The document discusses algorithms for finding the shortest path using Dijkstra's algorithm, optimizing road lighting by finding a minimum spanning tree, determining if a shortest path or minimum spanning tree would change if edge costs were modified, and finding the median of two databases using logarithmically many queries. It provides problems, solutions, and analyses for each algorithm discussed.

1. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
BLG 336E – Analysis of Algorithms II
Practice Session 3
Atakan Aral
Istanbul Technical University – Department of Computer Engineering
25.03.2014
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

2. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Outline
1 Dijkstra Algorithm
Problem
Solution
2 Street Lights
Problem
Solution
3 MST and Shortest Path
True or False?
4 Median of Two DB’s
Problem
Solution
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

3. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Problem
[Midterm 2013] Compute the shortest path from node 4 to
node 1 in the following undirected graph using the greedy
algorithm we learned in the class.
3 5
2
4
6
1
2
7
9 3
5
4
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

4. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 5
2
4
6
1
2
7
9 3
5
4
d(4) : {∞, ∞, ∞, 0, ∞}, s : {4}
d(4) : {∞, 2, ∞, 0, 7}s : {4, 2}
d(4) : {6, 2, 11, 0, 5}, s : {4, 2, 5}
d(4) : {6, 2, 10, 0, 5}, s : {4, 2, 5, 1}
d(4) : {6, 2, 10, 0, 5}, s : {4, 2, 5, 1, 3}
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

5. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Problem
Local government wants to reduce operating costs of road
lighting.
Not every road will be illuminated at night.
For safety, there will be at least one illuminated path
between all junctions.
Suggest an algorithm to optimize the road lighting
What is the maximum saving without endangering the
citizens?
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

6. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Problem
Input contains the junctions and the roads connecting them
As well as, the cost of illuminating each road.
File format is as follows:
7 11
0 1 7
0 3 5
1 2 8
1 3 9
1 4 7
2 4 5
3 4 15
3 5 6
4 5 8
4 6 9
5 6 11
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

7. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

8. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

9. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

10. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

11. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

12. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
g
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

13. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
3 4
1 2
5
6
5
9
11
0
6
8
5
8
9 7
15
7
Total cost: 90 – Optimized cost: 39 – Saving: 51
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

14. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
True or False?
Let G be an arbitrary connected, undirected graph with a
positive distinct cost c(e) on every edge e.
Let T be a MST of G. If we replace edge cost c(e) with
c(e) ∗ c(e) , T must still be MST for G.
Same is valid for c(e) + 5.
It is also valid if negative costs are allowed.
2 4
3
MST depends only on the order of the costs, actual values
are not important as long as order is the same.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

15. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
True or False?
Let G be an arbitrary connected, directed graph with a
positive cost c(e) on every edge e.
Let P be the shortest path between node s and node t in
G. If we replace edge cost c(e) with c(e)2, P must still be
shortest path between node s and node t in G.
Same is valid for c(e) + 5.
e
t
2
s
2
3
e
t
2
s
2
5
For shortest paths, actual values of the costs do matter.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

16. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Problem
You are interested in analyzing some hard-to-obtain data
from two separate databases A and B.
Each database contains n numerical values and you may
assume that no two values are the same.
Determine the median of this set of 2n values
Only way you can access these values is through queries
to the databases: In a single query, you can specify a
value i to one of the two databases, and the chosen
database will return the ith smallest value that it contains.
Since queries are expensive, you would like to compute
the median using as few queries as possible.
Give an algorithm that finds the median value using at
most O(logn) queries.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

17. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
Formulation
Let k = n/2 then A(k) and B(k) are the medians of
databases A and B. We are looking for C(n) where C is the
merged database.
If A(k) < B(k) then:
B(k) > A(1), A(2), ..., A(k)
B(k) > B(1), B(2), ..., B(k − 1)
B(k) is greater than at least 2k − 1 elements.
B(k) ≥ C(n) so no need to consider
B(k + 1), B(k + 2), ..., B(n)
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

18. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
Formulation
Let k = n/2 then A(k) and B(k) are the medians of
databases A and B. We are looking for C(n) where C is the
merged database.
If A(k) < B(k) then:
A(k) < B(k), B(k + 1), ..., B(n)
A(k) < A(k + 1), A(k + 2), ..., A(n)
A(k) is less than at least 2n − 2k − 1 elements.
A(k) ≤ C(n) so no need to consider
A(1), A(2), ..., A(k − 1)
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

19. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
Otherwise, if A(k) > B(k) then:
A(k) > B(1), B(2), ..., B(k)
A(k) > A(1), A(2), ..., A(k − 1)
A(k) ≥ C(n) so no need to consider
A(k + 1), A(k + 2), ..., A(n)
Similarly,
B(k) < A(k), A(k + 1), ..., A(n)
B(k) < B(k + 1), B(k + 2), ..., B(n)
B(k) ≤ C(n) so no need to consider B(1), B(2), ..., B(k −1)
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II

20. Dijkstra Algorithm Street Lights MST and Shortest Path Median of Two DB’s
Solution
median: n, a = 0, b = 0
1 if n == 1 then
2 return min(A(1), B(1))
3 end
4 k = n/2 ;
5 if A(a + k) < B(b + k) then
6 return median(k, a + n/2 , b)
7 end
8 else
9 return median(k, a, b + n/2 )
10 end
Array size is halved at each recursion. So the complexity is
O(logn).
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II