[SOLVED] data structure CSC263H1, Fall 2019 Problem Set 5

CSC263H1, Fall 2019 Problem Set 5
General instructions
CSC263H1: Problem Set 5
Due Tuesday November 5 before 10pm
Please read the following instructions carefully before starting the problem set. They contain important information about general problem set expectations, problem set submission instructions, and reminders of course policies.
Your problem sets are graded on both correctness and clarity of communication. Solutions that are technically correct but poorly written will not receive full marks. Please read over your solutions carefully before submitting them.
Each problem set may be completed in groups of up to three. If you are working in a group for this problem set, please consult the articles on collaboration and plagiarism on posted on quercus.
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The required filename for this problem set is problem set5.pdf.
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CSC263H1, Fall 2019 Problem Set 5
1. 10 marks Amortized analysis: Binomial Heaps
a Using the accounting method, prove that in a binomial heap, the amortized cost of BINOMIAL HEAPINSERT is in O1 and amortized cost of BINOMIALHEAPEXTRACTMIN is in Olog n, where n is the number of elements in the binomial heap. Only consider sequences containing insert and extractmin operations.
In your analysis, let 1 denote the maximum of:
the time to create a binomial heap with one element,
the time to compare the sizes of two binomial trees, the value of their roots, and link them
together, and
the time to update enough pointers to move a tree from one linked list to another.
Your solution should include at least a clearly stated credit invariant, and clear arguments that the invariant is maintained by the operations.
Page allowance: Please neatly present your solution using 12 pages. Any additional pages will not be marked.
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CSC263H1, Fall 2019 Problem Set 5
2. 15 marks Amortized analysis: Sets
Consider the following data structure for representing a dynamic set, supporting only search and insert operations. The elements of the set are stored in a singly linked list of sorted arrays, where the number of elements in each array is a power of 2 and the sizes of all the arrays in the list are different. Each element in the set occurs in exactly one array. The arrays in the linked list are kept in order of increasing size.
To perform SEARCH, perform binary search separately on each array in the list until either the desired element is found or all arrays have been considered.
To insert a new element x into the set given the precondition that x is not already in the set,
a create a new array of size 1 containing x
b insert this new array at the beginning of the linked list
c while the linked list contains 2 arrays of the same size, merge the 2 arrays into one array of twice the size
a 5 marks What is the worstcase time, to within a constant factor, for performing SEARCH when the set has size n? Justify your answer.
b 5 marks What is the worstcase time, to within a constant factor, for performing INSERT when the set has size n? Justify your answer.
c 5 marks Use the accounting method to prove that the amortized insertion time in a sequence of n insertions, starting with an initially empty set, is log n.
Page allowance: Please neatly present your solution using 12 pages. Any additional pages will not be marked.
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