Computer science Given an array of $13,000$ integers, where the values range from $1$ to $110,000,$ use a bucket sort algorithm to sort the array. Determine the time complexity of bucket sort and the space complexity for storing the bucket array. A binary search tree $($BST$)$ with $500$ nodes has a maximum height of $50.$ If the BST is unbalanced, calculate the number of nodes in the longest path from the root to a leaf. For a graph with $120$ vertices and $360$ edges, calculate the number of Hamiltonian paths using a backtracking approach, assuming the graph is Hamiltonian. Given a hash table with $6000$ slots and a load factor of $0\mathrm{.}5,$ 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 $13,000$ 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 longest common subsequence problem for two sequences, each of length $160.$ Calculate the time complexity of the algorithm and the size of the DP table. For a graph with $150$ vertices and $450$ edges, use Kruskal$$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 $2600$ elements. Determine the time complexity for inserting a new element and removing the maximum element from the heap. Given a set of $110$ tasks with start and end times, use the interval scheduling maximization algorithm to find the maximum number of non$-$overlapping tasks that can be scheduled. A depth$-$first search $($DFS$)$ is performed on a graph with $200$ nodes and $700$ edges. Calculate the space complexity of the DFS algorithm considering both the recursion stack and the adjacency list. For a network flow problem with $160$ nodes and $480$ edges, apply the Edmonds$-$Karp algorithm to find the maximum flow. Determine the time complexity of the Edmonds$-$Karp algorithm in terms of the number of nodes and edges. A set of $30$ integers is sorted using heap sort. Calculate the number of comparisons performed by the heap sort algorithm and its time complexity. For a shortest path problem in a graph with $140$ nodes and $420$ edges, apply the Bellman$-$Ford algorithm to find the shortest path from a single source. Calculate the time complexity of the Bellman$-$Ford algorithm and the maximum number of iterations required.