1. Ch. 14 Graphs Graph traversals Applications of graphs
Depth-first search
Breadth-first search
Applications of graphs
Topological sorting
Spanning trees
Shortest paths

2. Definitions Graph G = (V, E) where Subgraph
V is the set of vertices (nodes) and E is the set of edges
Subgraph
A subset of a graph’s vertices and edges
Two vertices are adjacent if they are joined by an edge

3. Path Cycle Simple path Simple cycle A seq. of connected edges
A path whose starting vertex and ending vertex are the same
Simple path
A path that contains no cycle
Simple cycle
A cycle that contains no cycle in it

4. Directed graph (digraph) : undirected graph
Connected graph
Every pair of distinct vertices has a path bet’n them
Complete graph
Every pair of distinct vertices has an edge bet’n them
Directed graph (digraph) : undirected graph
Whether edges have direction ?
Weighted graph : unweighted graph
Whether edges have weights ?
Adjacency, multigraph, loop

5. A campus map as a graph
A subgraph

6. Graphs that are
Connected
Disconnected
Complete

7. Multigraph
Loop
Usually these are not considered as graphs.
They are called as a multigraph and a graph w/ loops.

8. A weighted graph

9. A directed graph
~11/10/2003
~5/13/2009

10. Graph의 예 영희 철수 준호 동건 승우 재상
사람들간의 친분 관계를 나타낸 그래프 10

11. 그래프의 예 영희 철수 준호 동건 승우 재상 9 6 5 5 7 9 1 5
친밀도를 가중치로 나타낸 친분관계 그래프 11

12. 그래프의 예 Directed Graph (Digraph, 유향 그래프) 방향을 고려한 친분관계 그래프 영희 철수 준호 동건승우 재상
방향을 고려한 친분관계 그래프 12

13. 그래프의 예 영희 철수 준호 동건 승우 재상 9 6 8 8 5 5 7 9 6 2 1 5
가중치를 가진 Digraph 13

14. Graph Representation
N: # of vetices

15. Adjacency Matrix Undirected Graph의 예 영희 철수 준호 동건 승우 재상 1 1 1 2 3 4 5 61 1 2 4 6 2 3 4 3 5 5 6
Undirected Graph의 예 15

16. Adjacency Matrix Weighted Undirected Graph의 예 영희 철수 준호 동건 승우 재상 9 7 51 1 2 3 4 5 6 9 7 5 6 1 9 6 1 5 2 4 6 2 5 7 3 9 1 4 3 5 5 5 6
Weighted Undirected Graph의 예 16

17. Adjacency Matrix Directed Graph의 예 영희 철수 준호 동건 승우 재상 1 1 1 2 3 4 5 6 11 1 2 4 6 2 3 4 3 5 5 6
Directed Graph의 예 17

18. Adjacency Matrix Weighted Digraph의 예 영희 철수 준호 동건 승우 재상 8 7 5 6 9 2 1 93 4 5 6 8 7 5 6 9 2 1 9 6 1 8 8 5 2 4 6 2 5 7 3 9 6 2 1 4 3 5 5 5 6
Weighted Digraph의 예 18

19. Adjacency Matrix
Digraph의 다른 예 19

20. Adjacency Matrix
Weighted Graph의 다른 예 20

21. Adjacency list A graph is represented by N linked lists
where list i is the list of vertices that is adjacent to vertex i.
In case of weighted graphs, the list also contains the weight values.

22. Adjacency List Undirected Graph의 예 영희 철수 준호 동건 승우 재상 2 3 4 6 1 3 1 2 522

23. Adjacency List Weighted Graph의 예 영희 철수 준호 동건 승우 재상 9 7 2 9 3 7 4 5 6 123

24. A Digraph and Its Adjacency Matrix

25. A Digraph and Its Adjacency List

26. A Weighted Digraph and Its Adjacency List
~5/21/2012

27. Graph Traversals Traversal in a graph G
starts at a vertex u and visits all vertices v for which there is a path bet’n u and v.
Two representatives
Depth-first search
Breadth-first search
~5/14/2007

28. Depth-First Search DFS (v) { Mark v as visited;
for (each unvisited vertex u adjacent to v)
DFS(u); }

29. Non-Recursive Version of Depth-First Search
DFS(v)
{ // All vertices are marked UNVISITED in the beginning
stack.push(v);
mark[v] = VISITED;
while (!stack.isEmpty( )) {
if (no unvisited vertices are adjacent to the stack-top vertex)
stack.pop( ); // backtracking
else {
Select an unvisited vertex w adjacent to the stack-top vertex;
stack.push(w);
mark[w] = VISITED; }
~11/17/2008

30. Breadth-First Search BFS(v)
{ // All vertices are marked UNVISITED in the beginning
queue.enqueue(v);
mark[v] = VISITED;
while (!queue.isEmpty( )) {
w = queue.dequeue( );
for each vertex u adjacent to w {
queue.enqueue(u);
mark[u] = VISITED; }

31. 동일한 트리를 각각 DFS/BFS로 방문하기

32. A Connected Graph with Cycles

33. The results of a depth-first traversal, beginning at vertex a, of the graph in the previous page

34. The results of a breadth-first traversal, beginning at vertex a, of the graph in the previous page
~5/23/2018(제23강)

35. Topological Sorting Condition Topological order
Directed graph without cycles
Topological order
An order of vertices in which vertex x precedes vertex y if there is an edge from x to y
Usually, there are many topological orders for a directed graph.

