Computer Science Given an array of $12,000$ integers, where the values range from $1$ to $100,000,$ use a counting sort algorithm to sort the array. Determine the time complexity of counting sort and the space complexity of the counting array used for sorting. A binary search tree $($BST$)$ with $450$ nodes has a maximum height of $45.$ If the BST is unbalanced, calculate the number of nodes in the longest path from the root to a leaf. For a graph with $110$ vertices and $330$ edges, calculate the number of Eulerian circuits using Fleury$$s algorithm, assuming the graph is Eulerian. Given a hash table with $5500$ slots and a load factor of $0\mathrm{.}6,$ estimate the number of elements in the hash table and the average number of elements per slot. A quicksort algorithm is applied to an array of $12,000$ elements with a median$-$of$-$three pivot strategy. Determine the average number of comparisons made by the quicksort algorithm and its expected time complexity. A dynamic programming approach is used to solve the matrix chain multiplication problem for a sequence of $18$ matrices. Calculate the time complexity of the algorithm and the size of the DP table used. For a graph with $140$ vertices and $420$ edges, use Prim$$s algorithm to find the minimum spanning tree. Calculate the time complexity of the algorithm and the number of edges in the resulting spanning tree. A binary heap is used to implement a priority queue with $2400$ elements. Determine the time complexity for inserting a new element and deleting the minimum element from the heap. Given a set of $100$ tasks with start and end times, use the greedy algorithm for interval scheduling to determine the maximum number of non$-$overlapping tasks that can be scheduled. A depth$-$first search $($DFS$)$ is performed on a graph with $180$ nodes and $600$ edges. Calculate the space complexity of the DFS algorithm with respect to the number of nodes and edges. For a network flow problem with $150$ nodes and $450$ edges, apply the Ford$-$Fulkerson method to find the maximum flow. Determine the time complexity of the Ford$-$Fulkerson algorithm in terms of the number of nodes and edges. A set of $28$ integers is sorted using radix sort. Calculate the time complexity of the radix sort algorithm and the maximum number of passes required for sorting. For a shortest path problem in a graph with $130$ nodes and $390$ edges, apply Dijkstra$$s algorithm using a priority queue implemented as a binary heap. Calculate the time complexity of the algorithm and the maximum number of heap operations required.