Depth First Search Using Stack Example
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Graph Traversals
Graph Traversals CS200 - Graphs 1 Tree traversal reminder Pre order A A B D G H C E F I In order B C G D H B A E C F I Post order D E F G H D B E I F C A Level order G H I A B C D E F G H I Connected Components n The connected component of a node s is the largest set of nodes reachable from s. A generic algorithm for creating connected component(s): R = {s} while ∃edge(u, v) : u ∈ R∧v ∉ R add v to R n Upon termination, R is the connected component containing s. q Breadth First Search (BFS): explore in order of distance from s. q Depth First Search (DFS): explores edges from the most recently discovered node; backtracks when reaching a dead- end. 3 Graph Traversals – Depth First Search n Depth First Search starting at u DFS(u): mark u as visited and add u to R for each edge (u,v) : if v is not marked visited : DFS(v) CS200 - Graphs 4 Depth First Search A B C D E F G H I J K L M N O P CS200 - Graphs 5 Question n What determines the order in which DFS visits nodes? n The order in which a node picks its outgoing edges CS200 - Graphs 6 DepthGraph Traversalfirst search algorithm Depth First Search (DFS) dfs(in v:Vertex) mark v as visited for (each unvisited vertex u adjacent to v) dfs(u) n Need to track visited nodes n Order of visiting nodes is not completely specified q if nodes have priority, then the order may become deterministic for (each unvisited vertex u adjacent to v in priority order) n DFS applies to both directed and undirected graphs n Which graph implementation is suitable? CS200 - Graphs 7 Iterative DFS: explicit Stack dfs(in v:Vertex) s – stack for keeping track of active vertices s.push(v) mark v as visited while (!s.isEmpty()) { if (no unvisited vertices adjacent to the vertex on top of the stack) { s.pop() //backtrack else { select unvisited vertex u adjacent to vertex on top of the stack s.push(u) mark u as visited } } CS200 - Graphs 8 Breadth First Search (BFS) n Is like level order in trees A B C D n Which is a BFS traversal starting E F G H from A? A. -
Adversarial Search
Adversarial Search In which we examine the problems that arise when we try to plan ahead in a world where other agents are planning against us. Outline 1. Games 2. Optimal Decisions in Games 3. Alpha-Beta Pruning 4. Imperfect, Real-Time Decisions 5. Games that include an Element of Chance 6. State-of-the-Art Game Programs 7. Summary 2 Search Strategies for Games • Difference to general search problems deterministic random – Imperfect Information: opponent not deterministic perfect Checkers, Backgammon, – Time: approximate algorithms information Chess, Go Monopoly incomplete Bridge, Poker, ? information Scrabble • Early fundamental results – Algorithm for perfect game von Neumann (1944) • Our terminology: – Approximation through – deterministic, fully accessible evaluation information Zuse (1945), Shannon (1950) Games 3 Games as Search Problems • Justification: Games are • Games as playground for search problems with an serious research opponent • How can we determine the • Imperfection through actions best next step/action? of opponent: possible results... – Cutting branches („pruning“) • Games hard to solve; – Evaluation functions for exhaustive: approximation of utility – Average branching factor function chess: 35 – ≈ 50 steps per player ➞ 10154 nodes in search tree – But “Only” 1040 allowed positions Games 4 Search Problem • 2-player games • Search problem – Player MAX – Initial state – Player MIN • Board, positions, first player – MAX moves first; players – Successor function then take turns • Lists of (move,state)-pairs – Goal test -
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search PROGRAMMING In this assignment, you will create an adversarial search agent to play the 2048-puzzle game. A demo of the game is available here: gabrielecirulli.github.io/2048. I. 2048 As A Two-Player Game II. Choosing a Search Algorithm: Expectiminimax III. Using The Skeleton Code IV. What You Need To Submit V. Important Information VI. Before You Submit I. 2048 As A Two-Player Game 2048 is played on a 4×4 grid with numbered tiles which can slide up, down, left, or right. This game can be modeled as a two player game, in which the computer AI generates a 2- or 4-tile placed randomly on the board, and the player then selects a direction to move the tiles. Note that the tiles move until they either (1) collide with another tile, or (2) collide with the edge of the grid. If two tiles of the same number collide in a move, they merge into a single tile valued at the sum of the two originals. The resulting tile cannot merge with another tile again in the same move. Usually, each role in a two-player games has a similar set of moves to choose from, and similar objectives (e.g. chess). In 2048 however, the player roles are inherently asymmetric, as the Computer AI places tiles and the Player moves them. Adversarial search can still be applied! Using your previous experience with objects, states, nodes, functions, and implicit or explicit search trees, along with our skeleton code, focus on optimizing your player algorithm to solve 2048 as efficiently and consistently as possible. -
