Lecture 18.1 Multi-Robot Path Planning (3)

Size: px
Start display at page:

Download "Lecture 18.1 Multi-Robot Path Planning (3)"

Transcription

1 CS 60/560 Introduction to Computational Robotics Fall 07, Rutgers University Lecture 8. Multi-Robot Path Planning () Instructor: Jingjin Yu

2 Outline Algorithms for optimal multi-robot path and motion planning Graph search based algorithm for MPP p Coupled search Decoupled search Integer linear programming models for MPP r

3 Optimal Formulations X I 5 6 X G 6 5 Given (G, X I, X G ), want collision free P = {p,, p n } Optimality objectives (minimization): Max time (makespan): min max time(p i) P P p i P Total time: min σ p P P i P time(p i ) Max distance: min max length(p i) P P p i P Total distance: min σ p P P i P length(p i )

4 Incompatibility of the Optimal Formulations Clockwise Counterclockwise Makespan Total time x + x + x + x + x Left path only Using right path Total distance x + 8 x + 0 Total time x + x + x x A pair of the four MPP r objectives on makespan, total time, max distance, and total distance demonstrates a Pareto-optimal structure.

5 Discrete Search Algorithms General methods Coupled search treat all robots as a single robot Decoupled search treat robots as individual ones as much as possible Recall the basic structure of search algorithms (still applies!) input: G = (V, E), x I, x G AddToQueue(x I, Queue); // Add x I to a queue of nodes to be expanded while(!isempty(queue)) x Front(Queue); // Retrieve the front of the queue if(x. expanded == true) continue; // Do not expand a node twice x. expanded = true; // Mark x as expanded if(x == x G ) return solution; // Return if goal is reached for each neighbor n i of x // Add all neighbors of to the queue if(n i. expanded == false) AddToQueue(n i, Queue) return failure;

6 Coupled Search Key: treat all robots as a whole For a single robot, # neighbors? Up to neighbors What about multi-robot case? Up to 5 neighbors per robot, including staying put The search works in a straightforward way For three robots, a neighboring node may be (east, north, stay) But, huge branching factor! For n robots, 5 n neighbors in the graph For robots, 5 neighbors per step Optimal, complete, but impractical (why exactly?)

7 Coupled Search Why Impractical? How large can a priority queue be? For a single robot, no more than V For n robots, V n Well, not exactly, a bit smaller, why? But close enough Suppose V = 0, n = 0 V n = 0 0! We cannot hope to even store the queue on hard disk So search will be extremely slow! S [f=0+7] S A 5 C G A [+6] B [+] C [6+] G [+0] B [+] B C [5+] G [8+0]

8 Decoupled Search Key idea: treat robots as individual ones as much as possible To start, plan optimal paths for individual robots This reduces branching factor: 5 n n Then, simulate the execution of the paths When there are conflicts, push all choices onto the priority queue Then continue the execution Initial paths may get updated/changed In the example One queue node corresponding to robot takes the junction first One queue node corresponding to robot takes the junction first Both will add an additional (makespan) cost of A rough sketch: practical implementations require lots of care-taking

9 Handling Different Objectives Different objectives cause the queue to be sorted differently x x Total distance Initially all choose left path Then - have conflict at t =, generating new nodes (robot i goes first) Then at t =, suppose we pick the node letting go first, three new nodes are created These three new nodes can be inserted into the front of the queue using a secondary heuristic After one more iteration, new nodes are generated Then one last iteration resolves all conflicts The total distance remains the same for all nodes, which is x + 8 Total time Initial node cost is x + 8 Here, at t =, new nodes, cost is now x + for all At t =, if goes through the right, cost is x +, otherwise, x +

10 Strengths and Weakness of Discrete Search Strengths of discrete search solutions When it works, the algorithm generally runs rather fast Because the overall algorithm is relatively simple due to its discrete nature Capable of solving large (sparse) problems Generally straightforward to implement and tweak Weaknesses As the interactions among the robots grow, performance degrades quickly As such, not suitable for solving very dense problems Not suitable for handling MPP r as the number of possible rotations can be very large; huge branching factor For 6-puzzle, >000 possible cycles Each cycle has two directions Enumerating becomes impossible

