1. Data Structures and Algorithms I
CMP 338
Data Structures and Algorithms I
Day 17, 11/1/11
(Undirected and Directed) Graphs

2. Homework 11 due: noon 11/4
Implement 2-3 trees directly (Algorithms pg 452)
public class Homework11
extends AbstractOrderedSymbolTable (or)
implements OrderedSymbolTable
Implement different classes for 2-Nodes and 3-Nodes
Preserve 2-3 tree invariants:
Keys are ordered in the 2-3 tree
All paths from leaves to root are the same length
Correctness test: FrequencyCounter on tale.txt

3. Homework 12 due: 11pm 11/8
Shortest path in a grid (Algorithms p.g. 689)
public class Homework12
public double length(double[][] grid)
grid: NxM two-dimensional array of doubles > 0
Nodes: <r, c> where 0<=r<N and 0<=c<M
Edges: <<r, c>, <r, c+1>> and <<r,c>, <r+1, c>>
weight (<r, c>, <r', c'>): grid(r, c) + grid(r', c')
return “length” of shortest path from <0, 0> to <N-1, M-1>
Extra credit: also handle edges from <r, c> to
<r-1, c> and <r, c-1>

4. Graphs (Mathematics) (Directed) Graph <V, E>
V is a set of vertices
E is a set of edges E  V x V
DAG Directed Acyclic Graph
Undirected Graph
E is symmetric
Edge-Weighted Graph
weight: E → R

5. Graph Vocabulary A path is a sequence edges connecting vertices
simple path: no vertex appears twice
A cycle is a path from a vertex to itself
simple cycle: removing final edge leaves a simple path
A connected component (undirected graph):
A maximal set of connected vertices
A strongly connected component (directed graph):
A maximal set of vertices such that there is a
directed path from any vertex to any other vertex

6. Spanning Tree (Undirected Graph)
A tree in an undirected graph:
A set of connected edges not containing a cycle
A spanning tree or an undirected graph:
A tree that connects each vertex of the graph
A spanning forest of an undirected graph:
Set of spanning trees of the connected components
A minimum spanning tree (MST) of a weighted graph
The spanning tree with minimum total weight

7. Depth-First Search Each node visited exactly once.
Visit each neighbor during visit to a node
void visit (Node n)
if (visited(n)) return;
mark n visited
do stuff
for each neighbor m of n
visit(m)
maybe do more stuff
Trace: (1(2(3)(4(5)(6))(7))(8(9)(A(B)(C))))(D(E)(F))
Example: ConnectedComponent.java

8. Breadth-First Search Each node visited exactly once.
Schedule visit to each neighbor during visit to a node
void visit (Node n)
if (visited(n)) return;
mark n visited
do stuff
for each neighbor m of n
put m on queue of nodes to visit
maybe do more stuff
Trace: (1)(2)(3)(6)(7)(8)(4)(5)(9)(A)(C)(B)(D)(E)(F)
Example: ShortestPath.java