Heuristics provide estimates of the distance to the goal that can help guide search algorithms like A*. Good heuristics are admissible (never overestimate cost) and consistent (monotone). Relaxed problems can provide heuristics by ignoring constraints of the full problem. Heuristics can also be learned from experience by using state features and regression on training data.
1. The document discusses strategies for students to score 10/10 grade in mathematics. It recommends practicing previous year question papers to understand concepts better.
2. It provides two sample question papers containing math problems like addition, subtraction, multiplication and division of integers, fractions and decimals. It notes that there are extra marks questions in both papers.
3. Studying the question papers carefully and understanding the logic behind extra marks questions will help students solve similar questions correctly and score full marks.
The document presents a new encryption method for elliptic curve cryptography based on matrices. It begins by generating an addition table containing all possible point combinations on the elliptic curve. The plaintext is then converted to multiple points on the curve. These points are arranged in a matrix and encrypted using matrix multiplication with a non-singular matrix. The resulting cipher matrix undergoes circular shifting. The decryption process recovers the points from the cipher and performs the inverse operations to obtain the original plaintext. An example is provided to demonstrate the encryption of the word "cipher" using this method.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Some topics in analysis of boolean functionsguest756c74
This document summarizes key points from a lecture on computational complexity and algorithms. It covers Fourier analysis and its applications to social choice functions, Arrow's impossibility theorem, and the hypercontractive inequality. The lecture also discusses algorithmic gaps, where the optimal solution differs from what can be achieved in polynomial time.
This document contains a chapter about mathematical descriptions of continuous-time signals. It includes examples of signal functions, operations like shifting and scaling on signals, derivatives and integrals of signals, properties of even and odd signals, and exercises with answers related to these topics. The exercises involve graphing signals, finding signal values at times, manipulating signals using operations, and identifying signal properties.
A common random fixed point theorem for rational inequality in hilbert spaceAlexander Decker
This document presents a common random fixed point theorem for four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. It begins with introducing basic concepts such as separable Hilbert spaces, random operators, and common random fixed points. It then defines a condition (A) that the four mappings must satisfy. The main result is Theorem 2.1, which proves the existence of a unique common random fixed point for the four operators under condition (A) and a rational inequality condition. The proof constructs a sequence of measurable functions and shows it converges to the common random fixed point. This establishes the common random fixed point theorem for these operators.
In this paper, I propose an algorithm capable of solving the problem of isomorphic graphs in polynomial time. First, I define a pseudo tree that allows us to define for each vertex a label or a label. Secondly, I apply the pseudo tree for the first graph then I calculate the labels of each vertex of the first graph, then I do the same for the second graph. Thirdly, I look for for each graph vertex1 the graph vertices2 which have the same label, if or less a first graph vertex its label is not in the second graph vertices we deduce that the two pseudo trees are not isomorphic. In other cases I generate solutions and check them in polynomial time... This algorithm therefore allows isomorphic graphs to calculate the image of each vertex in polynomial time.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
This document discusses solving fuzzy matrix games where the payoff elements are fuzzy numbers. It begins with definitions related to fuzzy sets and fuzzy numbers. A two-person zero-sum matrix game model is presented where the payoff matrix contains trapezoidal fuzzy numbers. The fuzzy game is converted to a crisp equivalent game using defuzzification techniques. Different defuzzification methods are applied to a numerical example and the results are compared. The key concepts of mixed strategies, maximin-minimax criteria and saddle points in fuzzy matrix games are also covered.
1. The document discusses strategies for students to score 10/10 grade in mathematics. It recommends practicing previous year question papers to understand concepts better.
2. It provides two sample question papers containing math problems like addition, subtraction, multiplication and division of integers, fractions and decimals. It notes that there are extra marks questions in both papers.
3. Studying the question papers carefully and understanding the logic behind extra marks questions will help students solve similar questions correctly and score full marks.
The document presents a new encryption method for elliptic curve cryptography based on matrices. It begins by generating an addition table containing all possible point combinations on the elliptic curve. The plaintext is then converted to multiple points on the curve. These points are arranged in a matrix and encrypted using matrix multiplication with a non-singular matrix. The resulting cipher matrix undergoes circular shifting. The decryption process recovers the points from the cipher and performs the inverse operations to obtain the original plaintext. An example is provided to demonstrate the encryption of the word "cipher" using this method.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Some topics in analysis of boolean functionsguest756c74
This document summarizes key points from a lecture on computational complexity and algorithms. It covers Fourier analysis and its applications to social choice functions, Arrow's impossibility theorem, and the hypercontractive inequality. The lecture also discusses algorithmic gaps, where the optimal solution differs from what can be achieved in polynomial time.
This document contains a chapter about mathematical descriptions of continuous-time signals. It includes examples of signal functions, operations like shifting and scaling on signals, derivatives and integrals of signals, properties of even and odd signals, and exercises with answers related to these topics. The exercises involve graphing signals, finding signal values at times, manipulating signals using operations, and identifying signal properties.
A common random fixed point theorem for rational inequality in hilbert spaceAlexander Decker
This document presents a common random fixed point theorem for four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. It begins with introducing basic concepts such as separable Hilbert spaces, random operators, and common random fixed points. It then defines a condition (A) that the four mappings must satisfy. The main result is Theorem 2.1, which proves the existence of a unique common random fixed point for the four operators under condition (A) and a rational inequality condition. The proof constructs a sequence of measurable functions and shows it converges to the common random fixed point. This establishes the common random fixed point theorem for these operators.
In this paper, I propose an algorithm capable of solving the problem of isomorphic graphs in polynomial time. First, I define a pseudo tree that allows us to define for each vertex a label or a label. Secondly, I apply the pseudo tree for the first graph then I calculate the labels of each vertex of the first graph, then I do the same for the second graph. Thirdly, I look for for each graph vertex1 the graph vertices2 which have the same label, if or less a first graph vertex its label is not in the second graph vertices we deduce that the two pseudo trees are not isomorphic. In other cases I generate solutions and check them in polynomial time... This algorithm therefore allows isomorphic graphs to calculate the image of each vertex in polynomial time.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
This document discusses solving fuzzy matrix games where the payoff elements are fuzzy numbers. It begins with definitions related to fuzzy sets and fuzzy numbers. A two-person zero-sum matrix game model is presented where the payoff matrix contains trapezoidal fuzzy numbers. The fuzzy game is converted to a crisp equivalent game using defuzzification techniques. Different defuzzification methods are applied to a numerical example and the results are compared. The key concepts of mixed strategies, maximin-minimax criteria and saddle points in fuzzy matrix games are also covered.
