1. Sorting “Example” with Insertion Sort
A E M P X L E
i = 1
E X A M P L E
i = 2
E X A M P L E
i = 6
A E L M P X E
i = 3
A E X M P L E
i = 7
A E E L M P X
i = 4
A E M X P L E

2. Closest Pair Pseudo-code
ClosestPair(P[1..n])
//input: P[1..n] a collection of points sorted by the x coordinates
//output: Points r and s, with r ≠ s, such that distance(r,s) is the shortest distance among all pairs of points in P
If (n == 2) then return (P[1],P[2])
C1  Select(P[1..mid],c,d)
C2  Select(P[mid+1..n],c,d)
for each k in C1 do
Cand  candidates(k,C2,d)
//at most 6 points will be in D
for each ca in Cand do
if distance(k,ca) < d) then
d  distance(k,ca)
(r,s)  (k,ca)
return (r,s)
mid  floor(n/2)
c  (P[mid].x + P[mid+1].x)/2
(a,b)  ClosestPair(P[1..mid])
(w,z) ClosestPair(P[mid+1..n])
d1  distance(a,b)
d2  distance(w,z)
if (d1 < d2) then
(r,s,d)  (a,b,d1)
else
(r,s,d) (w,z,d2)

3. Graph Representations
G = (E,V) D E A F C B G H
Adjacency lists vs. Adjacency Matrix (Pages 29, 30)
|V|2
Size
|V| + k × 2 × |E|
size in bits of a pointer

4. Depth-First Search
Initially all nodes are marked with 0 (non visited).
Every node that is visited will be marked with a value of 1 or more
The number marking a node indicates its depth relative to other marked nodes
Complexity:
Adjacency Lists: O(|V| + |E|)
Adjacency Matrix: O(|V|2)

5. Breadth-First Search
Initially all nodes are marked with 0 (non visited).
Every node that is visited will be marked with a value of 1 or more
Use a queue (FIFO policy) to control order of visits
Complexity:
Adjacency Lists: O(|V| + |E|)
Adjacency Matrix: O(|V|2)

6. Questions about Graphs
Is the graph connected?
Find the connected components of the graph
Is the graph acyclic?
Find articulation points
Find minimum-number of edges

7. Applications of Graphs
In many problems, the graphs are not constructed apriori because they would be extremely large.
In such situations while the system is solving a problem, it is constructing the graph as-needed.
Example of such a problem: Computer-generated Game plan

8. Planning: Simple ExampleB C C A B
Initial
Goal
(on A Table) (on C A) (on B Table) (clear B) (clear C)
(on C Table) (on B C) (on A B) (clear A)

9. Search Space: A Graph! C A B C A B C B A B A C B A C B A B C C B A B A

10. Homework (Optional)
5.2: 2, 1.b, 4, 6, 7, 10