List of top Algorithms Questions asked in GATE CS

Which of the following statements regarding Breadth First Search (BFS) and Depth First Search (DFS) on an undirected simple graph \( G \) is/are TRUE?

Let \( G \) be an edge-weighted undirected graph with positive edge weights. Suppose a positive constant \( \alpha \) is added to the weight of every edge. Which ONE of the following statements is TRUE about the minimum spanning trees (MSTs) and shortest paths (SPs) in \( G \) before and after the edge weight update?

Which one of the following statements is TRUE for all positive functions \( f(n) \)?

Suppose we are given \(n\) keys, \(m\) hash table slots, and two simple uniform hash functions \(h_1\) and \(h_2\). Further suppose our hashing scheme uses \(h_1\) for the odd keys and \(h_2\) for the even keys. What is the expected number of keys in a slot?

Consider a simple undirected weighted graph \(G\), all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of \(G\) is/are TRUE?

Let \(G\) be a connected undirected weighted graph. Consider the following two statements:
\(S_1:\) There exists a minimum weight edge in \(G\) which is present in every minimum spanning tree of \(G\).
\(S_2:\) If every edge in \(G\) has distinct weight, then \(G\) has a unique minimum spanning tree.
Which one of the following options is correct?

What is the worst-case number of arithmetic operations performed by recursive binary search on a sorted array of size \(n\)?

Consider a complete binary tree with 7 nodes. Let \( A \) denote the set of first 3 elements obtained by performing Breadth-First Search (BFS) starting from the root. Let \( B \) denote the set of first 3 elements obtained by performing Depth-First Search (DFS) starting from the root. The value of \( |A - B| \) is \(\underline{\hspace{2cm}}\).

Consider the string abbccddeee. Each letter in the string must be assigned a binary code satisfying the following properties:
1. For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter.
2. For any two letters of the same frequency, the letter which occurs earlier in the dictionary order is assigned a code whose length is at most the length of the code assigned to the other letter.
Among the set of all binary code assignments which satisfy the above two properties, what is the minimum length of the encoded string?

In a directed acyclic graph with source vertex \( s \), the quality-score of a directed path is the product of the weights of the edges on the path.
For a vertex \( v \neq s \), the quality-score of \( v \) is the maximum among the quality-scores of all paths from \( s \) to \( v \). The quality-score of \( s \) is assumed to be 1.
The sum of the quality-scores of all the vertices in the graph is \(\underline{\hspace{2cm}}\).

Consider the following directed graph:
Which of the following is/are correct about the graph?

Let \(P\) be an array containing \(n\) integers. Let \(t\) be the lowest upper bound on the number of comparisons of the array elements, required to find the minimum and maximum values in an arbitrary array of \(n\) elements. Which one of the following choices is correct?

Consider the following three functions:
\[ f_1 = 10^n, f_2 = n^{\log n}, f_3 = n^{\sqrt{n}} \] Which one of the following options arranges the functions in the increasing order of asymptotic growth rate?

Consider the following array:
\[ 23 32 45 69 72 73 89 97 \] Which algorithm out of the following options uses the least number of comparisons (among the array elements) to sort the above array in ascending order?

Consider the following undirected graph with edge weights as shown. The number of minimum-weight spanning trees of the graph is \(\underline{\hspace{2cm}}\).

Consider the following recurrence relation.
\[ T(n) = \begin{cases} T(n/2) + T(2n/5) + 7n, & \text{if } n > 0 \\ 1, & \text{if } n = 0 \end{cases} \] Which one of the following options is correct?

Define \(R_n\) to be the maximum amount earned by cutting a rod of length \(n\) meters into one or more pieces of integer length and selling them. For \(i>0\), let \(p[i]\) denote the selling price of a rod whose length is \(i\) meters. Consider the array of prices:
\[ p[1]=1,\; p[2]=5,\; p[3]=8,\; p[4]=9,\; p[5]=10,\; p[6]=17,\; p[7]=18 \] Which of the following statements is/are correct about \(R_7\)?

An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components. Let \(T\) be a DFS tree obtained by doing DFS in a connected undirected graph \(G\). Which of the following options is/are correct?

Consider a dynamic hashing approach for 4-bit integer keys:

There is a main hash table of size 4.
The 2 least significant bits of a key are used to index into the main hash table.
Initially, the main hash table entries are empty.
To resolve collisions, the set of keys corresponding to a main hash table entry is organized as a binary tree that grows on demand.
First, the 3^rd least significant bit is used to divide the keys into left and right subtrees.
Further collisions are resolved by subdividing based on the 4^th least significant bit.
A split is done only if needed, i.e., only when there is a collision.

Consider the following state of the hash table. Which of the following sequences of key insertions can cause the above state of the hash table (assume the keys are in decimal notation)?