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 16.3, Problem 5E
Program Plan Intro
To prove that if characters in an alphabet are ordered in a monotonically decreasing order of frequencies then an optimal code exists whose codeword length in monotonically increasing.
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1. If I prove that an algorithm takes Θ(k^2) worst-case time, can it take o(k^2) on some inputs?
2. If I prove that an algorithm takes Θ(k^2) worst-case time, can it take o(k^2) on all inputs?
Outline an algorithm that can partition a region of census blocks with different population sizes into 9sub-regions with approximately uniform populations. What is the maximum difference in population sizeof each sub-region from the optimal value?
Give a (simple but clear) example that shows that mutation may be necessary to find optimal solutions using Genetic Algorithms.
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- 1. If I prove that an algorithm takes Θ(n2) worst-case time, is it possible that it takes O(n) on some inputs? 2. If I prove that an algorithm takes Θ(n2) worst-case time, is it possible that it takes O(n) on all inputs?arrow_forwardFind the path from S to G using A* algorithm. Is it optimal ?arrow_forwardAssume that you are asked to write a pseudocode of a genetic algorithm to find an optimal road for a new Amazon delivery truck driver who is also new in town. He departs from Amazon warehouse and has to go through n houses by the end of his day (h1,h2,…,hn). Given that we know the distance between these houses (called dij = distance(hi, hj), and dsi = distance(s,hi) with s being the Amazon warehouse. At the end of the day, the truck will go back to the warehouse s. So a typical road would look like: s -> h2 -> h1 -> h3 ….-> hn-1 -> hn -> S. Please use genetic algorithm with the above framework to write a pseudo code to find a road that minimize the distance that he needs to travel. In your pseudocode, please feel free to suggest the size of the initial population, the fitness function, choose how you do cross-over and mutation. You don’t have to implement this pseudo code but will be graded on the details you could provide so that people who are implementing it would…arrow_forward
- Given an FA M = (Q,Σ,q0, A,δ) that accepts language L(M). Describe two algorithms for construction of (not necessarily optimal) FA’s that accept L(M).arrow_forwardProve Proposition U. Given a set of r symbols and frequencies, the Huffman algorithmbuilds an optimal prefix-free code.arrow_forwardIf a Genetic Algorithm only finds local optimal solutions, what should be done to find a better one globally?arrow_forward
- Question 2 Outline an algorithm that can partition a region of census blocks with different population sizes into 9 sub-regions with approximately uniform populations. What is the maximum difference in population size of each sub-region from the optimal value?arrow_forwardA* search is optimal with an admissible search heuristic Select one: True Falsearrow_forwardGiven the locations of the gas stations in some arbitrary order, describe an algorithm that finds an optimal placement of coffee shops (i.e., a placement of coffee shops that minimizes the number of coffee shops you’d need to open). Hint Try to come up with a greedy algorithm that moves along the turnpike for as long as possible before it’s forced to place a coffee shop.arrow_forward
- PRINT-OPTIMAL-PARENS(s, i, j){ if (i=j) then print “A”i else{ print “(” PRINT-OPTIMAL-PARENS(s,i,s[i, j]) PRINT-OPTIMAL-PARENS(s, s[i, j] + 1, j) print “)”} } b- Find the complexity of your program? c- Show that the parenthesization algorithm is loop invariant?arrow_forwardA ternary code for an alphabet is a wayof writing each of its symbols as a distinct ternary string (i.e as a sequence of 0's, 1's and 2's). Design a greedy algorithm that, given an alphabet and symbol frequencies as input, outputs a prefix-free ternary code with the minimum-possible average encoding length. Prove that your algorithm is correct.arrow_forwardA numeric array of length N is given. We need to design a function that finds all positive numbers in the array. a) Describe approaches for solving optimal worst case and optimal average case performance, respectively.arrow_forward
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