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computer science
introduction to algorithms

Questions and Answers of Introduction to Algorithms

Describe a sequence of n insertions in a heap that requires Ω(nlogn) time to process.
Draw an example of a heap whose keys are all the odd numbers from 1 to 59 (with no repeats), such that the insertion of an entry with key 32 would cause up-heap bubbling to proceed all the way up to
Illustrate all the steps of the adaptable priority queue call replaceKey(e, 18) for entry e storing (5, A) in the heap of Figure 9.1. (4,C) (5,A) (6,Z) (15,К) (9,F) (7,Q) (20,B) (16,X) (25,J)
Illustrate all the steps of the adaptable priority queue call remove(e) for entry e storing (16,X) in the heap of Figure 9.1. (4,C) (5,A) (6,Z) (15,К) (9,F) (7,Q) (20,B) (16,X) (25,J) (14,E) (12,H)
Let H be a heap storing 15 entries using the array-based representation of a complete binary tree. What is the sequence of indices of the array that are visited in a preorder traversal of H? What
Illustrate the execution of the in-place heap-sort algorithm on the following input sequence: (2, 5, 16, 4, 10, 23, 39, 18, 26, 15).
At which positions of a heap might the third smallest key be stored?
When Bob wants to send Alice a message M on the Internet, he breaks M into n data packets, numbers the packets consecutively, and injects them into the network. When the packets arrive at Alice’s
A useful operation in databases is the natural join. If we view a database as a list of ordered pairs of objects, then the natural join of databases A and B is the list of all ordered triples (x,y,
Redo Exercise C-7.51 assuming L is implemented with an array list.Exercise C-7.51There is a simple algorithm, called bubble-sort, for sorting a list L of n comparable elements. This algorithm scans
There is a simple algorithm, called bubble-sort, for sorting a list L of n comparable elements. This algorithm scans the list n−1 times, where, in each scan, the algorithm compares the current
Design a circular positional list ADT that abstracts a circularly linked list in the same way that the positional list ADT abstracts a doubly linked list.
An array is sparse if most of its entries are null. A list L can be used to implement such an array, A, efficiently. In particular, for each nonnull cell A[i], we can store a pair (i,e) in L, where e
Modify the LinkedPositionalList class to support amethod swap(p, q) that causes the underlying nodes referenced by positions p and q to be exchanged for each other. Relink the existing nodes; do not
Describe a method for performing a card shuffle of a list of 2n elements, by converting it into two lists. A card shuffle is a permutation where a list L is cut into two lists, L1 and L2, where L1 is
Describe a nonrecursive method for reversing a positional list represented with a doubly linked list using a single pass through the list.
Repeat the previous problem, but use knowledge of the size of the list to traverse from the end of the list that is closest to the desired index.Previous problemSuppose we want to extend the
Prove that when using a dynamic array that grows and shrinks as in the previous exercise, the following series of 2n operations takes O(n) time: n insertions at the end of an initially empty list,
Complete the previous exercise, except using a dynamic array to provide unbounded capacity.
Let L be a list of n items maintained according to the move-to-front heuristic. Describe a series of O(n) accesses that will reverse L.
Demonstrate how to use the java.util.Colletions.reverse method to reverse an array of objects.
Describe how to implement a method, alternateIterator( ), for a positional list that returns an iterator that reports only those elements having even index in the list.
Consider an implementation of the array list ADT using a dynamic array, but instead of copying the elements into an array of double the size (that is, fromN to 2N) when its capacity is reached, we
Give an implementation of the stack ADT using an array list for storage.
Give a complete implementation of the Deque ADT using a fixed-capacity array, so that each of the update methods runs in O(1) time.
Repeat the previous problem using a singly linked list for storage, and a maximum capacity specified as a parameter to the constructor.Previous problemThe introduction of Section 6.1 notes that
Design an ADT for a two-color, double-stack ADT that consists of two stacks—one “red” and one “blue”—and has as its operations color-coded versions of the regular stack ADT operations.
When a share of common stock of some company is sold, the capital gain (or, sometimes, loss) is the difference between the share’s selling price and the price originally paid to buy it. This rule
Suppose you have two nonempty stacks S and T and a deque D. Describe how to use D so that S stores all the elements of T below all of its original elements, with both sets of elements still in their
Describe how to implement the deque ADT using two stacks as the only instance variables. What are the running times of the methods?
Give a pseudocode description for an array-based implementation of the doubleended queue ADT. What is the running time for each operation?
