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Algorithm to determine visible buildings in a 2D plane with given positions and heights
Sort the buildings by their x-coordinates
Traverse the sorted buildings from left to right
For each building, check if it is visible by comparing its height with the maximum height of previously visited buildings
If visible, add it to the list of visible buildings
Return the list of visible buildings
In a group of people with a symmetric relation of knowing each other, there will always be at least two people who know the same number of people.
Consider the person who knows the maximum number of people in the group.
If there is no one who knows the same number of people, then everyone else must know a different number of people.
But this would mean that the total number of people known by everyone else is different fr...
Prove that there exists a rectangle with all corners of the same color in a 2D plane with blue and red points.
Divide the plane into a grid of squares.
By the pigeonhole principle, there must be at least one row or column with four points of the same color.
Consider the pairs of points in that row or column and check if any of them form a rectangle.
Prove that F_nk is divisible by F_n where F_i is the ith Fibonacci number. with f_0 = 0
Use mathematical induction to prove the statement
Base case: F_n0 = 0, F_n is also 0, so 0 is divisible by 0
Inductive step: Assume F_nk is divisible by F_n, prove F_n(k+1) is divisible by F_n
F_n(k+1) = F_nk + F_n(k-1), use the assumption to show that F_nk is divisible by F_n
Therefore, F_n(k+1) is also divisible by F_n
Hence, the statem
Find the largest substring formed from repetition of a smaller string.
Identify all possible substrings of the given string.
Check if each substring can be formed by repeating a smaller string.
Return the largest substring that can be formed from repetition of a smaller string.
Tower Research Capital LLC interview questions for popular designations
I would prefer to develop mathematical models.
I enjoy the process of creating and testing mathematical models.
Developing trading infrastructure is important, but not my primary interest.
Mathematical models can be applied to various industries, not just trading.
Examples of mathematical models I have developed include predictive models for customer behavior and financial forecasting models.
At a party with 7 guests and a host, each guest shakes hands with a certain number of people. How many people did the host shake hands with?
The first guest shakes hands with 1 person, the second with 2 people, and so on.
The kth guest shakes hands with k people.
The host shakes hands with all 7 guests, so the number of people the host shakes hands with is the sum of 1 to 7.
The sum of 1 to n is n(n+1)/2, so the host shake
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Expected length of segment containing (1,0) on a unit circle with three random points.
Use law of cosines to find length of each segment.
Calculate expected value using probability density function.
Answer is (4/pi) + (2/3).
If dist(u,v)>n/2 in an undirected graph, there exists a vertex x such that removing x makes u and v go to different connected components.
Find the shortest path between u and v
If the path length is greater than n/2, then there must be a vertex x on the path
Removing x will separate u and v into different connected components
For any subset of size n+1 of the set S of the first 2n numbers, there exists 2 numbers u and v such that u divides v.
Divide the set S into two subsets of n numbers each.
By the pigeonhole principle, at least one of the subsets contains two numbers whose ratio is an integer.
If the subset contains n+1 numbers, then one of the numbers must be in the subset with the two numbers whose ratio is an integer.
Therefore, there ex
For an odd-sized grid, there cannot be a Hamiltonian cycle in the graph.
A Hamiltonian cycle is a path that visits every vertex exactly once and ends at the starting vertex.
In an n*n grid, there are n^2 vertices and each vertex has degree 4.
For an odd n, the total degree of all vertices is odd, which means there cannot be a Hamiltonian cycle.
I was interviewed in Dec 2016.
A graph has a unique topological sort if and only if it is a directed acyclic graph (DAG).
A topological sort is a linear ordering of the vertices of a graph such that for every directed edge (u, v), vertex u comes before vertex v in the ordering.
To check if a graph has a unique topological sort, we can use depth-first search (DFS) or breadth-first search (BFS) algorithms.
If during the DFS or BFS traversal, we encounter...
Check if a string containing parenthesis is balanced
Use a stack to keep track of opening parenthesis
Iterate through the string and push opening parenthesis onto the stack
If a closing parenthesis is encountered, check if the stack is empty or if the top of the stack is the corresponding opening parenthesis
If the stack is empty or the top of the stack doesn't match, the string is not balanced
After iterating through the s...
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