36. A graph and examples of topological orders
~11/10/2010
~11/21/2011

37. topologicalSort (G)
// G: graph {
for i = 1 to G.size( ) {
Select a vertex v that has no successors;
vertex[i] = v;
Delete from G vertex v and the edges to v; }
List vertex[G.size( )],…,vertex[1]; // a topological order

38. Remove b from theGraph;
add it to aList

39. Spanning Trees Condition Tree Spanning tree of graph G
Undirected connected graph
Tree
A connected graph with no cycles
A tree with n vertices always has n-1 edges.
Spanning tree of graph G
A tree as a subgraph of G that contains all of G’s vertices
Mention observation
~11/19/2008

40. A Spanning Tree

41. DFS/BFS Spanning Trees
DFSTree (v) {
Mark v as visited;
for (each unvisited vertex u adjacent to v) {
Mark the edge (u, v);
DFSTree(u); }
Mention observations.
The edges marked in DFSTree() constitute a DFS spanning tree.

42. The edges marked in BFSTree() constitute a BFS spanning tree.
BFSTree(v)
{ // All vertices are marked UNVISITED in the beginning
queue.enqueue(v);
mark[v] = VISITED;
while (!queue.isEmpty( )) {
w = queue.dequeue( );
for each vertex u adjacent to w {
queue.enqueue(u);
mark[u] = VISITED;
Mark the edge (w, u); }
The edges marked in BFSTree() constitute a BFS spanning tree.

43. A DFS spanning tree rooted at vertex a

44. A BFS spanning tree rooted at vertex a

45. Minimum Spanning Trees
Condition
Weighted undirected graph
The cost of a spanning tree
The sum of the edge weights in the spanning tree
Minimum spanning tree
A spanning tree with the minimum cost

46. Prim Algorithm PrimAlgorithm (v) {
Mark v as visited and include it in the m.s.t.;
while (there are unvisited vertices) {
Find a least-cost edge (x,u) from a visited vertex x
to an unvisited vertex u;
Mark u as visited;
Add the vertex u and the edge (x,u) to the m.s.t.; }
~5/23/2012
Prim’s algorithm is an example of greedy algorithm.
It is a rare case that a greedy algorithm guarantees global optimality.

47. A weighted, connected, undirected graph

48. Prim Algorithm의 작동 예 1

52. ~5/19/2004
~11/25/2004
~5/21/2007
~5/25/2009

53. Prim Algorithm의 작동 예 2 (a) (b) (c) (d) : 방금 S에 포함된 정점 : 방금 이완이 일어난 정점8 8 8 ∞ 10 10 r ∞ ∞ 9 9 5 5 11 ∞ 9 11 13 12 13 12 ∞ ∞ ∞ 11 8
11/13 수정됨 7 8 7 ∞ ∞
(d)
(c) 8 8 10 8 10 S 10 S 5 9 9 5 11 5 9 11 9 13 12 13 12 ∞ 11 12 11 8 7 ∞ 8 7 ∞
: 방금 S에 포함된 정점
: 방금 이완이 일어난 정점

54. (e)
(f) 8 8 5 5 5 S 11 9 11 9 13 12 12 12
(g) 11 12 11 8 8 7 S 8 7 ∞ 8
11/13 수정됨
~5/21/2008 10 9
(i) S
(h) 12 8 8 7 11 5 7 10 8 8 9 9 7 11 7 11 7 8 8 S

55. Shortest Paths Condition Shortest path bet’n two vertices
Weighted digraph
Undirected graph can be also thought to be a digraph in which undirected edge (u,v) means two directed edges (u,v) and (v,u).
Shortest path bet’n two vertices
A path that has the smallest sum of edge weights

56. Dijkstra Algorithm Relaxation!
DijkstraAlgorithm (G, s) // s: the starting vertex {
S = {s} ;
for v = 1 to G.size( )
distance[v] = weight(s,v);
for i = 2 to G.size( ) {
Find the smallest distance[v] s.t. v is not in S;
S = S ∪{v};
for each vertex u not in S {
if (distance[u] > distance[v] + weight(v,u))
distance[u] = distance[v] + weight(v,u); }
Relaxation!
Dijkstra’s algorithm is an example of greedy algorithm.
It is another rare case that a greedy algorithm guarantees global optimality.

57. A weighted directed graph and its adjacency matrix
Manual trace.
A weighted directed graph and its adjacency matrix

58. An example relaxation

59. Dijkstra Algorithm의 작동 예∞ 8 10 8 8 10 8 10 6 ∞ ∞ 2 6 2 18 6 2 9 1 9 1 11 ∞ ∞ 11 1 ∞ 5 9 5 9 ∞ 3 12 3 12 3 12 5 ∞ ∞ ∞ 8 11 ∞ 8 11 8 8 7 4 8 7 4 7 ∞ 8 4 ∞ ∞ 8 8 8 6 10 10 6 10 2 2 1 ∞ 9 12 9 12 9 12 5 12 5 12 5 3 19 ∞ ∞ 11 11 11 8 8 8 7 8 7 4 8 7 4 8 4 19 ∞ ∞

60. 8 8 6 10 10 9 12 9 12 12 5 19 19 11 8 11 8 7 4 19 16
~11/21/2007(top. Sort와 Prim alg.과 순서 바꿈)
~11/15/2010 8 8 10 8 10 2 10 9 1 11 9 12 3 12 5 9 12 5 12 19 8 8 10 11 19 11 7 4 10 2 16 9 1 16 11 9 12 3 12 5 19 11 7 4 16

61. Euler Circuit : The Oldest Graph Application
Is there a cycle that uses every edge exactly once?
To have an Euler circuit, every vertex must have an even degree.

62. Pencil and paper drawings

63. Other Graph Problems Traveling salesman problem Graph planarity
Graph coloring
Graph bisection
Max flow
VLSI circuit placement & routing
Too many more …
~11/28/2005
~5/26/2008
~11/24/2008
~5/27/2009