1. Data Structures and Algorithms I Day 19, 11/3/11 Edge-Weighted Graphs
CMP 338
Data Structures and Algorithms I
Day 19, 11/3/11
Edge-Weighted 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. 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

5. Prim's MST Algorithm Use priority queue to keep track of the edges
mark any node
while exists an edge from marked to unmarked
pick the shortest such edge
add the edge to the MST
mark the unmarked vertex
Use priority queue to keep track of the edges
Optimization: only 1 edge per unmarked node
Need to be able to reduce a key in a priority queue
Running-time: ||E|| + ||V|| lg ||V||

6. Kruskal's MST Algorithm
while exists an unconsidered edge
consider the shortest unconsidered edge
if it would not create a cycle
add the edge to MST
Use priority queue to keep track of the edges
Cycle detection: Disjoint Union / Find algorithm
Edge will create a cycle iff both end-points in the same set
Adding an edge to MST requires union of two sets
Running-time: ||E|| lg ||E||

7. Shortest Path Algorithms
Shortest paths in edge-weighted directed graphs
Problem ill-formed if any negative cycle is reachable
If graph is a DAG
Relax nodes in topological order O(||E|| + ||V||)
If all edges are non-negative (Dijkstra)
Mark and relax nearest unmarked node O(||E||+||V|| lg ||V||)
General edge-weighted directed graphs (Bellman-Ford)
Repeat up to ||V|| times:
Relax nodes changed in previous iteration O(||E|| ||V||)

8. Relaxation void relax (Node n) for Node m in edgeFrom(n) relax(n, m)
void relax (Node n, Node m)
If dist(s, n) + w(n, m) < dist(s, m)
dist(s, m) = dist(s, n) + w(n, m)
parent(m) = n
dist is a Map<Node, Double>
parent is a Map<Node, Node>

9. Dijkstra's Shortest Path Algorithm
mark the source node
while exists an edge from marked to unmarked
pick closest unmarked node n to source
pick shortest edge from marked to n
add edge to Shortest-Path tree
mark and relax n
Use priority queue to order unmarked nodes
Running-time: ||E|| + ||V|| lg ||V||