11 An Integer Programming Based Solver for MPP r Transform MPP into multiflow over time steps G Constraints u v w Each edge is a binary variable u v u u v w v t t + t t + i (x uv,t,t+ i + x vv,t,t+ i + x wv,t,t+ ) i (x uv,t,t+ i + x vu,t,t+ ) i n i n

12 An Example r r, r r t = 0 T=

13 Additional Splitting Heuristic The ILP-base algorithm can require big models 0 5 grid, 0 steps, 0 robots ~ million variables We can use a divide-and-conquer like heuristic through splitting over time The algorithm is no longer complete and may yield sub-optimal solutions

14 The Approach Can Solves Some Tough Problem states > 0 branching factor A 7-step min makespan plan

15 Some Examples in D Continuous Domain No static obstacles, 75 robots.7 seconds to compute,.6-optimal Random obstacles, 50 robots.0 seconds to compute,.9-optimal

CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University. Lecture 08 Extras. A* In More Detail. Instructor: Jingjin Yu

CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University. Lecture 08 Extras. A* In More Detail. Instructor: Jingjin Yu CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University Lecture 08 Extras A* In More Detail Instructor: Jingjin Yu Outline A* in more detail Admissible and consistent heuristic

More information

CS 378: Computer Game Technology

CS 378: Computer Game Technology CS 378: Computer Game Technology Dynamic Path Planning, Flocking Spring 2012 University of Texas at Austin CS 378 Game Technology Don Fussell Dynamic Path Planning! What happens when the environment changes

More information

Lecture 11 Combinatorial Planning: In the Plane

Lecture 11 Combinatorial Planning: In the Plane CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University Lecture 11 Combinatorial Planning: In the Plane Instructor: Jingjin Yu Outline Convex shapes, revisited Combinatorial planning

More information

Structure and Intractability of Optimal Multi-Robot Path Planning on Graphs

Structure and Intractability of Optimal Multi-Robot Path Planning on Graphs Structure and Intractability of Optimal Multi-Robot Path Planning on Graphs Jingjin Yu Department of Electrical and Computer Engineering University of Illinois, Urbana, IL 680 jyu8@uiuc.edu Steven M. LaValle

More information

5. Hashing. 5.1 General Idea. 5.2 Hash Function. 5.3 Separate Chaining. 5.4 Open Addressing. 5.5 Rehashing. 5.6 Extendible Hashing. 5.

5. Hashing. 5.1 General Idea. 5.2 Hash Function. 5.3 Separate Chaining. 5.4 Open Addressing. 5.5 Rehashing. 5.6 Extendible Hashing. 5. 5. Hashing 5.1 General Idea 5.2 Hash Function 5.3 Separate Chaining 5.4 Open Addressing 5.5 Rehashing 5.6 Extendible Hashing Malek Mouhoub, CS340 Fall 2004 1 5. Hashing Sequential access : O(n). Binary

More information

LECTURE NOTES OF ALGORITHMS: DESIGN TECHNIQUES AND ANALYSIS

LECTURE NOTES OF ALGORITHMS: DESIGN TECHNIQUES AND ANALYSIS Department of Computer Science University of Babylon LECTURE NOTES OF ALGORITHMS: DESIGN TECHNIQUES AND ANALYSIS By Faculty of Science for Women( SCIW), University of Babylon, Iraq Samaher@uobabylon.edu.iq

More information

REFERENCE MATERIALS. Assignment, Display, and Input Evaluates expression and assigns the result to the variable a.

