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- The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S berepresented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) ofINSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtainedfrom insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n)for i = 1 to n determine j such that i ∈ Kj if j ≠ m + 1 extracted[j] = i let L be the smallest value greater than j for which KL exists KL = KL U Kj, destoying Kjreturn extracted Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where eachnumber stands for its insertion.…The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S berepresented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) ofINSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtainedfrom insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n)for i = 1 to n determine j such that i ∈ ?? if j ≠ m + 1 extracted[j] = i let L be the smallest value greater than j for which KL exists KL = KL U Kj, destroying ????return extracted (1) Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where eachnumber stands for its…The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S berepresented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) ofINSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtainedfrom insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n)for i = 1 to n determine j such that i ∈ Kj if j ≠ m + 1 extracted[j] = i let L be the smallest value greater than j for which KL exists KL = KL U Kj, destoying Kjreturn extracted Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where eachnumber stands for its insertion.…
- QUESTION 2 There are many algorithms exist for keeping binary search trees balanced, among them are AVL and Splay. Discuss the differences between the AVL and Splay algorithms in keeping a binary search tree balanced. In the big-O notation, what is the cost of splaying amortized over a sequence of retrievals, insertions and deletes? Given the following Figure 3. Show step by step splaying process when node d is accessed. Proceed with accessing node bConsider a sort of items, according to their keys, that inserts all the items one at a time into an initially empty regular binary search tree and then applies an in-order traversal to complete the sort. Assume that all items have distinct keys. Using big-Theta notation, what is the worst-case complexity of the sort? What is the average-case complexity of the sort? Now answer the same two questions if an AVL tree is used instead of a regular binary search tree.Assuming that the graph G = (V, E) is represented in Adjacency List format, justify in detail the fact Greedy independent sets can be implemented in O(n 2 + m) worst-case running time, where n = |V |, m = |E|. This will require you to take real care in how the adjustment of Adj is done in line 10. The key is to only update/delete what is really necessary for the Algorithm, rather than being concerned with an accurate representation of the residual graph.
- A hash table with ordered linear probing maintains an order amongkeys considered during the rehashing process. When the keys are encountered,say, in increasing order, the performance of a failed lookup approaches that ofa successful search. Describe how a key might be inserted into the orderedsequence of values that compete for the same initial table entry. ans in 30 min and give accurateNote: Your solution should have O(n) time complexity, where n is the number of elements in l, and O(1) additional space complexity, since this is what you would be asked to accomplish in an interview. Given a linked list l, reverse its nodes k at a time and return the modified list. k is a positive integer that is less than or equal to the length of l. If the number of nodes in the linked list is not a multiple of k, then the nodes that are left out at the end should remain as-is. You may not alter the values in the nodes - only the nodes themselves can be changed.Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (ii) Devise an algorithm that uses a hash table of length n to test whether S is a subset of T. What is the expected running time of your algorithm?
- With p processes, where p is a power of 2, derive formulas for the number of receives and additions that process 0 carries out using(a) the original pseudo-code for a global sum (Process 0 directly receives local sums from all other cores), and(b) the tree-structured global sum.Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (i) Describe a deterministic algorithm that checks whether S is a subset of T. What is the running time of your algorithm? (ii) Devise an algorithm that uses a hash table of length n to test whether S is a subset of T. What is the expected running time of your algorithm?For the traversal log: {X, Y, Z, W, Y, A, B, C, D, Y, C, D, E, F, D, E, X, Y, A, B, M, N}, a. Find maximal forward references. b. Find large reference sequences if the threshold value is 0.3 (or 30%). c. Find maximal reference sequences.