Computing solutions of the paintshop–necklace problem

2001

1993

PART II MATHEMATICAL APPLICATIONS OF THE THEORY OF ALGORITHMS 2.1 Investigations of mass problems. 121 2.1.0 Main definitions. .. .. .. 121 2.1.1 Seven unsolvable problems 2.1.2 Mass problems in mathematics 2.1.3 Mass problems in the sense of Medvedev 2.1.4 The correct terminology is important. .

1991

The deadline for this coursework is NOON on Thursday 25th November, 2010. Please submit your solutions electronically via submit. This is worth 50% of the coursework for A&DS. In this coursework, we consider the "knapsack counting" problem. Your mission is to understand, implement, and experiment with a suite of algorithms for this problem. In the knapsack counting problem, we are given as input a list of non-negative integer weights w 1 , w 2 ,. .. , w n ∈ N, and an upper bound B ∈ N. We say that some specific set S ⊆ {1,. .. , n} represents a feasible knapsack solution (wrt w 1 ,. .. , w n , B) if and only if i∈S w i ≤ B. The total number of feasible knapsack solutions (which we wish to count) is count(n, B) = S ⊆ {1, 2,. .. , n} : i∈S w i ≤ B. Note we may assume wlog that w i ≤ B for all 1 ≤ i ≤ n (we can delete any w i > B). §1 describes how to compute count(n, B) exactly via a recursive (exponential-time) algorithm. §2 describes a dynamic programming which computes count(n, B) exactly in Θ(nB) time. §3 introduces a very basic approximation algorithm, which runs in Θ(n 3) time on a "rounded" version of the input, and returns an "approximate count" within a factor of (n + 1) of count(n, B) (this is an n-approximation algorithm). §3 then shows we can improve the quality of this approximation (to within a factor of 1.25, say, or closer), by drawing uniform samples from the feasible solutions of the "rounded problem", and checking whether or not they also are solutions to the original problem 1. Your task in this coursework is to implement all three algorithms, prove some relevant facts (see §3.1), and write a short report on experimental results.

3.GRIN Verlag Publications, Munich, Germany, E-book, 14 pages, ISBN # 9783668983892, July, 2019, https://www.grin.com/document/491406

The paper presents an analytical exposition, critical context and integrative conclusion on the six major text books on Algorithms Design and Analysis. An algorithm is a sequence of unambiguous instructions for solving a problem, and is used for obtaining a required output for any legitimate input in a finite amount of time. Algorithms can be considered as procedural solutions to problems where the focus is on correctness and efficiency. The important problem types are sorting, searching, string processing, graph problems, combinatorial problems, geometric problems, and numerical problems. Divide-and-conquer is a general algorithm design technique that solves a problem by dividing it into several smaller sub-problems of the same type and about the same size, solving each of them recursively, and then combining their solutions to get a solution to the original problem. The brute-force approach to combinatorial problems is the exhaustive search, which generates each and every combinatorial object of the problem following a selection of those of them that satisfy all the constraints until a desired object is found. Although the main tool for analyzing the time efficiency of a nonrecursive algorithm is to set up a sum expressing the number of executions of its basic operation and ascertain the sum's order of growth, the main tool for analyzing the time efficiency of a recursive algorithm is to set up a recurrence relation expressing the number of executions of its basic operation and ascertain the solution's order of growth. A graph with directions on its edges is called a digraph. To list vertices of a digraph in an order such that for every edge of the digraph, the vertex it starts at is listed before the vertex it points to, is the topological sorting problem which has a solution if and only if a digraph is a dag (directed acyclic graph), i.e., it has no directed cycles. The topological sorting problem can be solved by two algorithms, one based on depth-first search; and the second based on a direct application of the decrease-by-one technique. The three major variations of decrease-and-conquer are: decrease-by-a-constant, most often by one (e.g., insertion sort) decrease-by-a-constant-factor, most often by the factor of two (e.g., binary search) variable-size-decrease (e.g., Euclid's algorithm).

Lecture Notes in Computer Science, 2019

The deadline for this coursework is NOON on Thursday 25th November, 2010. Please submit your solutions electronically via submit. This is worth 50% of the coursework for A&DS. In this coursework, we consider the "knapsack counting" problem. Your mission is to understand, implement, and experiment with a suite of algorithms for this problem. In the knapsack counting problem, we are given as input a list of non-negative integer weights w 1 , w 2 ,. .. , w n ∈ N, and an upper bound B ∈ N. We say that some specific set S ⊆ {1,. .. , n} represents a feasible knapsack solution (wrt w 1 ,. .. , w n , B) if and only if i∈S w i ≤ B. The total number of feasible knapsack solutions (which we wish to count) is count(n, B) = S ⊆ {1, 2,. .. , n} : i∈S w i ≤ B. Note we may assume wlog that w i ≤ B for all 1 ≤ i ≤ n (we can delete any w i > B). §1 describes how to compute count(n, B) exactly via a recursive (exponential-time) algorithm. §2 describes a dynamic programming which computes count(n, B) exactly in Θ(nB) time. §3 introduces a very basic approximation algorithm, which runs in Θ(n 3) time on a "rounded" version of the input, and returns an "approximate count" within a factor of (n + 1) of count(n, B) (this is an n-approximation algorithm). §3 then shows we can improve the quality of this approximation (to within a factor of 1.25, say, or closer), by drawing uniform samples from the feasible solutions of the "rounded problem", and checking whether or not they also are solutions to the original problem 1. Your task in this coursework is to implement all three algorithms, prove some relevant facts (see §3.1), and write a short report on experimental results.