REFERENCE MATERIALS. Assignment, Display, and Input Evaluates expression and assigns the result to the variable a. a expression Assignment, Display, and Input Evaluates expression and assigns the result to the variable a. DISPLAY (expression) Displays the value of expression, followed by a space. INPUT () Accepts a

More information

Sampling-based Planning 2

Sampling-based Planning 2 RBE MOTION PLANNING Sampling-based Planning 2 Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Problem with KD-tree RBE MOTION PLANNING Curse of dimension

More information

Digital Image Processing (CS/ECE 545) Lecture 5: Edge Detection (Part 2) & Corner Detection

Digital Image Processing (CS/ECE 545) Lecture 5: Edge Detection (Part 2) & Corner Detection Digital Image Processing (CS/ECE 545) Lecture 5: Edge Detection (Part 2) & Corner Detection Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Recall: Edge Detection Image processing

More information

Sung-Eui Yoon ( 윤성의 )

Sung-Eui Yoon ( 윤성의 ) Path Planning for Point Robots Sung-Eui Yoon ( 윤성의 ) Course URL: http://sglab.kaist.ac.kr/~sungeui/mpa Class Objectives Motion planning framework Classic motion planning approaches 2 3 Configuration Space:

More information

CS 1567 Intermediate Programming and System Design Using a Mobile Robot Aibo Lab3 Localization and Path Planning

CS 1567 Intermediate Programming and System Design Using a Mobile Robot Aibo Lab3 Localization and Path Planning CS 1567 Intermediate Programming and System Design Using a Mobile Robot Aibo Lab3 Localization and Path Planning In this lab we will create an artificial landscape in the Aibo pen. The landscape has two

More information

CAD Algorithms. Categorizing Algorithms

CAD Algorithms. Categorizing Algorithms CAD Algorithms Categorizing Algorithms Mohammad Tehranipoor ECE Department 2 September 2008 1 Categorizing Algorithms Greedy Algorithms Prim s Algorithm (Minimum Spanning Tree) A subgraph that is a tree

More information

Abstract Path Planning for Multiple Robots: An Empirical Study

Abstract Path Planning for Multiple Robots: An Empirical Study Abstract Path Planning for Multiple Robots: An Empirical Study Charles University in Prague Faculty of Mathematics and Physics Department of Theoretical Computer Science and Mathematical Logic Malostranské

More information

Algorithm Performance Factors. Memory Performance of Algorithms. Processor-Memory Performance Gap. Moore s Law. Program Model of Memory I

Algorithm Performance Factors. Memory Performance of Algorithms. Processor-Memory Performance Gap. Moore s Law. Program Model of Memory I Memory Performance of Algorithms CSE 32 Data Structures Lecture Algorithm Performance Factors Algorithm choices (asymptotic running time) O(n 2 ) or O(n log n) Data structure choices List or Arrays Language

More information

Computational Geometry

Computational Geometry Lecture 1: Introduction and convex hulls Geometry: points, lines,... Geometric objects Geometric relations Combinatorial complexity Computational geometry Plane (two-dimensional), R 2 Space (three-dimensional),

More information

EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS

EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 3, 4/6/2005 University of Washington, Department of Electrical Engineering Spring 2005 Instructor: Professor Jeff A. Bilmes 4/6/2005 EE562 1 Today: Basic

More information

Chapter 18 Indexing Structures for Files. Indexes as Access Paths

Chapter 18 Indexing Structures for Files. Indexes as Access Paths Chapter 18 Indexing Structures for Files Indexes as Access Paths A single-level index is an auxiliary file that makes it more efficient to search for a record in the data file. The index is usually specified

More information

Data Mining. Part 2. Data Understanding and Preparation. 2.4 Data Transformation. Spring Instructor: Dr. Masoud Yaghini. Data Transformation

Data Mining. Part 2. Data Understanding and Preparation. 2.4 Data Transformation. Spring Instructor: Dr. Masoud Yaghini. Data Transformation Data Mining Part 2. Data Understanding and Preparation 2.4 Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Normalization Attribute Construction Aggregation Attribute Subset Selection Discretization

More information

Lecture 19 Sorting Goodrich, Tamassia

Lecture 19 Sorting Goodrich, Tamassia Lecture 19 Sorting 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 2004 Goodrich, Tamassia Outline Review 3 simple sorting algorithms: 1. selection Sort (in previous course) 2. insertion Sort (in previous

