S500 Algorithms
Homework #1 Solution
1. (1) False
(2) False
2. Reduce the problem into a maxflow problem.
z
Construct flow network by given parameters. (see the above example)
z
Run maxflow algorithm on the constructed network.
z
If f is equal to the number of client, conclude that every client can be connected
simultaneously to a base station.
1
L
L
1
1
1
L
s
C
1
B
2
B
1
t
B
3
Base
C
2
Client
1
1
E = {(s, C
i
) : C
i
∈ Client}
∪ {(C
i
, B
j
) : C
i
∈ Client, B
j
∈ Base and dist(C
i
, B
j
) < r}
∪ {(B
i
, t) : B
i
∈ Base}
1/1
1/2
1/2
1/1
3/4
1/1
1/2
s
t
2/2
2/3
2/3
1/2
5/5
2/2
1/3
s
t
s
1/2
1/1
t
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3. The edge connectivity
z
Algorithm
Construct flow network; directed graph G’=(V’, E’) by given undirected graph G=(V, E)
such that V’ = V and E’ = {(u
i
, v
i
), (v
i
, u
i
) : e
i
=(u
i
, v
i
)∈E} with each edge capacity 1
Choose a vertex u as source
Define f
uv

as maximum flow of G’ with source u and sink v
Min{f
uv
 : v ∈V{u} }
z
Analysis
The number of flow networks is V1
The number of vertices is O(V)
The number of vertices is O(E)
4. a. Separate the vertex into two pieces.
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 Spring '09
 rick
 Algorithms, Graph Theory, Flow network, Maximum flow problem, Maxflow mincut theorem, Network flow, Construct flow network

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