Questions and Answers of Data Structures Algorithms

Describe how to implement the TaskSchedule method to run in O(n log n) time.
Consider again the modification of the deterministic version of the quick-sort algorithm so that, instead of selecting the last element in an n-element sequence as the pivot, we choose the element at
In image-processing applications, such as for optical character recognition, it is often useful to group together contiguous sets of similarly colored pixels in an image. (See Figure 7.11.) For
Show that n3 log n is Ω(n3).
Show that n2 is ω(n).
Provide an example instance of the fractional knapsack problem where a greedy strategy based on repeatedly choosing as much of the highest-benefit item as possible results in a suboptimal solution.
Let S = {a, b, c, d, e, f, g} be a collection of objects with benefit-weight values, a: (12, 4), b : (10, 6), c : (8, 5), d: (11, 7), e: (14, 3), f : (7, 1), g : (9, 6). What is an optimal solution
Suppose instead of choosing a single pivot in the quick-select algorithm, we chose log n pivots. Show that the probability that at least one of them is good is at least 1 − 1/n.
Given an array, A, of n numbers in the range from 1 to n, describe an O(n)-time method for finding the mode, that is, the number that occurs most frequently in A.
Suppose you are given two sorted lists, A and B, of n elements each, all of which are distinct. Describe a method that runs in O(log n) time for finding the median in the set defined by the union of
Show how a deterministic O(n)-time selection algorithm can be used to design a quick-sort-like sorting algorithm that runs in O(n log n) worst-case time on an n-element sequence.
What does the weighted median algorithm return if the weights of all the elements are equal?
Given an unordered sequence S of n comparable elements, describe a lineartime method for finding the [√n ] items whose rank in an ordered version of S is closest to that of the median.
Explain where the induction proof for showing that deterministic selection runs in O(n) time would fail if we formed groups of size 3 instead of groups of size 5.
Show that the worst-case running time of quick-select on an n-element sequence is Ω(n2).
Suppose we are given a sequence, S, of n integers in the range from 1 to n3. Give an O(n)-time method for determining whether there are two equal numbers in S.
Give a pseudocode description of an in-place quick-select algorithm.
Suppose we are given a sequence S of n elements, each of which is an integer in the range [0, n2 − 1]. Describe a simple method for sorting S in O(n) time.
Suppose we are given two sequences A and B of n integers, possibly containing duplicates, in the range from 1 to 2n. Describe a linear-time algorithm for determining if A and B contain the same set
Show that any comparison-based sorting algorithm can be made to be stable, without affecting the asymptotic running time of this algorithm.
Which, if any, of the algorithms bubble-sort, heap-sort, merge-sort, and quicksort are stable?
Suppose you would like to sort n music files, but you only have an old, unreliable computer, which you have nicknamed “Rustbucket.” Every time Rustbucket compares two music files, x and y, there
Given a sequence of numbers, (x1, x2,...,xn), the mode is the value that appears the most number of times in this sequence. Give an efficient algorithm to compute the mode for a sequence of n
Let S be an array of n elements on which a total order relation is defined. An inversion in S is a pair of indices i and j such that i S[j]. Describe an algorithm running in O(n log n) time for
Many states require that candidate names appear on a ballot in random order, so as to minimize biases that can arise from the order in which candidate names appear on a ballot for a given election.
Suppose we are given a sequence S of n elements, on which a total order relation is defined. Describe an efficient method for determining whether there are two equal elements in S. What is the
In computer games and also in simulations of card-playing scenarios, we sometimes need to use a computer to simulate the way that person would shuffle a deck of cards. Given two decks of n cards
As mentioned above, for each word, w, in a collection of documents, an inverted file stores a list of documents that contain the word, w. In addition, search engines typically order the list for each
how that the running time of the merge-sort algorithm on an n-element sequence is O(n log n), even when n is not a power of 2.
Let A be a collection of objects. Describe an efficient method for converting A into a set. That is, remove all duplicates from A. What is the running time of this method?
Give a pseudocode description of the merge-sort algorithm assuming the input is given as a linked list.
Suppose you are given a new hardware device that can merge k > 2 different sorted lists of total size n into a single sorted list in O(n) time, independent of the value of k. Such a device could,
Describe a variation of the merge-sort algorithm that is given a single array, S, as input, and uses only an additional array, T, as a workspace. No other memory should be used other than a constant
Suppose we implement the tree-based union-find data structure, but we don’t use the union-by-size heuristic nor the path-compression heuristic. Show that the total running time for performing a
Answer the previous exercise assuming that we implement both the union-bysize and path compression heuristics.
