NPTEL MOOC: Algorithms for Big Data July — Sept, 2016
Assignment for Week 7 Total Marks: 20
Name: Roll No.:
Question: 1 2 3 4 5 6 7 8 9 10 Total
Marks: 2 2 2 2 2 2 2 2 2 2 20
Score:
1. (2 marks) Consider a simple non-bipartite 4-regular graph whose normalized adjacency matrix is M. Let the eigen values ofM be 1,0.5,0.4,0.2, . . . ,−0.05,−0.1,−0.6. What is the spectral gap ofM?
A. 0 B. .4 C. .5 D. .6
2. (2 marks) Consider a graph G= (V, E), where |V|=n, whose spectral gap is O(n1). What will be the n1c-mixing time of a random walk onG, where c >0 is a positive constant?
A. O(lognn) B. O(n) C. O(nlogn) D. O(n2)
3. (2 marks) When we want to test if a givend-regular graph is bipartite or not using random walks, we must utilize a variation known as a lazy random walk. For what reason do we use this variation?
A. To ensure that the resulting Markov Chain is aperiodic.
B. To ensure that the resulting Markov Chain is irreducible.
C. To ensure that the resulting Markov Chain is finite.
4. (2 marks) When we use random walks to test for bipartiteness, we use a procedure Odd-Cycle to detect if there exist nodes which we can reach in both an even number of steps and an odd number of steps from a given node. Does this count of steps include the number of self-loops traversed or exclude it?
A. Include.
B. Exclude.
C. Depends on the particular scenario.
5. (2 marks) Consider a graph G = (V, E) that is-far from being bipartite. Furthermore, assume that G is rapidly mixing in the sense that random walks reach near-uniform distribution inO(poly(logn)) steps. Consider using the random walks algorithm (discussed in the lecture) to detect bipartiteness. LetPvobe the probability that a random walk terminates atv and the walk length is odd. LetPvebe the probability that a random walk terminates atv and the walk length is even. Consider the statement “P
v∈V
PvePvo is 0”. Is the statement true or false?
A. True.
B. False.
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For questions 6 through 9, assume that the input stream is the following sequence of weighted edges on the vertex set {1,2, . . . ,10}. Each edge is represented by a triple (u, v, w), whereuand vare the two endpoints andw is the weight.
(2,10,8),(9,10,5),(6,10,2),(8,10,9),(5,9,4),(9,8,1),(5,8,16),(4,5,11),(6,5,12), (3,4,12),(2,5,15),(8,7,7),(2,3,6),(1,2,13),(6,2,14),(7,6,17),(3,1,10),(1,7,3).
6. (2 marks) The edges in the spanning tree (not in any particular order) output by the semi-streaming spanning tree algorithm discussed in the lecture is:
A. (1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10) B. (2,10),(10,9),(6,10),(5,9),(4,5),(8,9),(7,8),(1,7),(1,3) C. (2,10),(10,9),(10,6),(5,9),(6,7),(7,9),(1,2),(1,3),(3,4) D. (1,2),(2,10),(3,4),(4,5),(5,9),(6,10),(7,8),(8,10),(9,10)
7. (2 marks) The edges in the minimum spanning tree (not in any particular order) output by the semi-streaming minimum spanning tree algorithm discussed in the lecture is:
A. (1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10) B. (2,10),(10,9),(6,10),(5,9),(4,5),(8,9),(7,8),(1,7),(1,3) C. (2,10),(10,9),(10,6),(5,9),(6,7),(7,9),(1,2),(1,3),(3,4) D. (1,2),(2,10),(3,4),(4,5),(5,9),(6,10),(7,8),(8,10),(9,10)
8. (2 marks) The edges in the matching (not in any particular order) output by the semi-streaming (approx) maximum cardinality matching algorithm discussed in the lecture is:
A. (3,4),(7,8),(2,10),(5,9) B. (8,7),(2,6),(1,3),(5,4),(9,10) C. (1,2),(3,4),(6,7),(5,8),(2,10) D. (2,1),(5,6),(4,3),(10,9),(7,8)
9. (2 marks) The edges in the matching (not in any particular order) output by the semi-streaming (approx) maximum weight matching algorithm discussed in the lecture (with parameterγ= 2) is:
A. (3,4),(7,8),(2,10),(5,9) B. (8,7),(2,6),(1,3),(5,4),(9,10) C. (1,2),(3,4),(6,7),(5,8),(2,10) D. (2,1),(5,6),(4,3),(10,9),(7,8)
10. (2 marks) Consider the following two graphsGandH, both on the vertex set{1,2,3}. GraphGhas all three edges, each with weight 1. GraphH only has edges{1,2}and{2,3}, each with weight 0.75. IfH is a (1 +α) sparsification ofG, then we call (1 +α) as the sparsification factor ofH with respect toG. Which of the following is the smallest correct sparsification factor ofH with respect toG?
A. 1.25 B. 1.5 C. 1.75 D. 2
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