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Algorithms-Lecture,Section-2-Spring 2025

Algorithms-Lecture,Section-2-Spring 2025

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To find the strongly connected components (SCCs) in a directed graph G(V,E) eficiently, where V

represents the nodes and E represents the directed edges, which specific algorithms and/or operations

below are NOT employed?

Depth-First Search (DFS) on Transposed Graph

❌

Stack operations

❌

Depth-First Search (DFS)

Topological sorting

Graph Transposition

Following the discussion from our lectures, for the text of size n and the pattern of size m, if the number of valid occurrences of the pattern in the text is constant and the hash function is good, then the Rabin-Karp algorithm is expected to run in time _____________ (fill in the blank).

O(n+m)

Θ(n²)

O(m)

O(m logm)

Θ(mn)

Which of the following is an example of a function in Ω ( n² )?

logn

None of these

n²-10n

n logn

n^1.9999

What is the primary difference between BFS and DFS when traversing a graph?

BFS is faster than DFS in all cases

❌

BFS explores nodes level by level, while DFS explores as deep as possible before backtracking

✅

BFS uses a stack while DFS uses a queue

❌

DFS uses a priority queue, while BFS uses a simple list

❌

BFS works only on directed graphs

❌

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Which of the following statements is true about weakly connected graphs?

A weakly connected graph has a directed path between every pair of nodes,

regardless of edge direction.

A weakly connected graph must also be strongly connected.

In a weakly connected graph, there is an undirected path between every pair of

nodes when treating the graph as undirected.

A weakly connected graph has no directed path between any pair of nodes.

A weakly connected graph has a directed path between every pair of nodes, but

only in directed graphs.

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Consider the following statement:

"In order to solve the recurrence T[n]=3T[n/4]+n·log₂n by the Master method, you will need to check the regularity condition."

Select every correct answer below (More than one choice could be selected. You may lose points for wrong selections).

This statement is true and 5/4 would work as the necessary constant

This statement is true and 1/4 would work as the necessary constant

This statement is false

This statement is true and 1/3 would work as the necessary constant

This statement is true and 4/5 would work as the necessary constant

This statement is true and 5/6 would work as the necessary constant

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Solve the following recurrence and find the correct asymptotic bound in terms of k:

T(5^k) = 25⋅T(5^k−3)+k³⋅5^k

T[k]=Θ(5^2k)

T[k]=O(k⋅5^k)

T[k]=Θ(k²⋅5^k)

T[k]=Θ(k³⋅5^k)

T[k]=O(5^k)

Consider a double hashing scheme in which the primary hash function is

h₁(k) = (2k+3)mod 29

and the secondary hash function is

h₂(k) = 1+(3k mod 23).

Assume that the table size is 29.

Find the address returned by probe 1 in the probe sequence for the key value k = 75. Assume that the probe sequence begins at probe 0.

✅

❌

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Consider the following array: [3, 6, 8, 10, 1, 2, 4].

Suppose we perform quicksort on this array using the first element as the pivot. What is the state of the array after the first partitioning step based on our discussion in lectures?

[2, 1, 3, 6, 8, 10, 4]

❌

[1, 2, 3, 4, 6, 8, 10]

❌

[1, 2, 4, 3, 6, 8, 10]

❌

[1, 2, 3, 6, 8, 10, 4]

✅

[1, 2, 3, 6, 8, 4, 10]

❌

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What is the worst-case time complexity of the selection sort algorithm, and what is its primary operation?