## CS 2401 Final Exam

Date: Tuesday, December 7, 2010.
Name: ____________________________________________

1-2. Recursion:

1. Write a recursive method for transforming a decimal number into a binary number.
2. Trace your method on the example of n = 10.
3. Describe advantages and disadvantages of recursion vs. iteration in solving a problem, from the viewpoint of easiness to write, easiness to understand, and running time.
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3. Multi-D Arrays:
1. Write a method that, given a general square matrix A, returns the sum of the diagonal elements from the upper-left to lower-right; this sum is called a trace of the matrix. For example, for the matrix
```  1  2  3
4  5  6
7  8  9
```
your method should return the value 1 + 5 + 9 = 15.
2. Trace your method on the example of this matrix.
3. Using big-oh notation, what is the wost-case running time for your algorithm?

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4. Stacks:
1. Rewrite the expresion 12 * 7 - (20 + 10) in postfix form.
2. Show, step by step, how a stack can be used to compute the value of the resulting postfix expression.

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5. Queues:
1. Show, step by step, the state of a queue -- implemented as an 3-element array - when we first add 12 and 7, then dequeue, then add 20 and 1, dequeue again and add 0.
2. Write a method for dequeuing a queue represented as an array.
3. Using big-oh notation, what is the worst-case running time for your dequeuing method?

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6-7. Sorting:
1. Show, step by step, how bubble sort, selection sort, insertion sort, mergesort, quicksort, and heapsort will sort a list consisting of the elements 12, 7, 20, 1, and 0.
2. Write a code for one of these sorting algorithms.
3. Draw a table listing the worst-case and average-case complexity for each of the six sorting algorithms.
4. True or False: It is possible to implement a sorting algorithm that is much faster than heapsort. Justify your answer.

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8. Binary Search Trees:
1. Show, step by step, what will happen if we add elements 12, 7, 20, 1, and 0 to an initially empty AVL binary search tree; do not forget to balance at every step at which balancing is necessary.
2. Show how the resulting tree will look like in an array-based implementation of trees (leave blank spaces for any nodes that do not exist in the tree).
3. In 1-2 sentences, what is the purpose of balancing?
4. State the condition that every node in a binary search tree must satisfy.
No code is needed.

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9. Search:
1. Show, step by step, how sequential search and binary search will look for an element 9 in the sorted list 0, 1, 7, 12, and 20.
2. What is the worst-case and average complexity of each search?
3. What are relative advantages and disadvantages of these two search algorithms?
4. Write down a method implementing sequential search and another implementing binary search. Assume that the list is implemented as an array.

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10. Hash Tables:
1. Show, step by step, how the hash table with four "buckets" will looks like if we sequentially add to it elements 7, 12, 20, 1, and 0. Assume that as a hash function, we take remainder modulo 4:
h(n) = n % 4.