More information

Constraint (Logic) Programming

Constraint (Logic) Programming Constraint (Logic) Programming Roman Barták Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic bartak@ktiml.mff.cuni.cz Sudoku Combinatorial puzzle, whose goal is to enter

More information

AP Computer Science Principles Exam Reference Sheet

AP Computer Science Principles Exam Reference Sheet AP Computer Science Principles Exam Reference Sheet July 2015 As AP Computer Science Principles does not designate any particular programming language, this reference sheet provides instructions and explanations

More information

CSE Data Structures and Introduction to Algorithms... In Java! Instructor: Fei Wang. Mid-Term Exam. CSE2100 DS & Algorithms 1

CSE Data Structures and Introduction to Algorithms... In Java! Instructor: Fei Wang. Mid-Term Exam. CSE2100 DS & Algorithms 1 CSE 2100 Data Structures and Introduction to Algorithms...! In Java!! Instructor: Fei Wang! Mid-Term Exam CSE2100 DS & Algorithms 1 1. True or False (20%=2%x10)! (1) O(n) is O(n^2) (2) The height h of

More information

Locality- Sensitive Hashing Random Projections for NN Search

Locality- Sensitive Hashing Random Projections for NN Search Case Study 2: Document Retrieval Locality- Sensitive Hashing Random Projections for NN Search Machine Learning for Big Data CSE547/STAT548, University of Washington Sham Kakade April 18, 2017 Sham Kakade

More information

15-494/694: Cognitive Robotics

15-494/694: Cognitive Robotics 15-494/694: Cognitive Robotics Dave Touretzky Lecture 9: Path Planning with Rapidly-exploring Random Trees Navigating with the Pilot Image from http://www.futuristgerd.com/2015/09/10 Outline How is path

More information

Lecture 18. Questions? Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 1

Lecture 18. Questions? Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 1 Lecture 18 Questions? Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 1 Outline Chapter 6 - Constraint Satisfaction Problems Path Consistency & Global Constraints Sudoku Example Backtracking

More information

Computational Geometry Algorithmische Geometrie

Computational Geometry Algorithmische Geometrie Algorithmische Geometrie Panos Giannopoulos Wolfgang Mulzer Lena Schlipf AG TI SS 2013 !! Register in Campus Management!! Outline What you need to know (before taking this course) What is the course about?

More information

Design and Analysis of Algorithms Prof. Madhavan Mukund Chennai Mathematical Institute. Week 02 Module 06 Lecture - 14 Merge Sort: Analysis

Design and Analysis of Algorithms Prof. Madhavan Mukund Chennai Mathematical Institute. Week 02 Module 06 Lecture - 14 Merge Sort: Analysis Design and Analysis of Algorithms Prof. Madhavan Mukund Chennai Mathematical Institute Week 02 Module 06 Lecture - 14 Merge Sort: Analysis So, we have seen how to use a divide and conquer strategy, we

More information

Compiler Design. Register Allocation. Hwansoo Han

Compiler Design. Register Allocation. Hwansoo Han Compiler Design Register Allocation Hwansoo Han Big Picture of Code Generation Register allocation Decides which values will reside in registers Changes the storage mapping Concerns about placement of

More information

Assignment 1 is out! Due: 9 Sep 23:59! Can work in a group of 2-3 students.! NO cheating!!!! Submit in turnitin! Code + report!

Assignment 1 is out! Due: 9 Sep 23:59! Can work in a group of 2-3 students.! NO cheating!!!! Submit in turnitin! Code + report! Assignment 1 is out! Due: 9 Sep 23:59! Submit in turnitin! Code + report! Can work in a group of 2-3 students.! Please register your group in the website linked from the assignment description before tomorrow

More information

Heuristic Search and Advanced Methods

Heuristic Search and Advanced Methods Heuristic Search and Advanced Methods Computer Science cpsc322, Lecture 3 (Textbook Chpt 3.6 3.7) May, 15, 2012 CPSC 322, Lecture 3 Slide 1 Course Announcements Posted on WebCT Assignment1 (due on Thurs!)