The document discusses rule-based expert systems in Prolog, including examples of representing facts and rules, using variables, lists, and solving problems like factorials, membership, selection, and cryptography. It also provides an overview of expert systems, knowledge representation using rules, and the iterative development process of building and refining an expert system based on feedback from experts.
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
The document summarizes key concepts in machine learning including concept learning as search, general-to-specific learning, version spaces, candidate elimination algorithm, and decision trees. It discusses how concept learning can be viewed as searching a hypothesis space to find the hypothesis that best fits the training examples. The candidate elimination algorithm represents the version space using the most general and specific hypotheses to efficiently learn from examples.
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document discusses code optimization techniques at various levels including the design level, compile level, assembly level, and runtime level. It describes common subexpression elimination as an optimization that identifies identical expressions and replaces them with a single variable to improve efficiency. The document provides an example of applying common subexpression elimination to optimize a quicksort algorithm by removing redundant computations.
The document provides examples of solving mathematical expressions and equations using order of operations (BODMAS) and other simplification rules. It includes 25 word problems with step-by-step solutions showing the calculations and reasoning. The problems cover topics like percentages, ratios, proportions, time/work problems and involve setting up and solving equations. The document aims to help students practice simplifying complex expressions and solving different types of mathematical word problems.
Global Domination Set in Intuitionistic Fuzzy Graphijceronline
The document defines and discusses global domination sets in intuitionistic fuzzy graphs (IFGs). Some key points:
- A global domination set of an IFG G is a domination set that dominates both G and its complement. The global domination number γg(G) is the minimum cardinality of a global domination set.
- Bounds on γg(G) are established, such as Min{|Vi|+|Vj|} ≤ γg(G) ≤ p, where Vi and Vj are vertices.
- Properties of γg(G) are proved, including γg(G) = γg(Gc) where Gc is the complement of G.
- Special
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
This document provides an introduction to linear and integer programming. It defines key concepts such as linear programs (LP), integer programs (IP), and mixed integer programs (MIP). It discusses the complexity of different optimization problem types and gives examples of LP and IP formulations. It also covers common techniques for solving LPs and IPs, including the simplex method, cutting plane methods, branch and bound, and heuristics like beam search.
1. The document discusses machine learning concepts including what learning is, how to construct programs that automatically improve with experience, and designing learning systems.
2. It provides examples of learning problems involving chess games and handwriting recognition to classify days that a friend enjoys water sports.
3. Concept learning algorithms like FIND-S and version space algorithms like Candidate Elimination are introduced to learn concepts from examples in a restricted hypothesis space.
The document provides the questions and solutions for the JEE Advanced 2013 Mathematics Paper-II exam. It contains two sections - the first with multiple choice questions where one or more options may be correct, and the second with paragraph type questions where each paragraph is followed by two related questions having a single correct answer. The document gives the questions from the exam along with explanations for the answers. It addresses a total of 48 multiple choice questions spanning various mathematics topics such as trigonometry, calculus, complex numbers and functions.
This document contains a GATE exam question paper from 2000 with multiple choice questions in sections A and B testing knowledge of computer science topics. Section A contains 23 one-mark questions and section B contains 26 two-mark questions covering areas like algorithms, data structures, computer architecture, databases and more. The questions test understanding of concepts like binary trees, graphs, complexity analysis, regular expressions and more through matching, reasoning and problem solving questions.
Multi objective optimization and Benchmark functions resultPiyush Agarwal
The document summarizes a project on multi-objective optimization using the NSGA II and SPEA2 algorithms. A team of 5 students implemented the NSGA II and SPEA2 algorithms in MATLAB and tested them on various benchmark functions with 2 or more objectives. They compared the results of both algorithms on the benchmark functions and analyzed the Pareto fronts obtained.
The document provides examples of calculating angles between lines and planes in 3 dimensions. It includes calculating angles between a line and plane using tangent, and between two planes. It also provides practice questions involving finding angles between lines and planes given dimensional information about the lines and planes.
The document discusses problem-solving agents and uninformed search strategies. It introduces problem-solving agents as goal-based agents that try to find sequences of actions that lead to desirable goal states. It then discusses formulating problems by defining the initial state, actions, goal test, and cost function. Several examples of problems are provided, like the Romania tour problem. Uninformed search strategies like breadth-first search, uniform-cost search, and depth-first search are then introduced as strategies that use only the problem definition, not heuristics. Breadth-first search expands nodes in order of shallowest depth first, while depth-first search expands the deepest node in the frontier first.
The pure heuristic search algorithm maintains an open list of generated nodes that have not been expanded and a closed list of nodes that have. It begins with the initial state on the open list and at each cycle expands the node with the minimum heuristic value, generating its children and placing them on the open list in heuristic order. This continues until a goal state is expanded. Heuristic search sacrifices completeness for efficiency by using heuristics to guide the search towards the goal. Examples given include the 15-puzzle, maze navigation, and the missionaries and cannibals river crossing problem.
By integrating new techniques in data mining and operational research, we develop a novel travel planning system to design multi-day and multi-stay travel plans based on geo-tagged photos. Specifically, a modified Iterated Local Search heuristic algorithm is developed to find an approximate optimal solution for the multi-day and multi-stay travel planning problem using points of interests (POIs) and recurrence weights between POIs in a travel graph model, which are discovered from photos. To demonstrate the feasibility of this approach, we retrieved geo-tagged photos in Australia from the photo sharing website Panoromia.com to design experimental multi-day and multi-stay travel plans for tourists. The travel patterns that are mined using flow-mapping technique at different geographical scales are used to evaluate the experimental results.
A handout for our (Jo&Anita) seminar held on 31st May, 2013.
Unfortunately, the links towards the end are not working, so you have to type them into your browser. We've made a shorter version of the link to the spreadsheet so that you don't have to type a very long URL.
Any comments, ideas are welcome! :)
Heuristics are simple rules or mental shortcuts that allow humans to make decisions quickly and with limited information. The document discusses several types of heuristics including: the gaze heuristic, recognition heuristic, social heuristics like "do what the majority do", and heuristics based on reasons like take the best and tallying. It also covers cognitive biases like hindsight bias. Overall, the document examines how heuristics demonstrate bounded rationality and how humans use fast and frugal mental shortcuts to make decisions in an efficient manner.
