1. Graphs, BFS, DFS and More…
ctlDenon

2. Agenda Review More Graphs Graphs Representations DFS BFS
Topological-Sort
Shortest Path
Dijkstra’s , Bellman-Ford
All Pairs Shortest Path
Dijkstra’s, Floyd-WarShall

3. Graphs

4. Graphs (review) Definition. A directed graph (digraph)
G = (V, E) is an ordered pair consisting of
a set V of vertices (singular: vertex)
a set E ⊆ V × V of edges
In an undirected graph G = (V, E), the edge set E consists of unordered pairs of vertices.
More,
Connected, Disconnected
Tree, Forest
Cyclic, Acyclic
Weighted, un-weighted
Multigraph …

5. Graphs Representations - adjacency matrix

6. Graphs Representations - adjacency matrix
The adjacency matrix of a graph G = (V, E), where V = {1, 2, …, n}, is the matrix A[1 . . n, n] given by A[i, j] = 1 if (i, j) ∈ E, (What if multigraph ???) 0 if (i, j) ∉ E. Storage size??

7. Graphs Representations - adjacency list
An adjacency list of a vertex v ∈ V is the list Adj[v] of vertices adjacent to v.
Storage size??

8. DFS and BFS Review Possible ordering of traversing if DFS?
Possible ordering of traversing if BFS?

9. BFS
BFS(G) marks all Vertices in G as un-visited marks start vertex s as visited enqueue s into queue Q while queue not empty dequeue the first vertex u from queue Q for each vertex v directly reachable from u if v is unvisited enqueue v to queue Q mark v as visited

10. BFS Application while queue not empty
Shortest Path of un-weighted
while queue not empty
dequeue the first vertex u from queue
for each neighbor vertex v of u
if v is unvisited
enqueue v to queue
mark v as visited
parent[v] = u

11. DFS
DFS(G)
for each vertex u in G
color[u] = WHITE
if color[u] = WHITE
DFS-visit(u)
DFS-visit(u) color[u] = GRAY for each neighbor vertex v of u if color[v] = WHITE DFS-visit[v]

12. Topological Sort
Linear ordering of a directed acyclic graph (DAG)’ s nodes in which each node comes before all nodes to which it has outbound edges.

13. Topological Sort DFS(G)
for each vertex u in G
color[u] = WHITE
parent[u] = Nil
time = 0
if color[u] = WHITE
DFS-visit(u)
DFS-visit(u) color[u] = GRAY for each neighbor vertex v of u if color[v] = WHITE parent[v] = u DFS-visit[v] color[u] = BLACK time = time + 1 f[u] = time

14. Topological Sort
Call DFS(G) to compute the finishing times f[v] for each vertex v
As each vertex is finished (Mark as Black), insert it onto the front of a list
Return the list

15. Shortest Path

16. Shortest Path More Formal Definition
Consider a digraph G = (V, E) with edge-weight function w : E → R. The weight of path p = v1 → v2 → … → vk is defined to be
A shortest path from u to v is a path of minimum weight from u to v. The shortest-path weight from u to v is defined as (u, v) = min{w(p) : p is a path from u to v}.
Note: δ(u, v) = ∞ if no path from u to v exists.

17. Shortest Path Optimal substructure
Theorem. A subpath of a shortest path is a shortest path.

18. Shortest Path Triangle inequality
Theorem. For all u, v, x ∈ V, we have δ(u, v) ≤δ(u, x) + δ(x, v).
Relaxation: if d[v] > d[x] + w(x, v)
then d[v] ← d[x] + w(x, v)

19. Shortest Path
Un-weighted graph?

20. Shortest Path
Un-weighted graph? BFS

21. Shortest Path
Un-weighted graph? BFS
Non-negative edge weights?

22. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm

23. Shortest Path – Dijkstra’s algorithm
Suppose you create a knotted web of strings, with each knot corresponding to a node, and the strings corresponding to the edges of the web: the length of each string is proportional to the weight of each edge. Now you compress the web into a small pile without making any knots or tangles in it. You then grab your starting knot and pull straight up. As new knots start to come up with the original, you can measure the straight up-down distance to these knots: this must be the shortest distance from the starting node to the destination node. The acts of "pulling up" and "measuring" must be abstracted for the computer, but the general idea of the algorithm is the same: you have two sets, one of knots that are on the table, and another of knots that are in the air. Every step of the algorithm, you take the closest knot from the table and pull it into the air, and mark it with its length. If any knots are left on the table when you're done, you mark them with the distance infinity.

