List of top Computer Science and IT Engineering Questions on Algorithms asked in GATE CS

Let 𝐺 be a weighted directed acyclic graph with 𝑚 edges and 𝑛 vertices. Given 𝐺
and a source vertex 𝑠 in 𝐺, which one of the following options gives the worst case
time complexity of the fastest algorithm to find the lengths of shortest paths from 𝑠
to all vertices that are reachable from 𝑠 in 𝐺?

Let 𝐺(𝑉, 𝐸) be a simple, undirected, edge-weighted graph with unique edge weights.
Which of the following statements about the minimum spanning trees (MST)
of 𝐺 is/are true?

Let 𝐺(𝑉, 𝐸) be an undirected, edge-weighted graph with integer weights. The weight
of a path is the sum of the weights of the edges in that path. The length of a path is
the number of edges in that path.
Let 𝑠∈𝑉 be a vertex in 𝐺. For every 𝑢∈𝑉 and for every 𝑘 ≥0, let 𝑑𝑘(𝑢) denote
the weight of a shortest path (in terms of weight) from 𝑠 to 𝑢 of length at most 𝑘. If
there is no path from 𝑠 to 𝑢 of length at most 𝑘, then 𝑑𝑘(𝑢) = ∞.
Consider the statements:
S1:
For every 𝑘 ≥0 and 𝑢 ∈𝑉, 𝑑𝑘+1(𝑢) ≤𝑑𝑘(𝑢).
S2:
For every (𝑢, 𝑣) ∈𝐸, if (𝑢, 𝑣) is part of a shortest path (in terms of
weight) from 𝑠 to 𝑣, then for every 𝑘≥ 0, 𝑑𝑘(𝑢) ≤𝑑𝑘(𝑣).
Which one of the following options is correct?