The document discusses rule-based expert systems in Prolog, including examples of representing facts and rules, using variables, lists, and solving problems like factorials, membership, selection, and cryptography. It also provides an overview of expert systems, knowledge representation using rules, and the iterative development process of building and refining an expert system based on feedback from experts.
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
The document summarizes key concepts in machine learning including concept learning as search, general-to-specific learning, version spaces, candidate elimination algorithm, and decision trees. It discusses how concept learning can be viewed as searching a hypothesis space to find the hypothesis that best fits the training examples. The candidate elimination algorithm represents the version space using the most general and specific hypotheses to efficiently learn from examples.
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document discusses code optimization techniques at various levels including the design level, compile level, assembly level, and runtime level. It describes common subexpression elimination as an optimization that identifies identical expressions and replaces them with a single variable to improve efficiency. The document provides an example of applying common subexpression elimination to optimize a quicksort algorithm by removing redundant computations.
The document provides examples of solving mathematical expressions and equations using order of operations (BODMAS) and other simplification rules. It includes 25 word problems with step-by-step solutions showing the calculations and reasoning. The problems cover topics like percentages, ratios, proportions, time/work problems and involve setting up and solving equations. The document aims to help students practice simplifying complex expressions and solving different types of mathematical word problems.
Global Domination Set in Intuitionistic Fuzzy Graphijceronline
The document defines and discusses global domination sets in intuitionistic fuzzy graphs (IFGs). Some key points:
- A global domination set of an IFG G is a domination set that dominates both G and its complement. The global domination number γg(G) is the minimum cardinality of a global domination set.
- Bounds on γg(G) are established, such as Min{|Vi|+|Vj|} ≤ γg(G) ≤ p, where Vi and Vj are vertices.
- Properties of γg(G) are proved, including γg(G) = γg(Gc) where Gc is the complement of G.
- Special
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
This document provides an introduction to linear and integer programming. It defines key concepts such as linear programs (LP), integer programs (IP), and mixed integer programs (MIP). It discusses the complexity of different optimization problem types and gives examples of LP and IP formulations. It also covers common techniques for solving LPs and IPs, including the simplex method, cutting plane methods, branch and bound, and heuristics like beam search.
1. The document discusses machine learning concepts including what learning is, how to construct programs that automatically improve with experience, and designing learning systems.
2. It provides examples of learning problems involving chess games and handwriting recognition to classify days that a friend enjoys water sports.
3. Concept learning algorithms like FIND-S and version space algorithms like Candidate Elimination are introduced to learn concepts from examples in a restricted hypothesis space.
The document provides the questions and solutions for the JEE Advanced 2013 Mathematics Paper-II exam. It contains two sections - the first with multiple choice questions where one or more options may be correct, and the second with paragraph type questions where each paragraph is followed by two related questions having a single correct answer. The document gives the questions from the exam along with explanations for the answers. It addresses a total of 48 multiple choice questions spanning various mathematics topics such as trigonometry, calculus, complex numbers and functions.
This document contains a GATE exam question paper from 2000 with multiple choice questions in sections A and B testing knowledge of computer science topics. Section A contains 23 one-mark questions and section B contains 26 two-mark questions covering areas like algorithms, data structures, computer architecture, databases and more. The questions test understanding of concepts like binary trees, graphs, complexity analysis, regular expressions and more through matching, reasoning and problem solving questions.
Multi objective optimization and Benchmark functions resultPiyush Agarwal
The document summarizes a project on multi-objective optimization using the NSGA II and SPEA2 algorithms. A team of 5 students implemented the NSGA II and SPEA2 algorithms in MATLAB and tested them on various benchmark functions with 2 or more objectives. They compared the results of both algorithms on the benchmark functions and analyzed the Pareto fronts obtained.
The document provides examples of calculating angles between lines and planes in 3 dimensions. It includes calculating angles between a line and plane using tangent, and between two planes. It also provides practice questions involving finding angles between lines and planes given dimensional information about the lines and planes.
The document discusses problem-solving agents and uninformed search strategies. It introduces problem-solving agents as goal-based agents that try to find sequences of actions that lead to desirable goal states. It then discusses formulating problems by defining the initial state, actions, goal test, and cost function. Several examples of problems are provided, like the Romania tour problem. Uninformed search strategies like breadth-first search, uniform-cost search, and depth-first search are then introduced as strategies that use only the problem definition, not heuristics. Breadth-first search expands nodes in order of shallowest depth first, while depth-first search expands the deepest node in the frontier first.
The pure heuristic search algorithm maintains an open list of generated nodes that have not been expanded and a closed list of nodes that have. It begins with the initial state on the open list and at each cycle expands the node with the minimum heuristic value, generating its children and placing them on the open list in heuristic order. This continues until a goal state is expanded. Heuristic search sacrifices completeness for efficiency by using heuristics to guide the search towards the goal. Examples given include the 15-puzzle, maze navigation, and the missionaries and cannibals river crossing problem.
By integrating new techniques in data mining and operational research, we develop a novel travel planning system to design multi-day and multi-stay travel plans based on geo-tagged photos. Specifically, a modified Iterated Local Search heuristic algorithm is developed to find an approximate optimal solution for the multi-day and multi-stay travel planning problem using points of interests (POIs) and recurrence weights between POIs in a travel graph model, which are discovered from photos. To demonstrate the feasibility of this approach, we retrieved geo-tagged photos in Australia from the photo sharing website Panoromia.com to design experimental multi-day and multi-stay travel plans for tourists. The travel patterns that are mined using flow-mapping technique at different geographical scales are used to evaluate the experimental results.
A handout for our (Jo&Anita) seminar held on 31st May, 2013.
Unfortunately, the links towards the end are not working, so you have to type them into your browser. We've made a shorter version of the link to the spreadsheet so that you don't have to type a very long URL.
Any comments, ideas are welcome! :)
Heuristics are simple rules or mental shortcuts that allow humans to make decisions quickly and with limited information. The document discusses several types of heuristics including: the gaze heuristic, recognition heuristic, social heuristics like "do what the majority do", and heuristics based on reasons like take the best and tallying. It also covers cognitive biases like hindsight bias. Overall, the document examines how heuristics demonstrate bounded rationality and how humans use fast and frugal mental shortcuts to make decisions in an efficient manner.