Graph Traversal with DFS/BFS
Graph Traversal Graph Traversal with DFS/BFS One of the most fundamental graph problems is to traverse every Tyler Moore edge and vertex in a graph. For correctness, we must do the traversal in a systematic way so that CS 2123, The University of Tulsa we dont miss anything. For efficiency, we must make sure we visit each edge at most twice. Since a maze is just a graph, such an algorithm must be powerful enough to enable us to get out of an arbitrary maze. Some slides created by or adapted from Dr. Kevin Wayne. For more information see http://www.cs.princeton.edu/~wayne/kleinberg-tardos 2 / 20 Marking Vertices To Do List The key idea is that we must mark each vertex when we first visit it, We must also maintain a structure containing all the vertices we have and keep track of what have not yet completely explored. discovered but not yet completely explored. Each vertex will always be in one of the following three states: Initially, only a single start vertex is considered to be discovered. 1 undiscovered the vertex in its initial, virgin state. To completely explore a vertex, we look at each edge going out of it. 2 discovered the vertex after we have encountered it, but before we have checked out all its incident edges. For each edge which goes to an undiscovered vertex, we mark it 3 processed the vertex after we have visited all its incident edges. discovered and add it to the list of work to do. A vertex cannot be processed before we discover it, so over the course Note that regardless of what order we fetch the next vertex to of the traversal the state of each vertex progresses from undiscovered explore, each edge is considered exactly twice, when each of its to discovered to processed. -
Graph Traversal and Linear Programs October 6, 2016
CS 125 Section #5 Graph Traversal and Linear Programs October 6, 2016 1 Depth first search 1.1 The Algorithm Besides breadth first search, which we saw in class in relation to Dijkstra's algorithm, there is one other fundamental algorithm for searching a graph: depth first search. To better understand the need for these procedures, let us imagine the computer's view of a graph that has been input into it, in the adjacency list representation. The computer's view is fundamentally local to a specific vertex: it can examine each of the edges adjacent to a vertex in turn, by traversing its adjacency list; it can also mark vertices as visited. One way to think of these operations is to imagine exploring a dark maze with a flashlight and a piece of chalk. You are allowed to illuminate any corridor of the maze emanating from your current position, and you are also allowed to use the chalk to mark your current location in the maze as having been visited. The question is how to find your way around the maze. We now show how the depth first search allows the computer to find its way around the input graph using just these primitives. Depth first search uses a stack as the basic data structure. We start by defining a recursive procedure search (the stack is implicit in the recursive calls of search): search is invoked on a vertex v, and explores all previously unexplored vertices reachable from v. Procedure search(v) vertex v explored(v) := 1 previsit(v) for (v; w) 2 E if explored(w) = 0 then search(w) rof postvisit(v) end search Procedure DFS (G(V; E)) graph G(V; E) for each v 2 V do explored(v) := 0 rof for each v 2 V do if explored(v) = 0 then search(v) rof end DFS By modifying the procedures previsit and postvisit, we can use DFS to solve a number of important problems, as we shall see. -
Best-First and Depth-First Minimax Search in Practice
Best-First and Depth-First Minimax Search in Practice Aske Plaat, Erasmus University, [email protected] Jonathan Schaeffer, University of Alberta, [email protected] Wim Pijls, Erasmus University, [email protected] Arie de Bruin, Erasmus University, [email protected] Erasmus University, University of Alberta, Department of Computer Science, Department of Computing Science, Room H4-31, P.O. Box 1738, 615 General Services Building, 3000 DR Rotterdam, Edmonton, Alberta, The Netherlands Canada T6G 2H1 Abstract Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-®rst pro- cedure, for searching minimax trees. SSS*, with its best-®rst search strategy, reportedly offers the potential for more ef®cient search. However, the complex formulation of the al- gorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search ef®ciency of ªsmartº best-®rst SSS* has cast doubt on the effectiveness of ªdumbº depth-®rst Alpha-Beta. This paper presents a simple framework for calling Alpha-Beta that allows us to create a variety of algorithms, including SSS* and DUAL*. In effect, we formulate a best-®rst algorithm using depth-®rst search. Expressed in this framework SSS* is just a special case of Alpha-Beta, solving all of the perceived drawbacks of the algorithm. In practice, Alpha-Beta variants typically evaluate less nodes than SSS*. A new instance of this framework, MTD(ƒ), out-performs SSS* and NegaScout, the Alpha-Beta variant of choice by practitioners. 1 Introduction Game playing is one of the classic problems of arti®cial intelligence. -
