1
GATE CSE 2020
Numerical
+2
-0.67
Consider a graph G = (V, E), where V = {v1, v2, ...., v100},
E = {(vi, vj) | 1 ≤ i < j ≤ 100}, and weight of the edge (vi, vj) is |i - j|. The weight of the minimum spanning tree of G is ______.
2
GATE CSE 2018
Numerical
+2
-0
Consider the weights and values of items listed below. Note that there is only one unit of each item.

Item number Weight
(in Kgs)
Value
(in Rupees)
1 10 60
2 7 28
3 4 20
4 2 24

The task is to pick a subset of these items such that their total weight is no more than $$11$$ $$Kgs$$ and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $$V$$opt. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $$V$$greedy.

The value of $$V$$opt $$−$$ $$V$$greedy is ____________.

3
GATE CSE 2018
Numerical
+2
-0
Consider the following undirected graph $$G:$$

Choose a value for $$x$$ that will maximize the number of minimum weight spanning trees $$(MWSTs)$$ of $$G.$$ The number of $$MWSTs$$ of $$G$$ for this value of $$x$$ is ______.

4
GATE CSE 2016 Set 1
+2
-0.6
$$G = (V,E)$$ is an undirected simple graph in which each edge has a distinct weight, and e is a particular edge of G. Which of the following statements about the minimum spanning trees $$(MSTs)$$ of $$G$$ is/are TRUE?

$$\,\,\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,\,\,\,$$ If $$e$$ is the lightest edge of some cycle in $$G,$$ then every $$MST$$ of $$G$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$includes $$e$$
$$\,\,\,\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,\,\,\,\,\,$$ If $$e$$ is the heaviest edge of some cycle in $$G,$$ then every $$MST$$ of $$G$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$excludes $$e$$

A
$${\rm I}$$ only
B
$${\rm I}$$$${\rm I}$$ only
C
both $${\rm I}$$ and $${\rm II}$$
D
neither $${\rm I}$$ nor $${\rm I}$$$${\rm I}$$
GATE CSE Subjects
EXAM MAP
Medical
NEET