Show how to use a stack S and a queue Q to generate all possible subsets of an n-element set T nonrecursively.
Describe a nonrecursive algorithm for enumerating all permutations of the numbers {1,2, . . . ,n} using an explicit stack.
Postfix notation is an unambiguousway of writing an arithmetic expressionwithout parentheses. It is defined so that if “(exp1)op(exp2)” is a normal fully parenthesized expression whose operation
Give a simple adapter that implements the queue ADT while using an instance of a deque for storage.
Give a simple adapter that implements the stack ADT while using an instance of a deque for storage.
Provide a nonrecursive implementation of the drawInterval method for the English ruler project of Section 5.1.2. There should be precisely 2c −1 lines of output if c represents the length of the
Implement a recursive method with calling signature find(path, filename) that reports all entries of the file system rooted at the given path having the given file name.
Give a recursive definition of a singly linked list class that does not use any Node class.
Describe a recursive algorithm that will check if an array A of integers contains an integer A[i] that is the sum of two integers that appear earlier in A, that is, such that A[i] = A[ j]+A[k] for
Use recursion to write a Javamethod for determining if a string s hasmore vowels than consonants.
Write a short recursive Java method that determines if a string s is a palindrome, that is, it is equal to its reverse. Examples of palindromes include 'racecar' and 'gohangasalamiimalasagnahog'.
In the Towers of Hanoi puzzle, we are given a platform with three pegs, a, b, and c, sticking out of it. On peg a is a stack of n disks, each larger than the next, so that the smallest is on the top
Write a recursive method that will output all the subsets of a set of n elements (without repeating any subsets).
Describe an efficient recursive algorithm for solving the element uniqueness problem, which runs in time that is at most O(n2) in the worst case without using sorting.
Describe a recursive algorithm to compute the integer part of the base-two logarithm of n using only addition and integer division.
Describe a way to use recursion to compute the sum of all the elements in an n×n (two-dimensional) array of integers.
Draw the recursion trace for the execution of method PuzzleSolve(3,S,U), from Code Fragment 5.11, where S is empty and U = {a,b,c,d}. Algorithm PuzzleSolve(k, S, U): Input: An integer k, sequence S,
For each of the algorithms unique1 and unique2, which solve the element uniqueness problem, perform an experimental analysis to determine the largest value of n such that the given algorithm runs in
Perform an experimental analysis to test the hypothesis that Java’s Array.sort method runs in O(nlogn) time on average.
An array A contains n integers taken from the interval [0,4n], with repetitions allowed. Describe an efficient algorithm for determining an integer value k that occurs the most often in A. What is
Let p(x) be a polynomial of degree n, that is, p(x) = Σni=0 aixi.a. Describe a simple O(n2)-time algorithm for computing p(x).b. Describe an O(nlogn)-time algorithm for computing p(x), based upon a
Let S be a set of n lines in the plane such that no two are parallel and no three meet in the same point. Show, by induction, that the lines in S determine Θ(n2) intersection points.
Consider the following “justification” that the Fibonacci function, F(n) is O(n): Base case (n ≤ 2): F(1) = 1 and F(2) = 2. Induction step (n > 2): Assume claim true for n′ < n.
Determine the total number of grains of rice requested by the inventor of chess.
There is a well-known city (which will go nameless here) whose inhabitants have the reputation of enjoying a meal only if that meal is the best they have ever experienced in their life. Otherwise,
Given an n-element array X of integers, AlgorithmC executes an O(n)-time computation for each even number in X, and an O(logn)-time computation for each odd number in X. What are the best-case and
Given an n-element array X, Algorithm B chooses logn elements in X at random and executes an O(n)-time calculation for each. What is the worst-case running time of Algorithm B?
Algorithm A executes an O(logn)-time computation for each entry of an array storing n elements. What is its worst-case running time?
Show that ⌈ f (n)⌉ is O( f (n)), if f (n) is a positive nondecreasing function that is always greater than 1.
Show that n2 is W(nlogn).
Show that if p(n) is a polynomial in n, then log p(n) is O(logn).
Show that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).
Show that O(max{ f (n),g(n)}) = O( f (n)+g(n)).
Show that if d(n) is O( f (n)) and f (n) is O(g(n)), then d(n) is O(g(n)).
Show that if d(n) is O( f (n)) and e(n) is O(g(n)), then d(n)−e(n) is not necessarily O( f (n)−g(n)).
Show that if d(n) is O( f (n)) and e(n) is O(g(n)), then d(n)+e(n) is O( f (n) + g(n)).
