Algorithm GreedyIS(G=(V, E)) 1. n← |V| 2. Adj is an adjacency list structure for E; IS and deg are initialised to all-Os 3. for u 0 to n - 1 do 4. 5. 6. 7. 8. 9. 10. 11. for u 0 to n - 1 12. deg[u] = len (Adj[u]) while (some vertices remain in the graph) do u ← arg min{deg[v]; v € V, IS[v] == 0} IS[u] +1 for (w€ Nbd(u)) do 19 // (set the initial deg values for all nodes) IS[u] ← max{0, IS[u]} TO // (node with min residual degree // gets added to the IS) IS[w] -1 Update Adj, deg to reflect deletion of {u} U Nbd(u) // (mark u's neighbours as disallowed) // (tidy: delete marks of disallowed vertices)
Algorithm GreedyIS(G=(V, E)) 1. n← |V| 2. Adj is an adjacency list structure for E; IS and deg are initialised to all-Os 3. for u 0 to n - 1 do 4. 5. 6. 7. 8. 9. 10. 11. for u 0 to n - 1 12. deg[u] = len (Adj[u]) while (some vertices remain in the graph) do u ← arg min{deg[v]; v € V, IS[v] == 0} IS[u] +1 for (w€ Nbd(u)) do 19 // (set the initial deg values for all nodes) IS[u] ← max{0, IS[u]} TO // (node with min residual degree // gets added to the IS) IS[w] -1 Update Adj, deg to reflect deletion of {u} U Nbd(u) // (mark u's neighbours as disallowed) // (tidy: delete marks of disallowed vertices)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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![Algorithm GreedyIS (G=(V, E))
1. n← VI
2. Adj is an adjacency list structure for E; IS and deg are initialised to all-Os
3. for u
0 to n - 1 do
deg[u] = len (Adj[u])
while (some vertices remain in the graph) do
0}
4.
5.
6.
7.
8.
9.
10.
11. for u 0 to n - 1
12.
13. return IS
u ← arg min{deg[v]; v € V, IS[v] =
==
IS[u] ← 1
for (w
Nbd(u)) do
// (set the initial deg values for all nodes)
IS[u]max{0, IS[u]}
IS[w] -1
Update Adj, deg to reflect deletion of {u} U Nbd(u)
// (node with min residual degree
// gets added to the IS)
Algorithm
// (mark u's neighbours as disallowed)
// (tidy: delete marks of disallowed vertices)
A(iv) Assuming that the graph G = (V, E) is represented in Adjacency List format, justify in detail the
fact GreedylS can be implemented in O(n² + m) worst-case running time, where n = |V|, m = |E|.
=
note: this will require you to take real care in how the adjustment of Adj is done in line 10.
The key is to only update/delete what is really necessary for the Algorithm, rather than being
concerned with an accurate representation of the residual graph.
A(v) Consider an unweighted bipartite graph G = (LUR, E), with L = {u₁,..., u[n/3]}{W₁, W2, W3, W₁},
and R= {₁,..., Un}. The edge set E = E₁ UE2 consists of the following edges:
E₁ = {(ui, vi) i = 1,..., , [n/3]} U
{(ui, Vi+[n/31): i = 1,..., [n/3]} U
{(ui, Vi+2[n/3]): i = 1,..., [n/3]}
{(wi, vj) i = 1,..., 4, j = 1,...,n}
E2
Essentially, the E2 edges form a complete bipartite graph between the 4 special vertices {w₁, W2, W3, W₁}
and the set R, while the edge set E₁ is a subgraph where each of the vertices {u₁,...,un/3]} have
3 adjacent edges and all of the vertices in R have 1 adjacent edge.
Work out the independent set I which will be constructed by GreedyIS, justifying your answer with
respect to the degrees of the different vertices as the algorithm proceeds. Show that this set will be
approximately 1/3 of the size of the true maximum independent set for this graph?
note: You are welcome to assume n is a multiple of 3.
A(vi) How would you alter the graph construction of (v) to get progressively worse "approximation
factors" for the result returned by GreedyIS?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe45ffb40-fb97-4856-9f50-b44beb4208a0%2Fc0791299-a202-4fa2-8c72-e848a6cc7f71%2Fzpqr86l_processed.png&w=3840&q=75)
Transcribed Image Text:Algorithm GreedyIS (G=(V, E))
1. n← VI
2. Adj is an adjacency list structure for E; IS and deg are initialised to all-Os
3. for u
0 to n - 1 do
deg[u] = len (Adj[u])
while (some vertices remain in the graph) do
0}
4.
5.
6.
7.
8.
9.
10.
11. for u 0 to n - 1
12.
13. return IS
u ← arg min{deg[v]; v € V, IS[v] =
==
IS[u] ← 1
for (w
Nbd(u)) do
// (set the initial deg values for all nodes)
IS[u]max{0, IS[u]}
IS[w] -1
Update Adj, deg to reflect deletion of {u} U Nbd(u)
// (node with min residual degree
// gets added to the IS)
Algorithm
// (mark u's neighbours as disallowed)
// (tidy: delete marks of disallowed vertices)
A(iv) Assuming that the graph G = (V, E) is represented in Adjacency List format, justify in detail the
fact GreedylS can be implemented in O(n² + m) worst-case running time, where n = |V|, m = |E|.
=
note: this will require you to take real care in how the adjustment of Adj is done in line 10.
The key is to only update/delete what is really necessary for the Algorithm, rather than being
concerned with an accurate representation of the residual graph.
A(v) Consider an unweighted bipartite graph G = (LUR, E), with L = {u₁,..., u[n/3]}{W₁, W2, W3, W₁},
and R= {₁,..., Un}. The edge set E = E₁ UE2 consists of the following edges:
E₁ = {(ui, vi) i = 1,..., , [n/3]} U
{(ui, Vi+[n/31): i = 1,..., [n/3]} U
{(ui, Vi+2[n/3]): i = 1,..., [n/3]}
{(wi, vj) i = 1,..., 4, j = 1,...,n}
E2
Essentially, the E2 edges form a complete bipartite graph between the 4 special vertices {w₁, W2, W3, W₁}
and the set R, while the edge set E₁ is a subgraph where each of the vertices {u₁,...,un/3]} have
3 adjacent edges and all of the vertices in R have 1 adjacent edge.
Work out the independent set I which will be constructed by GreedyIS, justifying your answer with
respect to the degrees of the different vertices as the algorithm proceeds. Show that this set will be
approximately 1/3 of the size of the true maximum independent set for this graph?
note: You are welcome to assume n is a multiple of 3.
A(vi) How would you alter the graph construction of (v) to get progressively worse "approximation
factors" for the result returned by GreedyIS?
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