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Independent Set True of False? Flow Network
BLG 336E – Analysis of Algorithms IIPractice Session 5
Atakan Aral
Istanbul Technical University – Department of Computer Engineering
29.04.2014
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Outline
1 Independent Set
2 True of False?
3 Flow Network
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Problem
Find an independent set in a path G whose total weight isas large as possible.
Independent Set
Subset of nodes such that no two of them are adjacent(connected by an edge).
6 3
8
6
1
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Attempt 1: Odds vs. evens
1 + 6 + 6 vs. 8 + 3Yields 13 which is sub-optimal.
6 3
8
6
1
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Attempt 2: Heaviest-first
Take 8, remove first 3 nodes then take 6.Yields 14 which is optimal.
6 3
8
6
1
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Attempt 2: Heaviest-first
What about now?Yields 13 which is sub-optimal.
6 3
7
4
5
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Attempt 3: Dynamic Programming
Si : Independent set on v1, v2, ...vi
Xi : Weight of Si
X0 = 0X1 = w1
i > 1:1 vi does not belong to Si → Xi = Xi−12 vi belongs to Si → Xi = wi + Xi−2
So: Xi = max(Xi−1,wi + Xi−2)
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Attempt 3: Dynamic Programming
6 3
8
6
1
X0 = 0 S = ()
X1 = 1 S = (v1)
X2 = max(X1,w2 + X0) = max(1,8 + 0) = 8 S = (v2)
X3 = max(X2,w3 + X1) = max(8,6 + 1) = 8 S = (v2)
X4 = max(X3,w4 + X2) = max(8,3 + 8) = 11 S = (v2, v4)
X5 = max(X4,w5 + X3) = max(11,6 + 8) = 14 S = (v2, v5)
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Problem
Let G be an arbitrary flow network, with a source s, a sinkt , and a positive integer capacity ce on every edge e;let (A,B) be a minimum s − t cut with respect to thesecapacities ce.Now suppose we add 1 to every capacity; then (A,B) isstill a minimum s − t cut with respect to these newcapacities ce + 1.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
False:
b w
a
t
1
s
31
1
1
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Problem
Consider the flow network below.An s − t flow is computed.
What is the value of the flow?
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Problem
Consider the flow network below.An s − t flow is computed.
Is this a maximum s − t flow?
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Problem
Consider the flow network below.An s − t flow is computed.
Find the maximum s − t flow.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
Value is 18.We need to draw the residual graph to see if it is themaximum flow.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
itu
Independent Set True of False? Flow Network
Solution
It is not the maximum flow.An augmenting path with a value of 3 can be found.Since no more augmenting paths are available themaximum flow is 18 + 3 = 21.
Atakan Aral Istanbul Technical University – Department of Computer Engineering
BLG 336E – Analysis of Algorithms II
