Lecture 9 Graph Traversal
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1 Lecture 9 Graph Traversal Euiseong Seo (euiseong@skku.edu) SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
2 Need for Graphs One of unifying themes of computer science Closely related to many daily life problems Navigation Circuit generation Social network services Games Computer networks How can a problem be represented as a graph? How to solve a graph problem? SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
3 Graph Notation A graph G = (V, E) V is a set of vertices(nodes) E is a set of edges E = (x, y) where x, y V Ordered or unordered pairs of vertices from V Modeling of problems What are vertices and edges in the followings? Road networks Human interactions Program analysis SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
4 Flavors of Graphs Undirected or directed A graph is undirected if edge (x, y) E implies that (y,x) E, too Otherwise, the graph is directed w x w x y z y z SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
5 Flavors of Graphs Weighted or unweighted If each edge of a graph is assigned a numerical value, or weight, the graph is a weighted graph Otherwise, it is a unweighted graph w x w x y z y z SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
6 Flavors of Graphs Cyclic or acyclic An acyclic graph does not contain any cycles Trees are connected acyclic undirected graphs Directed acyclic graphs are called DAGs SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 6
7 Data Structures for Graphs Assume that a graph G = (V, E) contains n vertices and m edges Adjacency matrix Use a n x n matrix M M[i,j] =, if (i,j) E M[i,j] = 0, if (i,j) E Pros Easy to add or remove edges Easy to find a specific edge, (i,j), if exists Cons Waste of memory space for sparse graphs SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
8 Data Structures for Graphs Adjacency lists in lists / / / / / / / / / SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 8
9 Data Structures for Graphs Adjacency lists in matrices Use arrays instead of linked lists Looks like it combines the worst properties of both, b ut. SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 9
10 List in Array Representation An undirected edge (x,y) appears twice, once as y in x s list and once as x in y s list SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 0
11 Tree Traversal BFS (Breadth First Search) DFS (Depth First Search) SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
12 BFS 8 s 6 9 SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
13 Shortest path from s 8 0 s 6 9 Undiscovered Discovered Queue: s Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
14 8 0 s 6 9 Undiscovered Discovered Queue: s Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
15 8 0 s 6 9 Undiscovered Discovered Queue: s Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
16 8 0 s 6 9 Undiscovered Discovered Queue: Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 6
17 8 0 s already discovered: don't enqueue 6 9 Undiscovered Discovered Queue: Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
18 8 0 s 6 9 Undiscovered Discovered Queue: Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 8
19 8 0 s 6 9 Undiscovered Discovered Queue: Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 9
20 8 0 s 6 9 Undiscovered Discovered Queue: 6 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 0
21 8 0 s 6 9 Undiscovered Discovered Queue: 6 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
22 8 0 s 6 9 Undiscovered Discovered Queue: 6 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
23 8 0 s 6 9 Undiscovered Discovered Queue: 6 8 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
24 8 0 s 6 9 Undiscovered Discovered Queue: 6 8 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
25 8 0 s 6 9 Undiscovered Discovered Queue: Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
26 8 0 s 6 9 Undiscovered Discovered Queue: 8 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 6
27 8 0 s 6 9 Undiscovered Discovered Queue: 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
28 8 0 s 6 9 Undiscovered Discovered Queue: 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 8
29 8 0 s 6 9 Undiscovered Discovered Queue: 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 9
30 8 0 s 6 9 Undiscovered Discovered Queue: 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 0
31 8 0 s 6 9 Undiscovered Discovered Queue: 9 Top of queue Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
32 8 0 s 6 9 Undiscovered Discovered Queue: Top of queue Since Queue is empty, STOP! Finished SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
33 DFS Similar to Backtracking Go as deep as you can Next one is your siblings Stack is an ideal candidate DFS(G, v) for all edges e incident on v do if edge e is unexplored then w opposite(v, e) // return the end point of e distant to v if vertex w is unexplored then mark e as a discovered edge recursively call DFS(G, w) else mark e as a back edge SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
![Finding Paths BFS Tree from x is unique Parent[i] is the node that discovered node i duri ng the BFS originated from x Finding the shortest path from x to y](/docs-images/86/94248438/images/34-2.jpg "in a undirected graph By following the chain of ancestors backward from y to the root SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.")
34 Finding Paths BFS Tree from x is unique Parent[i] is the node that discovered node i duri ng the BFS originated from x Finding the shortest path from x to y in a undirected graph By following the chain of ancestors backward from y to the root SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
35 Connected Components Many seemingly complicated problems reduce t o finding connected components -Puzzle Connected components can be found by using repetitive application of DFS or BFS SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
36 Topological Sorting One of the fundamental operations on DAGs Construct an ordering of the vertices such that a ll directed edges go from left to right Cannot exist over cyclic graphs This gives us a way to process each vertex befor e any of its successors Suppose we seek the shortest (or longest) path from x to y No vertex appearing after y in the topological order can contribute to any such path SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 6
37 Topological Sorting Algorithm Definition A topological sort of a DAG G is a linear ordering of all its vertices such that if G contains a link (u,v), then node u appears before node v in the ordering b b c a c a d d SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
38 Topological Sorting Algorithm find source nodes (indegree = 0) if there is no such node, the graph is NOT DAG in_deg= a in_deg= f c in_deg= e b in_deg= d in_deg= c Queue Sorted: - SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 8
39 Topological Sorting Algorithm span c; decrement in_deg of a, b, e store a in Queue since in_deg becomes 0 a in_deg= f c in_deg= e b in_deg= d in_deg= a c Queue Sorted: c SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 9
40 Topological Sorting Algorithm span a; decrement in_deg of b, f store b, f in Queue since... a f c in_deg= e b d in_deg= f b a Queue Sorted: c a SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) 0
41 Topological Sorting Algorithm span b; store d in Queue a f c in_deg= e b d d f b Queue Sorted: c a b SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
42 Topological Sorting Algorithm span f; decrement in_deg of e no node with in_deg = 0 is found a f c in_deg= e b d d f Queue Sorted: c a b f SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
43 Topological Sorting Algorithm span d; store e in Queue. a f c e b d e d Queue Sorted: c a b f d SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
44 Topological Sorting Algorithm span e; Queue is empty a f c e b d e Queue Sorted: c a b f d e SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu)
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