This document summarizes a presentation on a new bidirectional A* search algorithm with shorter post-processing for solving 8-puzzle problems. It introduces bidirectional A* search and balanced heuristics. The new algorithm uses an inequality to reject nodes during search and trim the post-processing phase. Experimental results on the 8-puzzle and 15-puzzle show that the symmetric heuristic outperforms the balanced heuristic, reducing the number of states generated and solving problems faster.
This document provides an overview of various informed search algorithms including best-first search, greedy best-first search, A* search, local search algorithms like hill-climbing and simulated annealing, and genetic algorithms. It discusses concepts like heuristics, admissible heuristics, consistent heuristics, and how they relate to the optimality of A* search. Examples are provided for route finding and solving the 8-puzzle and n-queens problems.
This document discusses various heuristic search techniques, including generate-and-test, hill climbing, best first search, and simulated annealing. Generate-and-test involves generating possible solutions and testing them until a solution is found. Hill climbing iteratively improves the current state by moving in the direction of increased heuristic value until no better state can be found or a goal is reached. Best first search expands the most promising node first based on heuristic evaluation. Simulated annealing is based on hill climbing but allows moves to worse states probabilistically to escape local maxima.
16890 unit 2 heuristic search techniquesJais Balta
The document discusses heuristic search techniques for artificial intelligence. It covers greedy search which uses a heuristic function f(n) = h(n) to choose the successor node with the lowest estimated cost to reach the goal. An example of the travelling salesman problem is provided to illustrate greedy search.
This document discusses heuristic search algorithms. It begins by introducing heuristic search as trying to be smarter in how alternatives are chosen during search. It then discusses best-first search, which exploits state descriptions to estimate how promising each search node is using an evaluation function. The document focuses on constructing admissible heuristic functions and how A* search uses both the cost of the path found so far and an admissible heuristic estimate to guide its search. It proves that A* is complete and optimal if the heuristic is admissible and consistent.
Heuristic search algorithms use heuristics, or problem-specific knowledge, to guide the search for a solution. Some heuristics guarantee completeness while others may sacrifice completeness to improve efficiency. A heuristic function estimates the cost to reach the goal state from the current state. For example, in the 8-puzzle problem the Manhattan distance heuristic estimates this cost as the sum of the distances each misplaced tile would need to move to reach its goal position. The example shows applying the Manhattan distance heuristic to guide the search for a solution to instances of the 8-puzzle problem.
Solving problems by searching Informed (heuristics) Searchmatele41
This document discusses various informed (heuristic) search strategies for solving problems, including greedy best-first search, A* search, and memory-bounded variations. Greedy best-first search uses the heuristic function h(n) alone to select nodes for expansion. A* search combines the path cost g(n) and heuristic estimate h(n) to select nodes, guaranteeing an optimal solution if h is admissible. The document provides examples of applying these searches to route finding between cities in Romania. A* search is identified as finding the optimal solution for this problem if using an admissible heuristic like straight-line distance.
This presentation discusses various optimization heuristics, including genetic algorithms, hill climbing, tabu search, simulated annealing, and swarm intelligence. It defines heuristics as experience-based problem solving techniques and notes they are commonly used for optimization problems that are NP-hard or NP-complete. Each heuristic is explained, with examples like the traveling salesman problem provided to illustrate applications and techniques like local neighborhood searches, probabilistic acceptance of solutions, and mimicking natural processes through algorithms.
Lecture 14 Heuristic Search-A star algorithmHema Kashyap
A* is a search algorithm that finds the shortest path through a graph to a goal state. It combines the best aspects of Dijkstra's algorithm and best-first search. A* uses a heuristic function to evaluate the cost of a path passing through each state to guide the search towards the lowest cost goal state. The algorithm initializes the start state, then iteratively selects the lowest cost node from its open list to expand, adding successors to the open list until it finds the goal state. A* is admissible, complete, and optimal under certain conditions relating to the heuristic function and graph structure.
Constraint satisfaction problems (CSPs) define states as assignments of variables to values from their domains, with constraints specifying allowable combinations. Backtracking search assigns one variable at a time using depth-first search. Improved heuristics like most-constrained variable selection and least-constraining value choice help. Forward checking and constraint propagation techniques like arc consistency detect inconsistencies earlier than backtracking alone. Local search methods like min-conflicts hill-climbing can also solve CSPs by allowing constraint violations and minimizing them.
The document discusses various search techniques used in artificial intelligence including:
- Informed and uninformed searches that can use heuristics to guide the solution process.
- Common problems that use search techniques include pathfinding, constraint satisfaction, and two-player games.
- Depth-limited search avoids failures of depth-first search by limiting depth to avoid infinite loops.
- Backtracking search is a modified depth-first search used for constraint satisfaction problems that prunes unpromising branches.
- Adversarial search models multi-agent systems and is useful for games, employing techniques like minimax to determine the best move.
The document describes the firefly algorithm, a metaheuristic optimization algorithm inspired by the flashing behaviors of fireflies. The firefly algorithm works by simulating the flashing and attractiveness of fireflies, where the brightness of a firefly represents the quality of a solution. Fireflies move towards more bright fireflies and flash in synchrony in order to find near-optimal solutions to optimization problems. The document outlines the assumptions, formulas, pseudo-code, applications, and comparisons of the firefly algorithm to other algorithms like particle swarm optimization.
This document discusses various heuristic search algorithms including A*, iterative-deepening A*, and recursive best-first search. It begins by introducing the concept of using evaluation functions to guide best-first search and preferentially expand nodes with lower heuristic values. It then presents the general graph search algorithm and describes how A* specifically reorders nodes using an evaluation function that considers path cost and estimated cost to the goal. Consistency conditions for the heuristic function are discussed which guarantee A* finds optimal solutions.
The document provides an overview of problem spaces and problem solving through searching techniques used in artificial intelligence. It defines a problem space as a set of states and connections between states to represent a problem. Search strategies for finding solutions include breadth-first search, depth-first search, and heuristic search. Real-world problems discussed that can be solved through searching include route finding, layout problems, task scheduling, and the water jug problem is presented as a toy problem example.
