][{] }

31. not possible d[u] f[u] d[v] f[v] time time time time d[v] d[u] d[u] d[v] f[v] f[u] f[u] f[v] d[v] d[u] f[v] f[u] DFS: Discovery and finishing times TheoremIn any depth-first search of a (directed or undirected) graph G = (V, E), for any two vertices u and v, exactly one of the following three conditions holds: • the intervals [d[u], f[u]] and [d[v], f[v]] are entirely disjoint ➨ neither of u or v is a descendant of the other • the interval [d[u], f[u]] entirely contains the interval [d[v], f[v]] ➨ v is a descendant of u • the interval [d[v], f[v]] entirely contains the interval [d[u], f[u]] ➨ u is a descendant of v

32. DFS: Discovery and finishing times Proof (sketch): • assume d[u] < d[v] case 1: v is discovered in a recursive call from u ➨ v becomes a descendant of u recursive calls are finished before u itself is finished ➨ f[v] < f[u] case 2: v is not discovered in a recursive call from u ➨v is not reachable from u and not one of u’s descendants ➨ v is discovered only after u is finished ➨ f[u] < d[v] ➨ u cannot become a descendant of v since it is already discovered ■

33. DFS: Discovery and finishing times Corollaryv is a proper descendant of u if and only if d[u] < d[v] < f[v] < f[u]. Theorem (White-path theorem)v is a descendant of u if and only if at time d[u], there is a path u↝v consisting of only white vertices. (Except for u which was just colored gray.) (See the book for details and proof.)

34. Topological sort Using depth-first search …

35. execution order for processes process 4 process 1 process 6 process 3 process 5 process 2 Topological sort Input: directed, acyclic graph (DAG) G = (V, E) Output: a linear ordering of v1 ,v2 ,…, vn of the vertices such that if (vi ,vj ) ∈ E then i < j • DAGs are useful for modeling processes and structures that have a partial order Partial order • a > b and b > c➨ a > c • but may have a and b such that neither a > b nor b > a

36. Topological sort Input: directed, acyclic graph (DAG) G = (V, E) Output: a linear ordering of v1 ,v2 ,…, vn of the vertices such that if (vi ,vj ) ∈ E then i < j • DAGs are useful for modeling processes and structures that have a partial order Partial order • a > b and b > c➨ a > c • but may have a and b such that neither a > b nor b > a • a partial order can always be turned into a total order(either a > b or b > a for all a ≠ b) (that’s what a topological sort does …)

37. v1 v2 v3 v4 v5 v6 Topological sort Input: directed, acyclic graph (DAG) G = (V, E) Output: a linear ordering of v1 ,v2 ,…, vn of the vertices such that if (vi ,vj ) ∈ E then i < j • Every directed, acyclic graph has a topological order LemmaA directed graph G is acyclic if and only if DFS of G yields no back edges.

38. Topological sort TopologicalSort(V, E) • call DFS(V, E) to compute finishing time f[v] for all v ∈ E • output vertices in order of decreasing finishing time 17 11 socks underwear 18 16 9 watch 10 12 13 pants shoes 15 14 1 shirt 6 8 belt 7 2 tie 5 3 jacket 4

39. 17 11 socks underwear 18 16 9 watch 10 12 13 pants shoes 15 14 1 shirt 6 8 belt 7 2 tie 5 3 jacket 4 Topological sort TopologicalSort(V, E) • call DFS(V, E) to compute finishing time f[v] for all v ∈ E • output vertices in order of decreasing finishing time

40. Topological sort TopologicalSort(V, E) • call DFS(V, E) to compute finishing time f[v] for all v ∈ E • output vertices in order of decreasing finishing time LemmaTopologicalSort(V, E) produces a topological sort of a directed acyclic graph G = (V, E).

41. d[v] d[u] f[u] f[v] d[u] f[u] d[v] f[v] Topological sort LemmaTopologicalSort(V, E) produces a topological sort of a directed acyclic graph G = (V, E). Proof: To show: f[u] > f[v] Consider the intervals [d[∙], f[∙]] and assume f[u] < f[v] case 1: case 2: When DFS-Visit(u) is called, v has not been discovered yet. DFS-Visit(u) examines all outgoing edges from u, also (u, v). ➨v is discovered before u is finished. Let (u, v) ∈ E be an arbitrary edge. ➨u is a descendant of v ➨G has a cycle

42. Topological sort LemmaTopologicalSort(V, E) produces a topological sort of a directed acyclic graph G = (V, E). Running time? • we do not need to sort by finishing times • just output vertices as they are finished ➨Θ(V + E) for DFS and Θ(V) for output ➨Θ(V + E)

43. Tutorials this week Big tutorial on Thursday Small tutorial on Friday Next week Monday: Summary of course Thursday: Big tutorial for exam preparation Friday: Small tutorial