A
B
C
D
E
F
G
1) Use Dijkstra's Algorithm to determine the shortest distance from vertex A to all other vertices in the
graph with the weighted adjacency matrix given below:
A
B
C
D
E
F
G
0
12
∞
3
∞
6
∞
2
0
7
∞
10
∞
4
∞
∞
0
∞
∞
∞
1
7
8
13
0
∞
2
∞
∞
2
3
∞
0
∞
6
∞
5
12
∞
4
0
12
∞
∞
1
2
3
4
0
Fill in the chart below to indicate the steps of the algorithm. Each column stores the estimate of the shortest
distance from A to the labeled vertex.
Add to S
A
D
F
E
B
C
G
B
12
11
10
10
10
10
10
C
∞
16
16
12
12
12
12
D
3
3
3
3
3
3
3
E
∞
∞
9
9
9
9
9
F
6
5
5
5
5
5
5
G
∞
∞
17
15
14
13
13
What is the path reconstruction to get from A to G?
A D  F  E C  G
2) Use Prim's Algorithm starting at vertex B to find the minimum spanning tree of the graph with the
adjacency matrix below:
A
B
C
D
E
F
G
A
0
12
2
3
∞
6
1
B
12
0
7
∞
10
∞
4
C
2
7
0
∞
4
7
1
D
3
∞
∞
0
∞
2
∞
E
∞
10
4
∞
0
∞
6
F
6
∞
7
2
∞
0
12
G
1
4
1
∞
6
12
0
In particular, state each edge considered, and then whether or not that edge is added. (At each iteration of the
algorithm, consider only one edge.)
Added?
Added?
Edge #1: BG
YES
Edge #5: AD
YES
Edge #2: GA
YES
Edge #6: DF
YES
Edge #3: GC
YES
Edge #7: CE
YES
Edge #4: AC
NO, CYCLE
3) Do the same thing for Kruskal’s
Edge #1: AG
Edge #2: CG
Edge #3: AC
Edge #4: DF
Added?
YES
YES
NO, CYLE
YES
Edge #5: AD
Edge #6: CE
Edge #7: BG
Added?
YES
YES
YES
4) Huffman Encoding
Character
A
B
C
D
E
F
G
H
Frequency
16
4
5
31
86
18
10
30
Huffman Code
0110
00000
00001
010
1
0111
0001
001
Construct the Huffman tree for this file below. Afterwards, add the Huffman codes for each of the characters in
the chart above.
Note: There are multiple possible answers to this question based on which side you choose when merging
nodes.
200
/
114
/
49
/
\
19 30,h
/ \
9 10,g
/ \
4,b 5,c
\
86,e
\
65
/
\
31,d 34
/ \
16,a 18,f
Assuming that the file was previously stored using three bit codes for each character, and that it takes no space
to store the Huffman codes themselves, how many bits are saved when encoding this particular file using
Huffman coding?
Old file = 600 bits
New file = 4x16 + 5x4 + 5x5 + 3x31 + 1x86 + 4x18 + 4x10 + 3x30 = 490
Bits saved = 110
5) (5 pts) Show the order in which the nodes in the graph below would be visited in a breadth first search
starting at node number 1. Note: Whenever there is a choice of where to visit first, always visit the lower
numbered vertex.
_________________
|
|
1 ------- 2-------- 5-------8-----11-------------15
|
| \_____6—10—13 /
/
|
|
\ /
/
/
3--------4----------7----------9 ----- 12----14
1, 2, 3, 4, 5, 6, 11, 7, 8, 10, 9, 15, 13, 12, 14
6) (5 pts) Show the order in which the nodes in the graph from question #5 would be visited in a depth first
search starting at node number 1. . Note: Whenever there is a choice of where to visit first, always visit the
lower numbered vertex.
1, 2, 4, 3, 7, 6, 10, 13, 9, 11, 8, 5, 15, 14, 12
7) (10 pts) Write down the solution for each of these recurrence relations utilizing the Master Theorem:
n
a) T (n)  4T ( )  n
2
(n)
e) T(n) = 4T(n/2) + n2
(n2 log n)
e) T(n) = 4T(n/2) + n3
(n3)
d) T(n) = 4T(n/4) + n
(n log n)
e) T(n) = 7T(n/2) + n2
(n log27)
Other Problems solved in the slides:
Divide and Conquer: Skyline, Subset Sum Recursive, Integer Multiplication, Tromino Tiling
Greedy: Single Room Scheduling, Multiple Room Scheduling, Fractional Knapsack
Dynamic Programming: Fibonacci and Change Problem
Topological Sort
Graph Two-Coloring