Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 8, Problem 6P
(a)
Program Plan Intro
To compute the number of possible ways to divide
(b)
Program Plan Intro
To show that any
Program Plan Intro
To show that if two elements are consecutive in the sorted order and from different list then they must be compared.
(d)
Program Plan Intro
To show that a lower bond of
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1. How many ways can I make a list of length n out of n elements if I allow repeats? (For example, if n=2, if there are no repeats my possible lists are 12 and 21. But if I allow repeats the possible lists are 11, 12, 21, and 22.
Solve the following problem using balanced binary search trees (TreeSet or TreeMap whichever is appropriate). Given array A of integers, and
integer x, find a pair of integers i #j with A[i]+ A[j] = x. The array is not sorted, but it does not contain duplicate elements. If there are multiple pairs
that sum to x, choose a pair in which the smaller of two elements is as big as possible. For example, if A = {3, 6, 4, 1, 5, 2} then sumOf(A, 8) returns
{0, 4} corresponding to the pair A[0] = 3, A[4] = 5, whose sum is 8. Note that this pair is preferred to (1, 5}, because min(A[0], A[4]) = 3, which is
bigger than min(A[1], A[5]) = 2.
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Class Pair {
int one,
two;
Pair (int a, int b) {one
a; two
b;}
Pair sumOf (int [] A, int x)
//code to solve the problem; RT should be 0(n log n)
A linked list has a cycle such that if you start at any node p and follow a sufficient number of subsequent links, you will end up at node p. Not that p does not have to be the list's initial node. Assume you are given an N-node linked list. The value of N, on the other hand, remains unknown.
1. Create an O(N) algorithm to check if a list includes a cycle. You may add O(N) more spaces.
2. Rep the previous question, this time using just O(1) more space. (Hint: Use two pointers that start at the beginning of the list but progress at separate rates.)
Chapter 8 Solutions
Introduction to Algorithms
Knowledge Booster
Similar questions
- Correct answer will be upvoted else downvoted. Computer science. You are given an exhibit a comprising of n (n≥3) positive integers. It is realized that in this exhibit, every one of the numbers with the exception of one are something very similar (for instance, in the cluster [4,11,4,4] all numbers aside from one are equivalent to 4). Print the list of the component that doesn't rise to other people. The numbers in the cluster are numbered from one. Input The main line contains a solitary integer t (1≤t≤100). Then, at that point, t experiments follow. The main line of each experiment contains a solitary integer n (3≤n≤100) — the length of the exhibit a. The second line of each experiment contains n integers a1,a2,… ,an (1≤ai≤100). It is ensured that every one of the numbers aside from one in the an exhibit are something very similar. Output For each experiment, output a solitary integer — the list of the component that isn't equivalent to other people.arrow_forwardA linked list contains a cycle if, starting from some node p , following a sufficient number of next links brings us back to node p . p does not have to be the first node in the list. Assume that you are given a linked list that contains N nodes. However, the value of N is unknown. Design an O (N ) algorithm to determine if the list contains a cycle. You may use O (N ) extra space. Repeat part (a), but use only O (1) extra space. (Hint: Use two iterators that are initially at the start of the list, but advance at different speeds.)arrow_forward3. The following is a recursive algorithm for a preorder binary tree traversal. Determine the worst-case runtime of this algorithm where the proper binary tree t has n internal nodes (non null nodes). Do this by specify the recurrence equation for the runtime T(n) and then deriving the closed form. Count the comparison v != null as 1 primary operation and the call processNode(t,v) as c primary operations, where c is some constant. Algorithm preorder(Tree t, TreeNode v) Input: Binary tree t of size n and node v of t Output: None if v!= null then end end processNode(t.v) preorder(t,v.getLeft()) preorder(t,v.getRight())arrow_forward
- Algorithm of Preis in AlgebraThe following steps make up the algorithm.A weighted graph G = as the first input (V, E, w)A maximum weighted matching M of G as the output.While E =, choose any v V at random, let e E be the largest edge incident to v, and then perform the following calculations: 3. M e; 4. E e; 5. V V; 6.11. Two different Python implementations of this algorithm are shown by E, E, and all edges that are adjacent to e.arrow_forwardConsider the following problem:L is a sorted list containing n signed integers (n being big enough), for example [-5, -2, -1, 0, 1, 2, 4] (here, n has a value of 7). If L is known to contain the integer 0, how can you find the index of 0 ?arrow_forwardComputer Science please solve this problem as soon as possible Write a function to insert the element into the linked list the given number X. E.g., for a given set of integers, [2, 4, 12, 6, 5, 3], the linked list is 2->4->12->6->5->3. For a given number, (X=15), the count will be 11 that are (2,4), (2,12), (2,6), (2,5), (2,3), (4,6), (4,5), (4,3), (6,5), (6,3), (5,3).arrow_forward
- Consider the following problem:L is a sorted list containing n signed integers (n being big enough), for example [-5, -2, -1, 0, 1, 2, 4] (here, nhas a value of 7). If L is known to contain the integer 0, how can you find the index of 0 ?arrow_forwardFor the operation below, provide the worst case running time in terms of n (Big O notation). Briefly justify your answer >>> Using binary search in a sorted singly linked list (as you completed in LAB 4). The implementation for __len__ traverses the list to count each nodearrow_forwardThe intersection of two sets contains the list of elements that are common to both, without repetition. As such, you are required to write an algorithm that finds and returns the intersection of two sets A and B. The sets A and B and the resulting intersection set are all lists. Assume the size of A is n and that of B is m. (a) Write an iterative algorithm that finds the intersection. (b) Prove the correctness of your algorithm; i.e. state its loop invariant and prove it by induc- tion. (c) Analyze the time complexity of your algorithm. (d) Think of a way to enhance the complexity of your algorithm by sorting both lists A and B before calculating their intersection. Write the enhanced version and prove that it has a better time complexity.arrow_forward
- Consider a (singly) linked list. Describe an algorithm to remove every second elementfrom the list.(HINT: The algorithm is similar to the one that is used when we delete a single node. Here,we need to skip an element after every deletion, and then if we have not reached the end ofthe list repeat this procedure until the list is exhausted.).arrow_forwardWrite a binary search tree method that takes two keys, low and high, and prints all elements X that are in the range specified by low and high. Your program should run in O(K + log N) average time, where K is the number of keys printed. Thus, if K is small, you should be examining only a small part of the tree. Use a hidden recursive method and do not use an in-order iterator. Bound the running time of your algorithmarrow_forwardConsider the following recursive definition of list reversal. For a list L, rev(L) is defined: rev([]) = []rev(x : L) = rev(L) + [x] Please prove by induction on L that concatenation is left cancellative: L+M = L+N implies M = N. Please prove by induction on L that rev(rev(L)) = L for all lists L. Please prove by induction on L that rev(L + M ) = rev(M ) + rev(L) for all lists L and M . (Can utilize fact that concatenation = associative) Using 1, 2, and 3 to prove that concatenation is right cancellative as well: L + N = M + N implies L = M.arrow_forward
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