This document provides an introduction to using the Google Test framework for unit testing C++ code. It begins with an example of a simple test for a function called calc_isect. It then demonstrates how to add assertions to tests, use test fixtures to reduce duplicated setup code, and generate parameterized tests. The document also covers best practices for test organization, installing and using Google Test, and some key features like XML output and selecting subsets of tests. Overall, the document serves as a tutorial for getting started with the Google Test framework for writing and running unit tests in C++ projects.
The document discusses functional programming concepts in Ruby. It begins by stating that functional programming and Enumerable methods can be useful in Ruby. It then provides examples of various Enumerable methods like zip, select, partition, map, and inject. It encourages thinking functionally by avoiding side effects, mutating values, and using functional parts of the standard library. The document concludes by suggesting learning a true functional language to further improve functional programming skills.
The document summarizes various greedy algorithms and optimization problems that can be solved using greedy approaches. It discusses the greedy method, giving the definition that locally optimal decisions should lead to a globally optimal solution. Examples covered include picking numbers for largest sum, shortest paths, minimum spanning trees (using Kruskal's and Prim's algorithms), single-source shortest paths (using Dijkstra's algorithm), activity-on-edge networks, the knapsack problem, Huffman codes, and 2-way merging. Limitations of the greedy method are noted, such as how it does not always find the optimal solution for problems like shortest paths on a multi-stage graph.
Recursion is a technique where a method calls itself directly or indirectly. It is useful for solving problems that involve repeating patterns or combinatorial algorithms. The document provides examples of calculating factorials, generating all binary vectors, and finding all paths in a labyrinth recursively. It discusses how to avoid harmful recursion that uses excessive memory and discusses when recursion is preferable to iteration, such as for problems that require exploring multiple continuations at each step. Exercises are provided to help practice implementing various recursive algorithms.
A* and Min-Max Searching Algorithms in AI , DSA.pdfCS With Logic
A* and Min-Max Searching Algorithms in AI. Search algorithms are algorithms designed to search for or retrieve elements from a data structure, where they are stored. It is a searching algorithm that is used to find the shortest path between an initial and a final point. Mini-Max algorithm is a recursive or backtracking algorithm that is used in decision-making and game theory.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by defining dynamic programming as avoiding recalculating solutions by storing results in a table. It then describes Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, calculating shortest paths that pass through each intermediate node. It takes O(n3) time for a graph with n nodes. Finally, it discusses the multistage graph problem and provides forward and backward algorithms to find the minimum cost path from source to destination in a multistage graph in O(V+E) time, where V and E are the numbers of vertices and edges.
1- please please please don't send me an incomplete program this code.docxEvandWyBurgesss
1- please please please don't send me an incomplete program this code has an error please solve it.(Find attached the code to implement the following two functions:
- Insert in Interval Heap
- Display minimum and maximum values
The interval heap is implemented as a one-dimensional array. The even indices store the minimum heap while odd indices store the max heap.
You can extend this code to add the function for deleting minimum or maximum values.)
#include <stdio.h>
void swap(int a[], int n1, int n2) {
int temp = a[n1];
a[n1] = a[n2];
a[n2] = temp;
}
int get_parent(int n) {
if (!(n % 4)) return (n - 4) / 2;
if (!(n % 2)) return (n - 2) / 2;
if (((n - 1) % 4)) return (n - 3) / 2;
else return (n - 1) / 2;
}
void heapify_max(int a[], int n) {
int parent = get_parent(n);
while (n > 1 && a[n] > a[parent]) {
swap(a, n, parent);
n = parent;
parent = get_parent(n);
}
}
void heapify_min(int a[], int n) {
int parent = get_parent(n);
while (n > 0 && a[n] < a[parent]) {
swap(a, n, parent);
n = parent;
parent = get_parent(n);
}
}
void insert_intervalheap(int a[], int *pos, int val) {
int n = *pos, parent, parent1;
(*pos)++;
a[n] = val;
if (n == 0) return;
if (n == 1) {
if (a[0] > a[1]) swap(a, 0, 1);
return;
}
if (!(n % 2)) {
parent = get_parent(n);
if (a[n] < a[parent]) {
swap(a, n, parent);
heapify_min(a, parent);
}
} else {
parent1 = get_parent(n) + 1;
if (a[n] < a[parent]) {
swap(a, n, parent);
heapify_min(a, parent);
}
if (a[n] > a[parent1]) {
swap(a, n, parent1);
heapify_max(a, parent1);
}
}
}
int get_min(int a[], int n) {
return (a[0]);
}
int get_max(int a[], int n) {
if (n > 1) {
return (a[1]);
} else {
return (a[0]);
}
}
int delete_min(int a[], int *pos) {
if (*pos == 0) {
printf("Interval heap is empty. Cannot delete minimum value.\n");
return -1;
}
int min = a[0];
*pos -= 1;
a[0] = a[*pos];
heapify_min(a, 0);
return min;
}
int delete_max(int a[], int *pos) {
if (*pos == 0) {
printf("Interval heap is empty. Cannot delete maximum value.\n");
return -1;
}
if (*pos == 1) {
int max = a[1];
*pos -= 1;
return max;
}
int max = a[1], n = *pos;
if (a[n-1] > max) {
a[1] = a[n-1];
heapify_max(a, 1);
}
else {
a[1] = a[n-2];
heapify_max(a, 1);
}
*pos -= 1;
return max;
}
int main() {
int heap[MAX_SIZE];
int size = 0;
insert_intervalheap(heap, &size, 10);
insert_intervalheap(heap, &size, 15);
insert_intervalheap(heap, &size, 20);
insert_intervalheap(heap, &size, 25);
insert_intervalheap(heap, &size, 30);
insert_intervalheap(heap, &size, 35);
insert_intervalheap(heap, &size, 40);
printf("Interval heap: ");
for (int i = 0; i < size; i++) {
printf("%d ", heap[i]);
}
printf("\n");
int min = delete_min(heap, &size);
printf("Minimum value: %d\n", min);
int max = delete_max(heap, &size);
printf("Maximum value: %d\n", max);
printf("Interval heap after deletion: ");
for (int i = 0; i < size; i++) {
printf("%d ", heap[i]);
}
printf("\n");
return 0;
}
2- Find a.
This document discusses backtracking algorithms and provides examples for solving problems using backtracking, including:
1) Generating all subsets and permutations of a set using backtracking.
2) The eight queens problem, which can be solved using a backtracking algorithm that places queens on a chessboard one by one while checking for threats.
