Under what condition does a directed acyclic graph (DAG) have a unique topological ordering?

Consider two recursive algorithms A and B to solve a problem. The recurrences corresponding to their running times are

A. T(m) = 2T(m/2)+m²

B. T(m) = 8T(m/4)+10m²​

Which of these algorithms is faster (i.e., more efficient)?

What is the time complexity of the Floyd-Warshall all shortest path algorithm for a graph with V vertices? Select the tightest bound.

Consider the following statement: 

"In order to solve the recurrence T[n]=7·T[n/8]+n by the Master method, you will need to check the regularity condition."

Select every correct answer below (More than one choice could be selected. You may lose points for wrong selections). 

• Counting Sort is a comparison-based sorting algorithm.
• Counting Sort is inefficient when input elements are drawn from a small range.
• Counting Sort is a stable sorting algorithm.
• Counting Sort runs in linear time when the range of input values is not significantly larger than the number of elements.
• Counting Sort can efficiently sort floating-point numbers.
• Counting Sort has a worst-case time complexity of O(n log n) and this bound is tight.

How many of the following statements about Counting Sort are true?

Which statement is true about the recurrence relation T(m)=9T(m/9)+m·log(m)?

You are given the array [10, 15, 9, 7, 12], and you will sort this array in increasing order.

If QuickSort is applied using the first element as the pivot and the algorithm follows the partitioning scheme as discussed in the lectures, what will the array look like after the first partitioning step?

What is the space complexity of the dynamic programming solution (done in the classroom) for the TSP with n cities?

Given the array [88, 56, 100, 2, 25], and the selection sort algorithm to sort this array in increasing order, what will the array look like after the first two elements are placed in their correct sorted order?