You have two algorithms, A and B, for the same problem. A has a time complexity of O(n log n) and Ω(n), while B has a time complexity of O(n^2) and Ω(log n). Which of the following statements is always true?
Algorithm A is faster than Algorithm B for all input sizes.
Algorithm B is faster than Algorithm A for all input sizes.
We cannot definitively compare the performance of the two algorithms based on the given information.
Algorithm A is faster than Algorithm B for sufficiently large input sizes.
You need to find the smallest k elements in an unsorted array of size n. What is the most efficient time complexity achievable in the average case?
O(k^2)
O(n log n)
O(n^2)
O(n + k log n)
An algorithm processes an input list of size n. It first performs an O(n log n) sorting operation. Then, for each element in the sorted list, it performs a nested loop iterating over all pairs of elements in the list. What is the overall time complexity of this algorithm?
O(n^3)
O(n^2 log n)
You are designing a data structure that supports two operations: 'insert' and 'find median'. 'Insert' adds an element in O(log n) time. Which of the following allows the 'find median' operation to also be achieved in O(log n) time?
A self-balancing binary search tree
A standard binary search tree
A hash table
A sorted array
What is the time complexity of the following code snippet?
def mystery(n): if n <= 1: return 1 return mystery(n-1) + mystery(n-2)
O(log n)
O(2^n)
O(n)
What is the time complexity of inserting an element at the beginning of a dynamically sized array that needs to resize by doubling its capacity when full?
O(1)
Consider a dynamic array implementation where resizing to double the capacity takes O(n) time. If we perform 'n' insertions sequentially, what is the amortized time complexity per insertion?
You are tasked with optimizing a critical function in a real-time trading system. Your theoretical analysis suggests Algorithm A (O(log n)) is superior to Algorithm B (O(n)). However, benchmarking reveals Algorithm B performs better for your typical data size. What is the MOST likely reason for this discrepancy?
The trading system's hardware is optimized for linear-time algorithms like Algorithm B.
Algorithm A's hidden constant factors are significantly larger than Algorithm B's, making it slower for smaller input sizes.
Benchmarking is flawed and does not accurately represent the real-world scenario.
Theoretical analysis is unreliable and cannot predict real-world performance.
Why is time complexity generally considered more important than space complexity in algorithm analysis?
Time complexity is always more difficult to analyze.
Modern computers have abundant memory, while efficient time utilization often remains a bottleneck.
Space complexity is irrelevant in modern computing.
Space complexity cannot be expressed using Big-O notation.
You need to design an algorithm to solve a variation of the knapsack problem with a capacity C and N items. However, you can take multiple copies of the same item. Which algorithmic paradigm best addresses this scenario and provides an efficient solution?
Greedy Algorithm
Divide and Conquer
Depth First Search with Backtracking
Dynamic Programming with a 2D table