Millennium Management Quantitative Researcher Interview Questions and Answers

Round duration - 60 minutes
Round difficulty - Medium

Technical Interview round with questions based on DSA.

Q1. 
Duplicate Integer in Array
Given an array ARR of size N, containing each number between 1 and N-1 at least once, identify the single integer that appears twice.
Input:
The first line contains an integer...read more
Ans. 
Identify the duplicate integer in an array containing numbers between 1 and N-1.
• Iterate through the array and keep track of the frequency of each element using a hashmap.

• Return the element with a frequency greater than 1 as the duplicate integer.

• Ensure the constraints are met and a duplicate number is guaranteed to be present.
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Q2. 
Sum Root to Leaf Numbers
You are given an arbitrary binary tree consisting of N nodes, each associated with an integer value from 1 to 9. Each root-to-leaf path can be considered a number formed by concat...read more
Ans. 
Calculate the total sum of all root to leaf paths in a binary tree formed by concatenating node values.
• Traverse the binary tree from root to leaf nodes, keeping track of the current path sum

• Add the current node value to the path sum and multiply by 10 for each level

• When reaching a leaf node, add the final path sum to the total sum

• Return the total sum modulo (10^9 + 7)
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Q3. 
Topological Sort Problem Statement
You are given a directed acyclic graph (DAG). Your task is to perform topological sorting of the graph and return any valid ordering.
Explanation:
A directed acyclic g...read more
Ans. 
Implement a function to perform topological sorting on a directed acyclic graph (DAG) and return any valid ordering.
• Create a graph data structure to represent the DAG

• Use depth-first search (DFS) to perform topological sorting

• Maintain a visited array to keep track of visited nodes

• Return the ordering of nodes after DFS traversal
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Round duration - 60 minutes
Round difficulty - Easy

Technical interview round with questions based on DSA.

Q1. 
LCA of Binary Tree Problem Statement
You are given a binary tree consisting of distinct integers and two nodes, X and Y. Your task is to find and return the Lowest Common Ancestor (LCA) of these two nodes...read more
Ans. 
Find the Lowest Common Ancestor (LCA) of two nodes in a binary tree.
• Traverse the binary tree to find the paths from the root to nodes X and Y.

• Compare the paths to find the last common node, which is the LCA.

• Handle cases where one node is an ancestor of the other.

• Consider edge cases like when X or Y is the root node.

• Implement a recursive or iterative solution to find the LCA efficiently.
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Q2. 
Rotated Array Minimum Finder
You are provided with a sorted array that has undergone 'K' rotations (the exact value of 'K' is unknown). A rotation involves shifting each element of the array to the right,...read more
Ans. 
Implement a function to find the minimum number in a rotated sorted array efficiently.
• Use binary search to find the minimum element in the rotated array.

• Compare the mid element with the start and end elements to determine which half of the array to search next.

• Continue the binary search until the minimum element is found.
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Q3. 
Maximum Binary Tree Construction Problem
Given an array TREE of 'N' unique integers, construct a maximum binary tree using the following rules:
1. The root of this tree is the maximum number in TREE.
2. T...read more
Ans. 
Construct a maximum binary tree from an array of unique integers following specific rules.
• Find the maximum number in the array to set as the root of the tree.

• Recursively construct the left subtree with elements before the maximum number.

• Recursively construct the right subtree with elements after the maximum number.
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Round duration - 90 minutes
Round difficulty - Medium

There were 33 questions in total. The objective questions were simple.

Q1. 
Sort Big List Dates Problem Statement
Mary is an enthusiastic party-goer who struggles with remembering event dates. Help Mary by sorting a given list of event dates in an ascending order.
Example:
Inpu...read more
Ans. 
Sort a list of event dates in ascending order based on year, month, and day.
• Sort the list of dates based on year, then month, and finally day.

• Use a sorting algorithm to rearrange the dates in ascending order.

• Ensure the constraints are met for each date in the list.
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Q2. 
Pair Sum Problem Statement
You are given an integer array 'ARR' of size 'N' and an integer 'S'. Your task is to find and return a list of all pairs of elements where each sum of a pair equals 'S'.
Note:
...read more
Ans. 
Given an array and a target sum, find all pairs of elements that add up to the target sum.
• Iterate through the array and for each element, check if the complement (target sum - current element) exists in a hash set.

• If the complement exists, add the pair to the result list.

• Sort the pairs based on the first element and then the second element.

• Handle edge cases like duplicate elements and pairs with the same values.

• Example...read more
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Q3. 
Maximum Sum Problem Statement
Given an integer N, your task is to recursively break it into three integer parts: N / 2, N / 3, and N / 4. You need to compute the maximum sum possible by dividing the numbe...read more
Ans. 
Given an integer N, recursively break it into three parts and find the maximum sum possible.
• Recursively divide N into N/2, N/3, and N/4 to find the maximum sum

• Compare the sum obtained by dividing N with the sum of N itself

• Return the maximum sum for each test case
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