Suppose we implement the tree-based union-find data structure using the unionby-size heuristic and path-compression heuristics. Show that the total running time for performing a sequence of m union
Suppose we have 20 singleton sets, numbered 0 through 19, and we call the operation union(find(i),find(i + 5)), for i = 0, 1, 2,..., 14. Draw a picture of a tree-based representation of the sets that
Suppose we implement the tree-based union-find data structure using the unionby-size and path-compression heuristics. Show that the total running time for performing a sequence of m union and find
Suppose we have 20 singleton sets, numbered 0 through 19, and we call the operation union(find(i),find(i + 5)), for i = 0, 1, 2,..., 14. Draw a picture of a list-based representation of the sets that
Consider the game of Hex, as in the previous exercise, but now with a twist. Suppose some number, k, of the cells in the game board are colored gold and if the set of stones that connect the two
Suppose we implement the tree-based union-find data structure using the unionby-size heuristic and a partial path-compression heuristic. The partial path compression in this case means that, after
One additional feature of the list-based implementation of a union-find structure is that it allows for the contents of any set in a partition to be listed in time proportional to the size of the
The game of Hex is said to have, as one of its inventors, the mathematician John Nash, who is the subject of the book and movie A Beautiful Mind. In this game, two players, one playing black and the
Let A be a collection of objects. Describe an efficient method for converting A into a set. That is, remove all duplicates from A. What is the running time of this method?
For the sake of analysis, if we have a sequence of union, find, and makeSet operations, why can we can assume without loss of generality that all the makeSet operations come first?
Suppose that we implement a union-find structure by representing each set using a balanced search tree. Describe and analyze algorithms for each of the methods for a union-find structure so that
How many walls were erased to construct the maze in Figure 7.3, not counting the start and finish walls? Figure 7.3
One of the tasks for an operating system is the job of scheduling computations to be performed by the processor(s) that are part of that system. A subtask that comes up in some processor scheduling
Consider a method, remove(e), which removes e from whichever list it belongs to, in a list-based implementation of a union-find structure. Describe how to modify the list-based implementation so that
Another problem of interest in percolation theory is to determine the threshold probability where a liquid will permeate a porous material. One way to model this is to consider the barriers between
Describe how to implement a union-find structure using extendable arrays, which each contains the elements in a single set, instead of linked lists. Show how this solution can be used to process a
Suppose we have a social network with members A, B, C, D, E, F, and G, and the set of friendship ties, {(A, B),(B,C),(C, A),(D, E),(F, G)}. What are the connected components?
Most modern text processing systems have a built-in spelling checker, which checks to make sure words are spelled correctly and offers suggested corrections when words are misspelled. Suppose you
A popular tool for visualizing the themes in a speech is to draw a word cluster diagram, where the unique words from the speech are drawn in a group, with each word’s size being in proportion to
Imagine that you work for an insurance company that is insuring people against identity theft. You have just learned about a major security breach at a prominent bank used by many of your customers.
One way to measure the reading difficulty of a book is to count the number of unique words it contains. For example, Green Eggs and Ham, by Dr. Seuss, contains 50 unique words, whereas the book of
Suppose you would like to build a hash table for images, where the key for each image is a “thumbnail” image of 75×75 pixels, with each pixel being one of 256 possible colors. Describe a hash
It is well known that in a room of n people, the probability that at least two of them have the same birthday is over 1/2 if n > 23, which is a phenomenon known as the birthday paradox. Suppose,
A multimap is data structure that allows for multiple values to be associated with the same key. It has a put(k, v) method, which inserts an item with key k and value v even if there is already an
Imagine that you are building an online plagiarism checker, which allows teachers in the land of Edutopia to submit papers written by their students and check if any of those students have copied
Describe a different way of detecting an eviction cycle in the cuckoo hashing scheme than in counting iterations. You are allowed to use additional memory or to mark the cells of the two tables.
Sports announcers are expected to keep talking during a broadcast of a sporting event even when there is nothing actually happening, such as during half-time. One common way to fill empty time is
Dr. Wayne has a new way to do open addressing, where, for a key k, if the cell h(k) is occupied, then he suggests trying (h(k) + i · f(k)) mod N, for i = 1, 2, 3,..., until finding an empty cell,
In a double-entry accounting system, every business transaction has to be entered as two separate transactions, in different two accounts, once as a debit and once as a credit. For example, if a
Suppose that both the hash function, h, and the hash function, f, used in the double hashing open addressing scheme are random functions. Show that the expected time to perform the get(k) operation
Describe the limitations of using a linked list to store a collection of key-value pairs subject to put(k, v) and get(k).