More information

CS 245 Midterm Exam Solution Winter 2015

CS 245 Midterm Exam Solution Winter 2015 CS 245 Midterm Exam Solution Winter 2015 This exam is open book and notes. You can use a calculator and your laptop to access course notes and videos (but not to communicate with other people). You have

More information

FINALTERM EXAMINATION Fall 2009 CS301- Data Structures Question No: 1 ( Marks: 1 ) - Please choose one The data of the problem is of 2GB and the hard

FINALTERM EXAMINATION Fall 2009 CS301- Data Structures Question No: 1 ( Marks: 1 ) - Please choose one The data of the problem is of 2GB and the hard FINALTERM EXAMINATION Fall 2009 CS301- Data Structures Question No: 1 The data of the problem is of 2GB and the hard disk is of 1GB capacity, to solve this problem we should Use better data structures

More information

Ordered Indices To gain fast random access to records in a file, we can use an index structure. Each index structure is associated with a particular search key. Just like index of a book, library catalog,

More information

/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Priority Queues / Heaps Date: 9/27/17

/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Priority Queues / Heaps Date: 9/27/17 01.433/33 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Priority Queues / Heaps Date: 9/2/1.1 Introduction In this lecture we ll talk about a useful abstraction, priority queues, which are

More information

Chapter 6 Constraint Satisfaction Problems

Chapter 6 Constraint Satisfaction Problems Chapter 6 Constraint Satisfaction Problems CS5811 - Artificial Intelligence Nilufer Onder Department of Computer Science Michigan Technological University Outline CSP problem definition Backtracking search

More information

implementing the breadth-first search algorithm implementing the depth-first search algorithm

implementing the breadth-first search algorithm implementing the depth-first search algorithm Graph Traversals 1 Graph Traversals representing graphs adjacency matrices and adjacency lists 2 Implementing the Breadth-First and Depth-First Search Algorithms implementing the breadth-first search algorithm

More information

Table ADT and Sorting. Algorithm topics continuing (or reviewing?) CS 24 curriculum

Table ADT and Sorting. Algorithm topics continuing (or reviewing?) CS 24 curriculum Table ADT and Sorting Algorithm topics continuing (or reviewing?) CS 24 curriculum A table ADT (a.k.a. Dictionary, Map) Table public interface: // Put information in the table, and a unique key to identify

More information

17/05/2018. Outline. Outline. Divide and Conquer. Control Abstraction for Divide &Conquer. Outline. Module 2: Divide and Conquer

17/05/2018. Outline. Outline. Divide and Conquer. Control Abstraction for Divide &Conquer. Outline. Module 2: Divide and Conquer Module 2: Divide and Conquer Divide and Conquer Control Abstraction for Divide &Conquer 1 Recurrence equation for Divide and Conquer: If the size of problem p is n and the sizes of the k sub problems are

More information

CS 161 Fall 2015 Final Exam

CS 161 Fall 2015 Final Exam CS 161 Fall 2015 Final Exam Name: Student ID: 1: 2: 3: 4: 5: 6: 7: 8: Total: 1. (15 points) Let H = [24, 21, 18, 15, 12, 9, 6, 3] be an array of eight numbers, interpreted as a binary heap with the maximum

More information

Trees. Courtesy to Goodrich, Tamassia and Olga Veksler

Trees. Courtesy to Goodrich, Tamassia and Olga Veksler Lecture 12: BT Trees Courtesy to Goodrich, Tamassia and Olga Veksler Instructor: Yuzhen Xie Outline B-tree Special case of multiway search trees used when data must be stored on the disk, i.e. too large

More information

Motivation for B-Trees

Motivation for B-Trees 1 Motivation for Assume that we use an AVL tree to store about 20 million records We end up with a very deep binary tree with lots of different disk accesses; log2 20,000,000 is about 24, so this takes

More information

Introduction. for large input, even access time may be prohibitive we need data structures that exhibit times closer to O(log N) binary search tree

Introduction. for large input, even access time may be prohibitive we need data structures that exhibit times closer to O(log N) binary search tree Chapter 4 Trees 2 Introduction for large input, even access time may be prohibitive we need data structures that exhibit running times closer to O(log N) binary search tree 3 Terminology recursive definition

More information

Outline. Graphs. Divide and Conquer.