General Branch and Bound, and Its Relation to A* and AO*
ARTIFICIAL INTELLIGENCE 29 General Branch and Bound, and Its Relation to A* and AO* Dana S. Nau, Vipin Kumar and Laveen Kanal* Laboratory for Pattern Analysis, Computer Science Department, University of Maryland College Park, MD 20742, U.S.A. Recommended by Erik Sandewall ABSTRACT Branch and Bound (B&B) is a problem-solving technique which is widely used for various problems encountered in operations research and combinatorial mathematics. Various heuristic search pro- cedures used in artificial intelligence (AI) are considered to be related to B&B procedures. However, in the absence of any generally accepted terminology for B&B procedures, there have been widely differing opinions regarding the relationships between these procedures and B &B. This paper presents a formulation of B&B general enough to include previous formulations as special cases, and shows how two well-known AI search procedures (A* and AO*) are special cases o,f this general formulation. 1. Introduction A wide class of problems arising in operations research, decision making and artificial intelligence can be (abstractly) stated in the following form: Given a (possibly infinite) discrete set X and a real-valued objective function F whose domain is X, find an optimal element x* E X such that F(x*) = min{F(x) I x ~ X}) Unless there is enough problem-specific knowledge available to obtain the optimum element of the set in some straightforward manner, the only course available may be to enumerate some or all of the elements of X until an optimal element is found. However, the sets X and {F(x) [ x E X} are usually tThis work was supported by NSF Grant ENG-7822159 to the Laboratory for Pattern Analysis at the University of Maryland. -
Graphs, Connectivity, and Traversals
Graphs, Connectivity, and Traversals Definitions Like trees, graphs represent a fundamental data structure used in computer science. We often hear about cyber space as being a new frontier for mankind, and if we look at the structure of cyberspace, we see that it is structured as a graph; in other words, it consists of places (nodes), and connections between those places. Some applications of graphs include • representing electronic circuits • modeling object interactions (e.g. used in the Unified Modeling Language) • showing ordering relationships between computer programs • modeling networks and network traffic • the fact that trees are a special case of graphs, in that they are acyclic and connected graphs, and that trees are used in many fundamental data structures 1 An undirected graph G = (V; E) is a pair of sets V , E, where • V is a set of vertices, also called nodes. • E is a set of unordered pairs of vertices called edges, and are of the form (u; v), such that u; v 2 V . • if e = (u; v) is an edge, then we say that u is adjacent to v, and that e is incident with u and v. • We assume jV j = n is finite, where n is called the order of G. •j Ej = m is called the size of G. • A path P of length k in a graph is a sequence of vertices v0; v1; : : : ; vk, such that (vi; vi+1) 2 E for every 0 ≤ i ≤ k − 1. { a path is called simple iff the vertices v0; v1; : : : ; vk are all distinct. -
Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke
Trees Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke Department of Computer Science & Engineering University of Nebraska|Lincoln Lincoln, NE 68588, USA [email protected] http://cse.unl.edu/~cbourke 2015/01/31 21:05:31 Abstract These are lecture notes used in CSCE 156 (Computer Science II), CSCE 235 (Dis- crete Structures) and CSCE 310 (Data Structures & Algorithms) at the University of Nebraska|Lincoln. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License 1 Contents I Trees4 1 Introduction4 2 Definitions & Terminology5 3 Tree Traversal7 3.1 Preorder Traversal................................7 3.2 Inorder Traversal.................................7 3.3 Postorder Traversal................................7 3.4 Breadth-First Search Traversal..........................8 3.5 Implementations & Data Structures.......................8 3.5.1 Preorder Implementations........................8 3.5.2 Inorder Implementation.........................9 3.5.3 Postorder Implementation........................ 10 3.5.4 BFS Implementation........................... 12 3.5.5 Tree Walk Implementations....................... 12 3.6 Operations..................................... 12 4 Binary Search Trees 14 4.1 Basic Operations................................. 15 5 Balanced Binary Search Trees 17 5.1 2-3 Trees...................................... 17 5.2 AVL Trees..................................... 17 5.3 Red-Black Trees.................................. 19 6 Optimal Binary Search Trees 19 7 Heaps 19 -
Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space
