Write in python Longest Increasing Subsequence problemto find the length of the longest subsequence of a given sequence such that all elements of theThe longest increasing subsequence problem issubsequence are sorted in increasing orderFor example, the length of LIS for {10, 22, 9, 33, 21, 50, 41, 60, 80} is 6 and LIS is {10, 22, 33, 50, 60,80}Solution:1 + max(L,) for 0

Given: We have to write a Python program for the longest increasing Subsequence problem to find…

Longest Increasing Subsequence problem The longest increasing subsequence problem is to find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in increasing order For example, the length of LIS for {10, 22, 9, 33, 21, 50, 41, 60, 80} is 6 and LIS is {10, 22, 33, 50, 60, 80} Solution: 1 + max(L,) for 0

Longest Increasing Subsequence problem The longest increasing subsequence problem is to find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in increasing order For example, the length of LIS for {10, 22, 9, 33, 21, 50, 41, 60, 80} is 6 and LIS is {10, 22, 33, 50, 60, 80} Solution: 1 + max(L,) for 0

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Weighted alternating subsequence Consider a sequence of integers ai, a2, ..., &n, and each element a; is weight Wi. A maximum weighted alternating subsequence is an alternating subsequence with the sum of weight of each element being maximized. For example, the sequence [1, 6, 7, 4, 2, 3, 5] with element weights as [-3, 4, 10, 3, -6, 9, -2] will have maximum weighted alternating sequence [6,7,3] as which has weight 23, i.e., 4+10+9. Design an algorithm to find a maximum alternating subsequence given a sequence. State your algorithm in English or Pseudo code. What is the running time of your algorithm in big-Oh?
Given an array of integers, find the longest non-decreasing subsequences (the subsequence does not need to be consecutive). For example, A = [8,5,2,10,3,6,9,7] contains the longest subsequences [2,3,6,9] and [2,3,6,7].a. Formulate the recursive relation of the optimal solution (do not miss the base case); b. Design a bottom-up (iterative) algorithm to calculate the length of the longest non-decreasing subsequences (pseudo code); c. Analyze the complexity of your algorithm.
Build the sum-of-subset solution using backtracking algorithm. List all nonpromising nodes; specify the node that ended the solution subset in the form (1/0, 1/0, 1/0, 1/0, 1/0,) and print the final subset for the problem: S = {3, 9, 14, 15, 20} and W = 38
On an ordered list, use the binary search and Fibonacci search algorithms. Search for the components in the list 3, 18, 1, 25 for the list L = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.The number of critical comparisons performed during the searches should be compared.
The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4
Consider the Fibonacci sequence F(0)=0, F(1)=1 and F(i)=F(i-1)+F(i-2) for i > 1. For the sake of this exercise we define the height of a tree as the maximum number of vertices of a root-to-leaf path. In particular, the height of the empty tree is zero, and the height of a tree with a single vertex is one. Prove that the number of nodes of an AVL tree of height h is at least F(h) and this inequality is tight only for two values of h.
solution should have O(l1.length + l2.length) time complexity, since this is what you will be asked to accomplish in an interview. Given two singly linked lists sorted in non-decreasing order, your task is to merge them. In other words, return a singly linked list, also sorted in non-decreasing order, that contains the elements from both original lists.
Algebraic Preis’ AlgorithmAlgorithm due to Preis provides a different way to solve the maximal weightedmatching problem in a weighted graph. The algorithm consists of the followingsteps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e} show two ways of implementing this algorithm in Python
Algebraic Preis’ AlgorithmAlgorithm due to Preis provides a different way to solve the maximal weightedmatching problem in a weighted graph. The algorithm consists of the followingsteps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e}show two ways of implementing this algorithm in Python
26. Solve the following a) For the given set U, assume a family of subsets (F) on your own and apply the approximation algorithm to find the set cover. U={2,3,4,6,7,8} b) For the following weighted graph, apply approximation algorithm for solving traveling salesman problem. 3 1 4 2 5 5 27. For the following undirected graph, apply the approximation algorithm to find vertex cover. 2 6 1 3 5 4 7
Algebraic Preis’ AlgorithmThe algorithm consists of the followingsteps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e} show two ways of implementing this algorithm in Python.
Consider the following algorithm for the maximum cut problem, based on the technique of local search. Given a partition of V into sets, the basic step of the algorithm, called flip, is that of moving a vertex from one side of the partition to the other. The following algorithm finds a locally optimal solution under the flip operation, i.e., a solution which cannot be improved by a single flip. The algorithm starts with an arbitrary partition of V. While there is a vertex such that flipping it increases the size of the cut, the algorithm flips such a vertex. (Observe that a vertex qualifies for a flip if it has more neighbors in its own partition than in the other side.) The algorithm terminates when no vertex qualifies for a flip. Show that this algorithm terminates in polynomial time, and achieves an approximation guarantee of 1/2.