Outline. Graphs. Divide and Conquer. GRAPHS COMP 321 McGill University These slides are mainly compiled from the following resources. - Professor Jaehyun Park slides CS 97SI - Top-coder tutorials. - Programming Challenges books. Outline Graphs.

More information

Lecture 3: Art Gallery Problems and Polygon Triangulation

Lecture 3: Art Gallery Problems and Polygon Triangulation EECS 396/496: Computational Geometry Fall 2017 Lecture 3: Art Gallery Problems and Polygon Triangulation Lecturer: Huck Bennett In this lecture, we study the problem of guarding an art gallery (specified

More information

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe. Slide 14-1

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe. Slide 14-1 Slide 14-1 Chapter 14 Indexing Structures for Files Chapter Outline Types of Single-level Ordered Indexes Primary Indexes Clustering Indexes Secondary Indexes Multilevel Indexes Dynamic Multilevel Indexes

More information

Lecture 3, Review of Algorithms. What is Algorithm?

Lecture 3, Review of Algorithms. What is Algorithm? BINF 336, Introduction to Computational Biology Lecture 3, Review of Algorithms Young-Rae Cho Associate Professor Department of Computer Science Baylor University What is Algorithm? Definition A process

More information

Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 25, 2013 The following is a simple example of a connected, undirected graph with 5 vertices

More information

INSTITUTE OF AERONAUTICAL ENGINEERING

INSTITUTE OF AERONAUTICAL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 COMPUTER SCIENCE AND ENGINEERING TUTORIAL QUESTION BANK Course Name Course Code Class Branch DATA STRUCTURES ACS002 B. Tech

More information

Chapter 18 Indexing Structures for Files

Chapter 18 Indexing Structures for Files Chapter 18 Indexing Structures for Files Copyright 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Disk I/O for Read/ Write Unit for Disk I/O for Read/ Write: Chapter 18 One Buffer for

More information

11.1 Facility Location

11.1 Facility Location CS787: Advanced Algorithms Scribe: Amanda Burton, Leah Kluegel Lecturer: Shuchi Chawla Topic: Facility Location ctd., Linear Programming Date: October 8, 2007 Today we conclude the discussion of local

More information

R16 SET - 1 '' ''' '' ''' Code No: R

R16 SET - 1 '' ''' '' ''' Code No: R 1. a) Define Latency time and Transmission time? (2M) b) Define Hash table and Hash function? (2M) c) Explain the Binary Heap Structure Property? (3M) d) List the properties of Red-Black trees? (3M) e)

More information

Computer Science Spring 2005 Final Examination, May 12, 2005

Computer Science Spring 2005 Final Examination, May 12, 2005 Computer Science 302 00 Spring 2005 Final Examination, May 2, 2005 Name: No books, notes, or scratch paper. Use pen or pencil, any color. Use the backs of the pages for scratch paper. If you need more

More information

9/29/2016. Chapter 4 Trees. Introduction. Terminology. Terminology. Terminology. Terminology

9/29/2016. Chapter 4 Trees. Introduction. Terminology. Terminology. Terminology. Terminology Introduction Chapter 4 Trees for large input, even linear access time may be prohibitive we need data structures that exhibit average running times closer to O(log N) binary search tree 2 Terminology recursive

More information

Scribes: Romil Verma, Juliana Cook (2015), Virginia Williams, Date: May 1, 2017 and Seth Hildick-Smith (2016), G. Valiant (2017), M.