24. Shortest Path – Dijkstra’s algorithm

25. Shortest Path – Dijkstra’s algorithm

26. Shortest Path – Dijkstra’s algorithm

27. Shortest Path – Dijkstra’s algorithm

28. Shortest Path – Dijkstra’s algorithm

29. Shortest Path – Dijkstra’s algorithm

30. Shortest Path – Dijkstra’s algorithm

31. Shortest Path – Dijkstra’s algorithm

32. Shortest Path – Dijkstra’s algorithm

33. Shortest Path – Dijkstra’s algorithm

34. Shortest Path – Dijkstra’s algorithm

35. Shortest Path – Dijkstra’s algorithm

36. Shortest Path – Dijkstra’s algorithm
int dist[N];
int pred[N];
struct C{
int u,c; C(){} C(int a,int b):u(a),c(b){}
bool operator<(const C&a) const {
return c > a.c; } };
// the graph is represented by
// a adjacency list
// a vertex a has adjc[a] neighbors
// adj[a][0] … adj[a][adjc[a]-1] are
// neighbors of vertex
// cost[a][b] has edge cost for (a,b)
int adjc[N];
int adj[N][N];
int cost[N][N];
bool visit[N];
void dijkstra(int s){
memset(visit,0,sizeof(visit));
memset(dist,0x7f,sizeof(dist));
priority_queue<C> pq;
pq.push(C(s,0)); dist[s] = 0;
while(pq.size()>0){
C c = pq.top(); pq.pop(); int u = c.u;
if(visit[u]) continue; visit[u] = true;
for(int i=0;i<adjc[u];++i){
int v = adj[u][i]; if(visit[v]) continue;
if(c.c+cost[u][v]<dist[v]){
dist[v] = dist[u] + cost[u][v];
pred[v] = u;
pq.push(C(v, dist[v])); }

37. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm

38. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights?

39. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights? Bellman-Ford algorithm

40. Shortest Path – Bellman Ford
If a graph G = (V, E) contains a negative-
weight cycle, then some shortest paths may not exist.

41. Shortest Path – Bellman Ford
Bellman-Ford algorithm:
Finds all shortest-path lengths from a sources∈V to all v∈V or determines that a negative-weight cycle exists.

42. Shortest Path – Bellman Ford

43. Shortest Path – Bellman Ford

44. Shortest Path – Bellman Ford

45. Shortest Path – Bellman Ford

46. Shortest Path – Bellman Ford

47. Shortest Path – Bellman Ford

48. Shortest Path – Bellman Ford

49. Shortest Path – Bellman Ford

50. Shortest Path – Bellman Ford

51. Shortest Path – Bellman Ford

52. Shortest Path – Bellman Ford

53. Shortest Path – Bellman Ford

54. Shortest Path – Bellman Ford

55. Shortest Path – Bellman Ford

56. Shortest Path – Bellman Ford

57. Shortest Path – Bellman Ford

58. Shortest Path – Bellman Ford

59. Shortest Path – Bellman Ford

60. Shortest Path – Bellman Ford

61. Shortest Path – Bellman Ford

62. Shortest Path – Bellman Ford

63. Shortest Path – Bellman Ford

64. Shortest Path – Bellman Ford
Theorem.
If G= (V, E)contains no negative-weight cycles, then after the Bellman-Ford algorithm executes, d[v] = δ(s, v)for all v∈V.
Proof.
Let v∈V be any vertex, and consider a shortest path p from s to v with the minimum number of edges.
Since p is a shortest path, we have δ(s, vi) = δ(s, vi–1) + w(vi–1, vi) .

65. Shortest Path – Bellman Ford
If G contains no negative-weight cycles, p is simple. Longest simple path has ≤|V|–1edges.
Initially, d[v0] = 0 = δ(s, v0), and d[v0] is unchanged by subsequent relaxations (because of the lemma from Lecture 14 that d[v] ≥δ(s, v)).
After 1pass through E, we have d[v1] = δ(s, v1).
After 2passes through E, we have d[v2] = δ(s, v2).M
After k - passes through E, we have d[vk] = δ(s, vk).

66. Shortest Path – Bellman Ford
If a value d[v] fails to converge (stop changing) after |V|–1passes, there exists a negative-weight cycle in G reachable from s

67. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights? Bellman-Ford algorithm
DAG?

68. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights? Bellman-Ford algorithm
DAG? Topological Sort + One Round of Bellman-Ford

69. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights? Bellman-Ford algorithm
DAG? Topological Sort + One Round of Bellman-Ford
All Pairs Shortest Path?

70. Shortest Path Un-weighted graph? BFS
Non-negative edge weights? Dijkstra’s algorithm
Negative edge weights? Bellman-Ford algorithm
DAG? Topological Sort + One Round of Bellman-Ford
All Pairs Shortest Path?

71. All Pairs Shortest Path
Non-negative edge weights

72. All Pairs Shortest Path
Non-negative edge weights
Dijkstra’s algorithm |V| times

73. All Pairs Shortest Path
Non-negative edge weights
Dijkstra’s algorithm |V| times
Negative edge

74. All Pairs Shortest Path
Non-negative edge weights
Dijkstra’s algorithm |V| times
Negative edge
Floyd-WarShall (Not the fastest one, but good enough in most cases)

75. All Pairs Shortest Path - Floyd-WarShall
Define Cij(k)=weight of a shortest path from I to j with intermediate vertices belonging to the set {1, 2, …, k}.
Thus, δ(i, j) = Cij(n). Also, Cij(0) = w(i, j) .

76. All Pairs Shortest Path - Floyd-WarShall
Cij(k)= min { Cij(k–1),Cik(k–1)+ C kj(k–1) }
intermediate vertices in {1, 2, …, k}

77. All Pairs Shortest Path - Floyd-WarShall

78. Reference
Depth-First Search
Graph Algorithms
Topological Sorting
Greedy Algorithms (and Graphs)
Shortest Paths I
Shortest Paths II
Shortest Paths III