Given an array of nnn integers, where nnn is $2000,$ you need to find the median using a linear time algorithm. Calculate the time complexity of the algorithm and the number of comparisons required in the worst case. A binary search tree $($BST$)$ contains $60$ nodes, with the maximum height being $12.$ If the tree is not balanced, determine the number of nodes in the longest path from the root to a leaf. For a graph with $30$ vertices and $90$ edges, calculate the number of edges that can be removed while maintaining the graph$$s connectivity. Given a hash table with $500$ slots and a load factor of $0\mathrm{.}8,$ determine the number of collisions that would be expected if each slot is initially empty. A merge sort algorithm is applied to an array of $2000$ elements. Calculate the number of comparisons performed by the merge sort algorithm and its time complexity. A dynamic programming solution is used to solve the longest common subsequence problem for two sequences, each of length $60.$ Determine the time complexity of the algorithm and the size of the DP table. A graph with $120$ nodes and $360$ edges needs to be checked for bipartiteness. Using a BFS$-$based algorithm, calculate the number of additional edges needed to make the graph bipartite. For a depth$-$first search $($DFS$)$ on a graph with $80$ nodes and $200$ edges, determine the space complexity of the DFS algorithm, considering the use of a recursion stack. A priority queue is implemented using a Fibonacci heap with $500$ elements. Calculate the time complexity for inserting a new element and decreasing the key of an existing element. Given a directed graph with $40$ nodes and $100$ edges, determine the time complexity of finding all strongly connected components using Kosaraju's algorithm. A set of $10$ integers is to be sorted using radix sort. Calculate the time complexity of the radix sort algorithm and the maximum number of passes required. For a network flow problem with $50$ nodes and $200$ edges, apply the Ford$-$Fulkerson method to find the maximum flow. Determine the time complexity of the Ford$-$Fulkerson algorithm with respect to the number of nodes and edges. A binary tree with $60$ nodes is traversed using in$-$order traversal. Calculate the number of recursive calls and the space complexity of the traversal algorithm.