3) Key components of backtracking algorithms including candidate construction, checking for solutions, and pruning search spaces for efficiency.
The document provides an overview of absolute value functions and how to solve absolute value equations and inequalities. It defines absolute value, discusses evaluating absolute value expressions, and solves absolute value equations by isolating the absolute value. It also explains how to solve absolute value inequalities by considering whether the absolute value is less than, greater than, or equal to the right side of the inequality and interpreting the solutions. Examples of each type of problem are worked out step-by-step.
This document summarizes various informed search algorithms including greedy best-first search, A* search, and memory-bounded heuristic search algorithms like recursive best-first search and simple memory-bounded A* search. It discusses how heuristics can be used to guide the search towards optimal solutions more efficiently. Admissible and consistent heuristics are defined and their role in guaranteeing optimality of A* search is explained. Methods for developing effective heuristic functions are also presented.
This document provides an overview of clustering techniques including k-means clustering, expectation maximization algorithms, and spectral clustering. It discusses how k-means clustering works by initializing random cluster centers, assigning data points to the closest centers, and adjusting the centers iteratively. Expectation maximization is presented as a way to learn the parameters of a Gaussian mixture model to cluster data. Finally, applications of clustering like document clustering using mixture models are briefly described.
The document discusses several ways to write recursive functions in Swift, including functions to calculate the factorial of a number, find the nth Fibonacci number, and determine if a number contains the digit 7. It also provides examples of using tail recursion to iterate over an array and reduce it to a single value, as well as using tail recursion with custom functions.
This document provides an introduction and overview of MATLAB (Matrix Laboratory), an interactive program for numerical computation and visualization. It discusses basic MATLAB commands and functions for creating variables and matrices, performing mathematical operations, plotting graphs, and working with polynomials.
Flink Forward Berlin 2017: David Rodriguez - The Approximate Filter, Join, an...Flink Forward
In this talk we introduce the notion of approximate filter, join, and groupby operations for arrays. Typically, Flink streams contain primitive types and tuples where filter, join, and groupby operate on exact matches. But, exact matches are sometimes limiting. For example, the objects Array(100, 0, 100) and Array(100, 0, 101) may be “close enough” to match. To solve this problem, we introduce locality sensitive hashing (LSH) for arrays of numeric and string types. This technique encodes arrays into strings so that similar arrays are encoded to the same string. In other words, we ensure matching when arrays are similar, up to a degree of error. Therefore, it is easy to incorporate new approximate filter, join, and groupby design patterns built on the notion of exact matches. In conclusion, we highlight how Cisco Umbrella streams large signals stored in arrays and then clusters them using approximate filter, join, groupby methods to detect waves of botnets and cybercrime online.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by explaining dynamic programming as an optimization technique that works bottom-up by solving subproblems once and storing their solutions, rather than recomputing them. It then presents Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, updating the shortest path lengths between all pairs that include that node by exploring paths through it. Finally, it discusses solving multistage graph problems using forward and backward methods that work through the graph stages in different orders.
The document discusses ways to accelerate Python programs using GPUs. It begins by explaining how Python programs work and the global interpreter lock. It then covers parallelizing operations using NumPy, Cython, and Nvidia's CUDA. Deep learning frameworks that can utilize GPU acceleration are also presented, such as TensorFlow, PyTorch, and Caffe. The summary concludes that gaining performance involves planning operations, vectorizing code, offloading to accelerators, and using specialized libraries.
Using an Array include ltstdiohgt include ltmpih.pdfgiriraj65
Using an Array:
#include <stdio.h>
#include <mpi.h>
int main(int argc, char** argv) {
int rank, size;
MPI_Init(&argc, &argv);
MPI_Comm_rank(MPI_COMM_WORLD, &rank);
MPI_Comm_size(MPI_COMM_WORLD, &size);
// Define topology
int topology[5][2] = {{1, 4}, {0, 2, 3}, {1,-1}, {0, 4}, {0, 3}};
// Initialize index and edge arrays
int index[5] = {0};
int edges[5][3] = {{0}};
// Add edges into index array
int i =0;
int j =0;
for (i = 0; i < 5; i++)
{
index[i] = sizeof(topology[i]) / sizeof(int);
for (j = 0; j < index[i]; j++)
{
edges[i][j] = topology[i][j];
}
}
// Display topology for each process
for (i = 0; i < size; i++)
{
if (rank == i)
{
printf("Process %d has %d neighbors: ", i, index[i]);
for (j = 0; j < index[i]; j++)
{
printf("%d ", edges[i][j]);
}
printf("n");
}
MPI_Barrier(MPI_COMM_WORLD);
}
MPI_Finalize();
return 0;
}
MPI Functions:
#include <mpi.h>
#include <stdio.h>
int main(int argc, char** argv)
{
int rank, size;
MPI_Init(&argc, &argv); // Initialize MPI
MPI_Comm_rank(MPI_COMM_WORLD, &rank); // Get the rank of the current process
MPI_Comm_size(MPI_COMM_WORLD, &size); // Get the total number of processes
int nnodes = 5; // Number of nodes in the topology
int nedges = 8; // Number of edges in the topolofy
// Index of the first edge for each node
int index[5] = {2, 5, 6, 8, 9};
int edges[9] = {4, 1, 0, 2, 3, 1, 4, 0, 3}; // List of all edges
MPI_Comm graph_comm; // Create a new communicator for the graph topology
MPI_Graph_create(MPI_COMM_WORLD, nnodes, index, edges, 0, &graph_comm); // Create the
graph topology
int count; // Number of neighbors
int* neighbors; // Neighbor Ranks
MPI_Graph_neighbors_count(graph_comm, rank, &count); // Get the number of neighbors for the
current process
neighbors = (int*) malloc(count * sizeof(int)); // Allocate memory for the array of neighbor ranks
MPI_Graph_neighbors(graph_comm, rank, count, neighbors); // Get the neighbor ranks for the
current process
// Display process and the node in the topologies neighbors
printf("Process %d has %d neighbors:", rank, count);
int i;
for (i = 0; i < count; i++)
{
printf(" %d", neighbors[i]); // Print the neighbor ranks
}
printf("n");
MPI_Finalize();
return 0;
}
ive provided my own code above in case that helps or makes it easier on you guys.