Scribes: Romil Verma, Juliana Cook (2015), Virginia Williams, Date: May 1, 2017 and Seth Hildick-Smith (2016), G. Valiant (2017), M. Lecture 9 Graphs Scribes: Romil Verma, Juliana Cook (2015), Virginia Williams, Date: May 1, 2017 and Seth Hildick-Smith (2016), G. Valiant (2017), M. Wootters (2017) 1 Graphs A graph is a set of vertices

More information

CS 4100/5100: Foundations of AI

CS 4100/5100: Foundations of AI CS 4100/5100: Foundations of AI Constraint satisfaction problems 1 Instructor: Rob Platt r.platt@neu.edu College of Computer and information Science Northeastern University September 5, 2013 1 These notes

More information

Horn Formulae. CS124 Course Notes 8 Spring 2018

Horn Formulae. CS124 Course Notes 8 Spring 2018 CS124 Course Notes 8 Spring 2018 In today s lecture we will be looking a bit more closely at the Greedy approach to designing algorithms. As we will see, sometimes it works, and sometimes even when it

More information

UNIVERSITY OF WATERLOO DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING ECE 250 ALGORITHMS AND DATA STRUCTURES

UNIVERSITY OF WATERLOO DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING ECE 250 ALGORITHMS AND DATA STRUCTURES UNIVERSITY OF WATERLOO DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING ECE 250 ALGORITHMS AND DATA STRUCTURES Final Examination Instructor: R.E.Seviora 9-12 AM, Dec 14, 2002 Name (last, first) Student

More information

16 Greedy Algorithms

16 Greedy Algorithms 16 Greedy Algorithms Optimization algorithms typically go through a sequence of steps, with a set of choices at each For many optimization problems, using dynamic programming to determine the best choices

More information

Register Allocation (via graph coloring) Lecture 25. CS 536 Spring

Register Allocation (via graph coloring) Lecture 25. CS 536 Spring Register Allocation (via graph coloring) Lecture 25 CS 536 Spring 2001 1 Lecture Outline Memory Hierarchy Management Register Allocation Register interference graph Graph coloring heuristics Spilling Cache

More information

Lecture 25 Notes Spanning Trees

Lecture 25 Notes Spanning Trees Lecture 25 Notes Spanning Trees 15-122: Principles of Imperative Computation (Spring 2016) Frank Pfenning 1 Introduction The following is a simple example of a connected, undirected graph with 5 vertices

More information

CS4733 Class Notes. 1 2-D Robot Motion Planning Algorithm Using Grown Obstacles

CS4733 Class Notes. 1 2-D Robot Motion Planning Algorithm Using Grown Obstacles CS4733 Class Notes 1 2-D Robot Motion Planning Algorithm Using Grown Obstacles Reference: An Algorithm for Planning Collision Free Paths Among Poyhedral Obstacles by T. Lozano-Perez and M. Wesley. This

More information

Constraint Satisfaction Problems

Constraint Satisfaction Problems Constraint Satisfaction Problems CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2013 Soleymani Course material: Artificial Intelligence: A Modern Approach, 3 rd Edition,

More information

CSE 2123 Recursion. Jeremy Morris

CSE 2123 Recursion. Jeremy Morris CSE 2123 Recursion Jeremy Morris 1 Past Few Weeks For the past few weeks we have been focusing on data structures Classes & Object-oriented programming Collections Lists, Sets, Maps, etc. Now we turn our

More information

Problem 1: Complexity of Update Rules for Logistic Regression

Problem 1: Complexity of Update Rules for Logistic Regression Case Study 1: Estimating Click Probabilities Tackling an Unknown Number of Features with Sketching Machine Learning for Big Data CSE547/STAT548, University of Washington Emily Fox January 16 th, 2014 1

More information

CS 532: 3D Computer Vision 14 th Set of Notes

CS 532: 3D Computer Vision 14 th Set of Notes 1 CS 532: 3D Computer Vision 14 th Set of Notes Instructor: Philippos Mordohai Webpage: www.cs.stevens.edu/~mordohai E-mail: Philippos.Mordohai@stevens.edu Office: Lieb 215 Lecture Outline Triangulating