Graph the functions 8n, 4nlogn, 2n2, n3, and 2n using a logarithmic scale for the x- and y-axes; that is, if the function value f (n) is y, plot this as a point with x-coordinate at logn and
In the children’s game, Duck, Duck, Goose, a group of children sit in a circle. One of them is elected “it” and that person walks around the outside of the circle. The person who is “it”
Design a RandomCipher class as a subclass of the SubstitutionCipher from Exercise P-3.40, so that each instance of the class relies on a random permutation of letters for its mapping.Exercise
Implement a class, SubstitutionCipher, with a constructor that takes a string with the 26 uppercase letters in an arbitrary order and uses that as the encoder for a cipher (that is, A is mapped to
Write a program that can perform the Caesar cipher for English messages that include both upper- and lowercase characters.
Perform the previous project, but use a doubly linked list. Moreover, your implementation of remove(i) should make the fewest number of pointer hops to get to the game entry at index i.
Write a Java program for a matrix class that can add and multiply arbitrary twodimensional arrays of integers.
Implement the clone( ) method for the DoublyLinkedList class.
Implement the clone( ) method for the CircularlyLinkedList class.
Solve the previous problemusing inheritance, such that a DoublyLinkedList class inherits from the existing CircularlyLinkedList, and the DoublyLinkedList.Node nested class inherits from
Implement a circular version of a doubly linked list, without any sentinels, that supports all the public behaviors of the original as well as two new update methods, rotate( ) and rotateBackward( ).
Our implementation of a doubly linked list relies on two sentinel nodes, header and trailer, but a single sentinel node that guards both ends of the list should suffice. Reimplement the
Given a circularly linked list L containing an even number of nodes, describe how to split L into two circularly linked lists of half the size.
Suppose you are given two circularly linked lists, L and M. Describe an algorithm for telling if L and M store the same sequence of elements (but perhaps with different starting points).
Write a Java method that takes two three-dimensional integer arrays and adds them componentwise.
Write a method, shuffle(A), that rearranges the elements of array A so that every possible ordering is equally likely. You may rely on the nextInt(n) method of the java.util.Random class, which
Implement the equals( ) method for the DoublyLinkedList class.
Implement the equals( ) method for the CircularlyLinkedList class, assuming that two lists are equal if they have the same sequence of elements, with corresponding elements currently at the front of
What is the difference between a shallow equality test and a deep equality test between two Java arrays, A and B, if they are one-dimensional arrays of type int? What if the arrays are
Implement a rotate( ) method in the SinglyLinkedList class, which has semantics equal to addLast(removeFirst( )), yet without creating any new node.
Write a Java method that repeatedly selects and removes a random entry from an array until the array holds no more entries.
Write a Java program that can “make change.” Your program should take two numbers as input, one that is a monetary amount charged and the other that is a monetary amount given. It should then
Write a Java program that inputs a list of words, separated by whitespace, and outputs how many times each word appears in the list. You need not worry about efficiency at this point, however, as
Define a Polygon interface that has methods area( ) and perimeter( ). Then implement classes for Triangle, Quadrilateral, Pentagon, Hexagon, and Octagon, which implement this interface, with the
Write a Java program that simulates a system that supports the functions of an ebook reader. You should include methods for users of your system to “buy” new books, view their list of purchased
Write a simulator as in the previous project, but add a boolean gender field and a floating-point strength field to each Animal object. Now, if two animals of the same type try to collide, then they
Write a Java programto simulate an ecosystem containing two types of creatures, bears and fish. The ecosystemconsists of a river, which is modeled as a relatively large array. Each cell of the array
Write a Java program that inputs a document and then outputs a bar-chart plot of the frequencies of each alphabet character that appears within that document.
Write a Java program that inputs a polynomial in standard algebraic notation and outputs the first derivative of that polynomial.
Write a set of Java classes that can simulate an Internet application in which one party, Alice, is periodically creating a set of packets that she wants to send to Bob. An Internet process is
Use a solution to Exercise C-2.25 to reimplement the FibonacciProgression subclass to rely on the BigInteger class, in order to avoid overflows all together.Exercise C-2.25Redesign the Progression
Use a solution to Exercise C-2.25 to create a new progression class for which each value is the square root of the previous value, represented as a Double. You should include a default constructor
Redesign the Progression class to be abstract and generic, producing a sequence of values of generic type T, and supporting a single constructor that accepts an initial value. Make all corresponding

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