Algorithms Classroom Activities

1
Algorithms Classroom Activities

• Richard Anderson
• University of Washington

2
What does it mean for an algorithm to be
efficient?

3
Polynomial vs. Exponential Complexity

• Suppose you have an algorithm which takes n!
  steps on a problem of size n
• If the algorithm takes one second for a problem
  of size 10, estimate the run time for the
  following problems sizes

12 14 16
18 20
10 1 second 12 2 minutes 14 6 hours 16 2
months 18 50 years 20 20K years
4
Find a topological order for the following graph
H
E
I
A
D
G
J
C
F
K
B
L
5
How many processors are needed for this example?
6
Prove that you cannot schedule this set of
intervals with two processors
7
Who was Dijkstra?

• What were his major contributions?

8
Solve by unrollingT(n) n 5T(n/2)
Answer nlog(5/2)
9
Classify the following recurrences(Increasing,
Decreasing, Balanced)

• T(n) n 5T(n/8)
• T(n) n 9T(n/8)
• T(n) n2 4T(n/2)
• T(n) n3 7T(n/2)
• T(n) n1/2 3T(n/4)

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Integer Arithmetic
971548028394508438309485670104364384579021796
5702956767 12424310982340990573290750971798984
30928779579277597977
Runtime for standard algorithm to add two n digit
numbers
209506709303468099431859684686877940976671713
3476767930 X 59201750917776347096776793429290970
12308956679993010921
Runtime for standard algorithm to multiply two n
digit numbers
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Subset Sum Problem

• Let w1,,wn 6, 8, 9, 11, 13, 16, 18, 24
• Find a subset that has as large a sum as
  possible, without exceeding 50

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Knapsack Grid
Opt j, K max(Opt j 1, K, Opt j 1, K
wj vj)
4
3
2
1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Weights 2, 4, 7, 10 Values 3, 5, 9, 16
13
Determine the LCS of the following strings
BARTHOLEMEWSIMPSON KRUSTYTHECLOWN
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How good is this algorithm?

• Is it feasible to compute the LCS of two strings
  of length 100,000 on a standard desktop PC? Why
  or why not.

15
Find two augmenting paths
2/5
2/2
0/1
2/4
3/4
3/4
3/3
3/3
1/3
1/3
s
t
3/3
3/3
2/2
1/3
3/3
1/3
2/2
1/3
16
Find a minimum value cut
6
6
5
8
10
3
6
t
2
s
7
4
5
3
8
5
4
17
Enumerate all finite s,t cuts and show their
capacities
s
2
2
-2
1
2
-2
1
1
2
1
t
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Baseball elimination

• Can the Dinosaurs win the league?
• Remaining games
• AB, AC, AD, AD, AD, BC, BC, BC, BD, CD

W L
Ants 4 2
Bees 4 2
Cockroaches 3 3
Dinosaurs 1 5
A team wins the league if it has strictly more
wins than any other team at the end of the
season A team ties for first place if no team has
more wins, and there is some other team with the
same number of wins
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Circuit SAT
Satisfying assignment x1 T, x2 F, x3 F x4
T, x5 T
AND
OR
OR
Find a satisfying assignment
AND
AND
AND
AND
NOT
OR
NOT
OR
AND
AND
OR
NOT
AND
NOT
OR
NOT
AND
x3
x4
x5
x1
x2
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Complement of a Graph

• Defn G(V,E) is the complement of G(V,E) if
  (u,v) is in E iff (u,v) is not in E

1
2
1
2
3
5
3
5
4
4
6
6
7
7
Construct the complement
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Traveling Salesman Problem
Minimum cost tour highlighted

• Given a complete graph with edge weights,
  determine the shortest tour that includes all of
  the vertices (visit each vertex exactly once, and
  get back to the starting point)

3
7
7
2
2
5
4
1
1
4
Find the minimum cost tour