applied sciences Article Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space Dawid Połap 1,* ID , Karolina K˛esik 1, Marcin Wo´zniak 1 ID and Robertas Damaševiˇcius 2 ID 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; [email protected] (K.K.); [email protected] (M.W.) 2 Department of Software Engineering, Kaunas University of Technology, Studentu 50, LT-51368, Kaunas, Lithuania; [email protected] * Correspondence: [email protected] Received: 16 December 2017; Accepted: 13 February 2018 ; Published: 16 February 2018 Abstract: The increasing exploration of alternative methods for solving optimization problems causes that parallelization and modification of the existing algorithms are necessary. Obtaining the right solution using the meta-heuristic algorithm may require long operating time or a large number of iterations or individuals in a population. The higher the number, the longer the operation time. In order to minimize not only the time, but also the value of the parameters we suggest three proposition to increase the efficiency of classical methods. The first one is to use the method of searching through the neighborhood in order to minimize the solution space exploration. Moreover, task distribution between threads and CPU cores can affect the speed of the algorithm and therefore make it work more efficiently. The second proposition involves manipulating the solutions space to minimize the number of calculations. In addition, the third proposition is the combination of the previous two. All propositions has been described, tested and analyzed due to the use of various test functions. -
Backtracking Search (Csps) ■Chapter 5 5.3 Is About Local Search Which Is a Very Useful Idea but We Won’T Cover It in Class
CSC384: Intro to Artificial Intelligence Backtracking Search (CSPs) ■Chapter 5 5.3 is about local search which is a very useful idea but we won’t cover it in class. 1 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ● The search algorithms we discussed so far had no knowledge of the states representation (black box). ■ For each problem we had to design a new state representation (and embed in it the sub-routines we pass to the search algorithms). ● Instead we can have a general state representation that works well for many different problems. ● We can build then specialized search algorithms that operate efficiently on this general state representation. ● We call the class of problems that can be represented with this specialized representation CSPs---Constraint Satisfaction Problems. ● Techniques for solving CSPs find more practical applications in industry than most other areas of AI. 2 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ●The idea: represent states as a vector of feature values. We have ■ k-features (or variables) ■ Each feature takes a value. Domain of possible values for the variables: height = {short, average, tall}, weight = {light, average, heavy}. ●In CSPs, the problem is to search for a set of values for the features (variables) so that the values satisfy some conditions (constraints). ■ i.e., a goal state specified as conditions on the vector of feature values. 3 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ●Sudoku: ■ 81 variables, each representing the value of a cell. ■ Values: a fixed value for those cells that are already filled in, the values {1-9} for those cells that are empty. -
Best-First Minimax Search Richard E
Artificial Intelligence ELSEVIER Artificial Intelligence 84 ( 1996) 299-337 Best-first minimax search Richard E. Korf *, David Maxwell Chickering Computer Science Department, University of California, Los Angeles, CA 90024, USA Received September 1994; revised May 1995 Abstract We describe a very simple selective search algorithm for two-player games, called best-first minimax. It always expands next the node at the end of the expected line of play, which determines the minimax value of the root. It uses the same information as alpha-beta minimax, and takes roughly the same time per node generation. We present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, but at significant time cost. Our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move, pruning subtrees below moves not chosen by either player. We also show how to efficiently generate a class of incremental random game trees. On uniform random game trees, best-first minimax outperforms alpha-beta, when both algorithms are given the same amount of computation. On random trees with random branching factors, best-first outperforms alpha-beta for shallow depths, but eventually loses at greater depths. We obtain similar results in the game of Othello. Finally, we present a hybrid best-first extension algorithm that combines alpha-beta with best-first minimax, and performs significantly better than alpha-beta in both domains, even at greater depths. In Othello, it beats alpha-beta in two out of three games. 1. Introduction and overview The best chess machines, such as Deep-Blue [lo], are competitive with the best humans, but generate billions of positions per move.