My output isnt quite right ive been at it for a while if someone could fix the output and explain it to
me id be very happy :)
4. Use Graph topology to create following one. Once you create your topology, use one process
(e.g., process 0 ) to display the number of neighbors and its neighbors at each node (i.e.,
process). Use following two methods to check your topology. (20 points) i) Use two arrays, index
and edges, to display number of neighbors and its neighbors for each node. ii) Use two functions
in MPI, "MPI Graph neighbors count" and "MPI Graph neighbors". llinn@scholar-fe06: /470 $
mpirun -n 5 ./hmwk5q4-b Process 0 has 2 neighbors: 41 Process 1 has 3 neighbors: 023 Process
2 has 1 neighbors: 1 Process 3 has 2 neighbors: 40.
SAT and SMT solvers can reason about large sets of facts and constraints, and are used for applications like planning, configuration checking, placement, and formal verification. They work by translating problems into Boolean logic or logic with theories like arithmetic that can be solved by a SAT solver. SMT solvers combine a SAT solver with theory solvers to handle problems with richer expressiveness.
This presentation is the full application of discrete mathematics throughout a course and includes Set Theory, Functions nd Sequences, Automata Theory, Grammars and algorithm building.
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A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
2. Goodness of heuristics
If a heuristic is perfect, search work is proportional to solution length
S = O(b*d) b is average branching, d depth of solution
If h1 and h2 are two heuristics, and h1 < h2 everywhere,
then A*(h2) will expand no more nodes than A*(h1)
If an heuristic never overestimates more than N % of the least cost,
then the found solution is not more than N% over the optimal
solution.
h()=0 is an admissible trivial heuristic, worst of all
In theory, we can always make a perfect heuristics, if we perform a
full breadth first search from each node, but that would be pointless.
3. Example of good heuristics for the 8
puzzle
Transparency shows the heuristic
f(n)=g(n) + h(n)
h(n)=# misplaced tiles
On an A* search for the 8 - puzzle
4.
5. Graph Search
%% Original version
function GRAPH-SEARCH(problem,fringe) returns a solution, or
failure
closed <- an empty set
fringe <- INSERT(MAKE-NODE(INITIAL-STATE[problem]),fringe)
loop do
if EMPTY?(fringe) then return failure
node <- REMOVE-FIRST(fringe)
if GOAL-TEST[problem](STATE[node]) then return SOLUTION(node)
if STATE[node] is not in closed then
add STATE[node] to closed
fringe <- INSERT-ALL(EXPAND(node,problem),fringe)
6. Heuristic Best First Search
%% f [problem] is the heuristic selection function of the problem
function BEST-FIRST-SEARCH([problem])returns a solution,or failure
OPEN <- an empty set // P1
CLOSED <- an empty set // P2
OPEN <- INSERT(MAKE-NODE(INITIAL-STATE[problem]),OPEN) // P3
repeat
if EMPTY?(OPEN) then return failure //P4
best <- the lowest f-valued node on OPEN // P5
remove best from OPEN // P6
if GOAL-TEST[problem](STATE[best])
then return SOLUTION(best) // P7
for all successors M of best
if STATE[M] is not in CLOSED then // P8
OPEN <-INSERT(successor,OPEN) // P9
add STATE[best] to CLOSED // P10
7. Heuristic Best First Search
(A*)Java Pseudocode
// Instantiating OPEN, CLOSED // Move selected node from OPEN to n // P6
OPEN = new Vector<Node>(); // P1 n = OPEN.elementAt(lowIndex);
CLOSED = new Vector<Node>(); // P2 OPEN.removeElement(n);
// Successful exit if n is goal node // P8
// Placing initial node on OPEN if (n.equals(goalnode)) return;
OPEN.add(0, initialnode); // P3 // Retrieve all possible successors of n
M = n.successors();
// After initial phase,we enter the main loop //Compute f-,g- and h-value for each successor
// of the A* algorithm for (int i = 0; i < M.size(); i++) {
while (true) { Node s = M.elementAt(i);
// Check if OPEN is empty s.g = n.g + s.cost;
if (OPEN.size() == 0) { // P4 s.h = s.estimate(goalnode);
System.out.println("Failure :"); s.f = s.g + s.h;
return; // Augmenting OPEN with suitable nodes from M
// Locate next node on OPEN with heuristic for (int i = 0; i < M.size(); i++)
lowIndex = 0; // P5 // Insert node into OPEN if not on CLOSED// P9
low = OPEN.elementAt(0).f; if (!(on CLOSED))
for (int i = 0; i < OPEN.size(); i++) { OPEN.add(0, M.elementAt(i));
number = OPEN.elementAt(i).f; // Insert n into CLOSED
if (number < low) { CLOSED.add(0, n); // P10
lowIndex = i;
low = number;
8. AStar
Java Code
See exercise 7
http://www.idi.ntnu.no/emner/tdt4136/PRO/Astar.java
9. Example
Mouse King Problem
(1,5) (5,5)
___________
| | | | | |
| | |X| | |
| | |X| | |
| | |X| | |
|M| |X| |C|
-----------
(1,1) (5,1)
There is a 5X5 board.
At (1,1) there is a mouse M which can move as a king on a chess board.
The target is a cheese C at (5,1).
There is however a barrier XXXX at (3,1-4) which the mouse cannot go
through, but the mouse heuristics is to ignore this.
10. Heuristics for Mouse King
public class MouseKingState extends State {
public int[] value;
public MouseKingState(int[] v) {value = v;}
public boolean equals(State state) {…}
public String toString() {…}
public Vector<State> successors() {…}
public int estimate(State goal) {
MouseKingState goalstate = (MouseKingState)goal;
int[] goalarray = goalstate.value;
int dx = Math.abs(goalarray[0] - value[0]);
int dy = Math.abs(goalarray[1] - value[1]);
return Math.max(dx, dy);
}
} // End class MouseKingState
12. Perfect Heuristics Behaviour
If the heuristics had been perfect, the expansion of the nodes would have
been equal to the solution path.
1,1 2,2 2,3 2,4 3,5 4,4 4,2 4,2 5,1
This means that a perfect heuristics ”encodes” all the relevant knowledge of the problem
space.
