Random Quiz #9

CPS 100, Spring 1997

Due: April 2

Problem 1: Tight Rope (4 points)

The function IsHeightBalanced below determines whether a tree is height-balanced. A tree is height-balanced if, for all nodes, the left and right subtrees differ in height by no more than one. int Height(TNode * tree) { if (tree == 0) return -1; else { int leftHeight = Height(tree->left); int rightHeight = Height(tree->right); return max(leftHeight, rightHeight) + 1; } } bool IsHeightBalanced(TNode * tree) // postcondition: returns true if tree is balanced, else returns false { if (tree != 0) { bool leftBalanced = IsHeightBalanced(tree->left); bool rightBalanced = IsHeightBalanced(tree->right); bool heightOk = abs(Height(tree->left) - Height(tree->right)) <= 1); return leftBalanced && rightBalanced && heightOk; } return true; } The complexity of this function is greater than O(n). What is the complexity and why?

Problem 2: Prime of Life (2 points)

A postive integer n is a prime number if its only divisors are 1 and n. There are infinitely many prime numbers. Using big-Oh notation, give any correct bound on the number of primes less than a positive integer p.

Problem 3: ginorst stuff (0 points)

These problems are from a previous tests. The answers can be found via the class web page. You are urged to try these to see if you understand sorting and other topics we've covered. There is no credit/no penalty for these problems. Part A

Suppose a struct Student for students in a class is (partially) declared as below. Each student has a name, a list of grades, the number of grades stored in the list, and a class identification number. In a class of N students, each student is given an ID number in the range 1 ... N, e.g., in a class of 20 students the numbers 1 ... 20 are assigned one per student.

struct Student { int classID; // ID number string name; // name of student "Jane Doe" Vector<int> grades; // list of grades int numGrades; // # of items stored in grades };

Write the function SortStudents whose header is given below. SortStudents sorts the information in list so that it is in order from smallest (1) to largest (N) student ID number.

Your solution MUST sort in O(N) time!!

(hint: in what slot does the student with ID number 16 belong?)

void SortStudents(Vector<Student> & list, int numElts) // precondition: numElts = # of students in list // ID numbers are unique and in range 1...numElts // postcondition: list is sorted into increasing order by // student ID number // performance: O(n)

Part B

You must solve the problem of determining if two numbers in an array of n numbers sum to a given number k. For example, if the array contains 8, 4, 1, 6, there are numbers that sum to 10 (4 and 6); to 9 (8 and 1); to 16 (use 8 twice); but NOT to 8 (exactly two numbers must be used). Trying all pairs of numbers yields an O(n^2) algorithm. Instead, you must describe how to:

implement an O (n log n) algorithm to solve this problem. You can use the array as storage, but only O(1) other storage for this (e.g., int variables are fine, no vectors).
Hint: given k (the number to sum to), if one of the two numbers is determined, the other one is known --- you need to find if it's in the array by searching efficiently.
implement an O(n) algorithm (average case) by using auxiliary storage to facilitate an efficient search.

In both cases, describe how your algorithm will work (briefly) and justify the running time.

Owen L. Astrachan

Last modified: Fri Mar 28 10:14:50 EST