In our description of hashing with the separate chaining rule, we assumed each cell in the array, A, was a pointer to a linked list, which wastes space in the case where the list holds only one item.
Bob says that a hash table with collisions handled using open addressing can have a load factor greater than 1. Alice says that this is impossible. Who is right, and why?
Suppose you are working in the information technology department for a large hospital. The people working at the front office are complaining that the software to discharge patients is taking too
Give the pseudocode description for performing insertion, searching, and removal from a hash table that uses linear probing to resolve collisions where we use a special marker to represent deleted
Alice says that a hash table with collisions handled using separate chaining can have a load factor greater than 1. Bob says that this is impossible. Who is right, and why?
Show the steps for replacing 5 with 18 in the heap of Figure 5.6. Figure 5.6 5 15 20 (16 (25 (14 (12 (11 8
Show the steps for removing key 16 from the heap of Figure 5.6. Figure 5.6. 5 15 20 (16 (25 (14 (12 (11 8
Define a min-max queue to be a data structure that supports the queue operations of enqueue() and dequeue() for objects that come from a total order, as well as operations min() and max(), which
Is there a heap T storing seven distinct elements such that a preorder traversal of T yields the elements of T in sorted order? How about an inorder traversal? How about a postorder traversal?
Define a min-max stack to be a data structure that supports the stack operations of push() and pop() for objects that come from a total order, as well as operations min() and max(), which return, but
Let T be a heap storing n keys. Give an efficient algorithm for reporting all the keys in T that are smaller than or equal to a given query key x (which is not necessarily in T). For example, given
Illustrate the performance of the heap-sort algorithm on the following input list: (2, 5, 16, 4, 10, 23, 39, 18, 26, 15).
Develop an algorithm that computes the kth smallest element of a set of n distinct integers in O(n + k log n) time.
Where may an item with largest key be stored in a heap?
Show that the problem of finding the kth smallest element in a heap takes at least Ω(k) time in the worst case.
One of the oldest applications used on the Internet is FTP, the file transfer protocol. The definition for this protocol traces its roots back to 1971, before the Internet even existed, and its
We can represent a path from the root to a node of a binary tree by means of a binary string, where 0 means “go to the left child” and 1 means “go to the right child.” Design a
Give an example of a worst-case list with n elements for insertion-sort, and show that insertion-sort runs in Ω(n2) time on such a list.
The problem of accurately summing a set S of n floating-point numbers, S = {x1, x2,...,xn}, on a real-world computer is more challenging than might first appear. For example, using the standard
Show that, for any n, there is a sequence of insertions in a heap that requires Ω(n log n) time to process.
Suppose you are designing a system for buying and selling stocks using a continuous limit order book strategy, as described in the beginning of this chapter. Describe how to use priority queues to
Suppose the binary tree T used to implement a heap can be accessed using only the methods of a binary tree. That is, we cannot assume T is implemented as an array. Given a reference to the current
Suppose you work for a major airline and are given the job of writing the algorithm for processing upgrades into first class on various flights. Any frequent flyer can request an upgrade for his or
In a discrete event simulation, a physical system, such as a galaxy or solar system, is modeled as it changes over time based on simulated forces. The objects being modeled define events that are
In data smoothing applications, such as in visualizing trends in stock averages over time, it is useful to keep track of the median of a set, S, of numbers as values are inserted or removed from S.
Design a linear-time algorithm for the previous problem. Data From Previous ProblemSuppose we are given a sorted sequence S of items (x0, x1,...,xn−1) such that each item xi in S is given a
Suppose we are given a sorted sequence S of items (x0, x1,...,xn−1) such that each item xi in S is given a positive integer weight ai. Let A denote the total weight of all elements in S. Construct
Describe how to implement the methods, insert(k, v) and remove(k), as well methods, and min() and max(), which return the key-value pair with smallest and largest key, respectively, in O(log n) time
How many trinode restructuring operations are needed to perform the zig-zig, zig-zag, and zig updates in splay trees? Use figures to explain your counting.
Show that at most one node in an AVL tree becomes unbalanced after operation removeAboveExternal is performed within the execution of a remove operation.
Give a pseudocode description of an algorithm to find the element with smallest key in a binary search tree. What is the running time of your method?
Show that 2n+1 is O(2n).
Perform a similar analysis for method Loop3 shown in Algorithm 1.21. Algorithm Loop1(n): s-0 for i + 1 to n do s-s+i Algorithm Loop2(n): p-1 for i - 1 to 2n do p-p.i Algorithm Loop3(n): p-1 for
What does a splay tree look like if its items are accessed in increasing order by their keys?

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