Solution Path:
start,f,g,h (1,5) (5,5) (1,5) (5,5)
_______________ _______________
(1,1),8,0,8 | | |5 | | | | | |5 | | |
(2,2),8,1,7 | | 4| |6 | | | | 4| |6 | |
(2,3),8,2,6 | | 3| |7 | | | | 3| |7 | |
| | 2| |8 | | | | 2| |8 | |
(2,4),8,3,5
| 1| | | |9 | | 1| | | |9 |
(3,5),8,4,4 ---------------- ----------------
(4,4),8,5,3 (1,1) (5,1) (1,1) (5,1)
(4,3),8,6,2
Order of expansion Solution Path
(4,2),8,7,1
(5,1),8,8,0
13. Monotone (consistent) heuristics
A heuristic is monotone if the f-value is non-decreasing
along any path from start to goal.
This is fulfilled if for every pair of nodes
n n’
f(n) <= f(n’) = g(n’) + h(n’)
= g(n) + cost(n,n’) + h(n’)
f(n) = g(n) + h(n)
Which gives the triangle inequality
h(n) <= cost(n.n’) + h(n’)
cost(n,n’)
h(n’)
goal
h(n)
14. Properties of monotone heuristics
1) All monotone heuristics are admissible
2) Monotone heuristic is admissible at all nodes (if h(G)=0)
3) If a node is expanded using a monotone heuristic, A* has
found the optimal route to that node
4) Therefore, there is no need to consider a node if it is already
found.
5) If this is assumed, and the heuristic is monotone, the
algorithm is still admissible
6) If the monotone assumption is not true, we risk a non
optimal solution
7) However, most heuristics are monotone
15. Some more notes on heuristics
If h1(n) and h2(n) are admissible heuristics, then the
following are also admissible heuristics
• max(h1(n),h2(n))
• α*h1(n) + β*h2(n) where α+β=1
16. Monotone heuristic repair
Suppose a heuristic h is admissible but not consistent, i.e.
h(n) > c(n,n’)+h(n’), which means
f(n) > f(n’) (f is not monotone)
In that case, f’(n’) can be set to f(n) as a better heuristic,
(higher but still underestimate), i.e. use
h’(n’) = max(h(n’),h(n)-c(n,n’))
17. An example of a heuristics
Consider a knight (”horse”) on a chess board.
It can move 2 squares in one direction and 1 to either side.
The task is to get from one square to another in fewest possible steps
(e.g. A1 to H8).
A proposed heuristics could be ManHattanDistance/2
Is it admissible ? Is it monotone ?
(Actually, it is not straightforward to find an heuristics that is both admissible and not
monotone)
8 | | | | | | |
|*|
7 | | | | | | |
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3 | | | | | | |
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2 | | | | | | |
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1 |*| | | | | |
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A B C D E F G H
18. Relaxed problems
Many heuristics can be found by using a relaxed (easier, simpler)
model of the problem.
By definition, heuristics derived from relaxed models are
underestimates of the cost of the original problem
For example, straight line distance presumes that we can move in
straight lines.
For the 8-puzzle, the heuristics
W(n) = # misplaced tiles
would be exact if we could move tiles freely
The less relaxed, (and therefore better) heuristic
P(n) = distance from home ( Manhattan distance)
would allow tiles to be moved to an adjacent square even though
there may already be a tile there.
20. Learning heuristics from experience
Where do heuristics come from ?
Heuristics can be learned as a computed (linear?) combination of features of the state.
Example: 8-puzzle
Features:
x1(n) : number of misplaced tiles
x2(n) : number of adjacent tiles that are also adjacent in the goal state
Procedure: make a run of searches from 100 random start states. Let h(ni) be the
found minimal cost.
n1 h(n1) x1(n1) x2(n1)
… …… …….. ……….
n100 h(n100) x1(n100) x2(n100)
From these, use regression to estimate
h(n) = c1*x1(n) + c2*x2(n)
21. Learning heuristics from
experience (II)
Suppose the problem is harder than the heuristic (h1) indicates but that the
hardness is assumed to be uniform over the state space. Then , it is an estimate to
let an improved heuristics h2(x) = *h1(x).
|S| | | | | | | | Problem: move a piece from S to G using ChessKing
| | | | | | | | | heuristics h1(x) = # horisontal/vertical/diagonal moves.
| | | | | | | | | h1(S)=7 h1(n) = 4
| | | |n| | | | |
| | | | | | | | | Assume problem is actual harder (in effect Manhattan
disance h2(x), but we dont’ know that). It means
| | | | | | | | |
g(n) = 6
| | | | | | | | | We then estimate = g(n)/(h1(s)-h1(n))
| | | | | | | | G|
h2(n) = g(n)/(h1(s)-h1(n)) * h1(n)
= 6/(7 – 4) *4 = 8 (correct)
22. Learning heuristics from
experience (III)
Suppose the problem is easier than the heuristic (h1) indicates but that the
easiness is assumed to be uniform over the state space. Then , it is an estimate to
let an improved heuristics h2(x) = *h1(x).
|S| | | | | | | | Problem: move a piece from S to G using Manhattan
| | | | | | | | | heuristics h1(x) = # horisontal/vertical moves
| | | | | | | | | h1(S)=14 h1(n) = 8
| | | |n| | | | |
| | | | | | | | | Assume problem is actual easier (in effect Chess King
disance h2(x), but we dont’ know that). It means
| | | | | | | | |
g(n) = 3
| | | | | | | | | We then estimate = g(n)/(h1(s)-h1(n))
| | | | | | | | G|
h2(n) = g(n)/(h1(s)-h1(n)) * h1(n)
= 3/(14 – 8) *8 = 4 (correct)
23. Practical example
(Bus world scenario)
Find an optimal route from one place to
another by public transport
Nodes: Bus passing events
A bus route passes a station at a time
Actions: - enter bus
- leave bus
- wait
24. Bus scenario search space
Search space space(2) X time(1)
Time
Bus 3
wait
Space Bus 5
Space
25. Heuristics for bus route planner
which route is best ?
T3 = Z N # bus transfers
K3 A wait time 1. departure
Z wait time before arrival
T2 T sum transfer waiting time
K2 K sum driving time
Equivalent
transfer
T1
K1
T0 = A
26. Planner discussion
1. (T1+T2) = T critical if rain
2. If Z is to be minimised, must search backwards
3. Many equivalent transfers (same T and K)
4. In practice, A* is problematic
5. Waiting time A maybe unimportant
Solution: Relaxation
a) Find trasees independent of time
b) Eliminate equivalent transfers
c) For each trasee, find best route plan
d) Keep the best of these solutions