More information

Algorithms for Decision Support. Integer linear programming models

Algorithms for Decision Support. Integer linear programming models Algorithms for Decision Support Integer linear programming models 1 People with reduced mobility (PRM) require assistance when travelling through the airport http://www.schiphol.nl/travellers/atschiphol/informationforpassengerswithreducedmobility.htm

More information

12/30/2013 S. NALINI,AP/CSE

12/30/2013 S. NALINI,AP/CSE 12/30/2013 S. NALINI,AP/CSE 1 UNIT I ITERATIVE AND RECURSIVE ALGORITHMS Iterative Algorithms: Measures of Progress and Loop Invariants-Paradigm Shift: Sequence of Actions versus Sequence of Assertions-

More information

Hi everyone. I hope everyone had a good Fourth of July. Today we're going to be covering graph search. Now, whenever we bring up graph algorithms, we

Hi everyone. I hope everyone had a good Fourth of July. Today we're going to be covering graph search. Now, whenever we bring up graph algorithms, we Hi everyone. I hope everyone had a good Fourth of July. Today we're going to be covering graph search. Now, whenever we bring up graph algorithms, we have to talk about the way in which we represent the

More information

Algorithm Design Techniques (III)

Algorithm Design Techniques (III) Algorithm Design Techniques (III) Minimax. Alpha-Beta Pruning. Search Tree Strategies (backtracking revisited, branch and bound). Local Search. DSA - lecture 10 - T.U.Cluj-Napoca - M. Joldos 1 Tic-Tac-Toe

More information

PROBLEM 1 : (And the winner is...(12 points)) Assume you are considering the implementation of a priority queue that will always give you the smallest

PROBLEM 1 : (And the winner is...(12 points)) Assume you are considering the implementation of a priority queue that will always give you the smallest CPS 100, Ramm Hour Exam #2 (11/1/99) Fall, 1999 NAME (print): Honor Acknowledgment (signature): DO NOT SPEND MORE THAN 10 OR SO MINUTES ON ANY OF THE OTHER QUESTIONS! If you don't see the solution to a

More information

CS 188: Artificial Intelligence Spring Today

CS 188: Artificial Intelligence Spring Today CS 188: Artificial Intelligence Spring 2006 Lecture 7: CSPs II 2/7/2006 Dan Klein UC Berkeley Many slides from either Stuart Russell or Andrew Moore Today More CSPs Applications Tree Algorithms Cutset

More information

Grad AI Fall, 2007 Homework 1

Grad AI Fall, 2007 Homework 1 Grad AI. 15-780 Fall, 2007 Homework 1 Homework deadline: 10:30am on October 4 Please print your code and hand it in with the hard copy of your homework. Also send a copy of your code by e-mail to both

More information

MVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms

MVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms MVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2018 04 24 Lecture 9 Linear and integer optimization with applications

More information

10/31/18. About A6, Prelim 2. Spanning Trees, greedy algorithms. Facts about trees. Undirected trees

10/31/18. About A6, Prelim 2. Spanning Trees, greedy algorithms. Facts about trees. Undirected trees //8 About A, Prelim Spanning Trees, greedy algorithms Lecture CS Fall 8 Prelim : Thursday, November. Visit exams page of course website and read carefully to find out when you take it (: or 7:) and what

More information

Spanning Trees, greedy algorithms. Lecture 20 CS2110 Fall 2018

Spanning Trees, greedy algorithms. Lecture 20 CS2110 Fall 2018 1 Spanning Trees, greedy algorithms Lecture 20 CS2110 Fall 2018 1 About A6, Prelim 2 Prelim 2: Thursday, 15 November. Visit exams page of course website and read carefully to find out when you take it

More information

Principles of Algorithm Design

Principles of Algorithm Design Principles of Algorithm Design When you are trying to design an algorithm or a data structure, it s often hard to see how to accomplish the task. The following techniques can often be useful: 1. Experiment

More information

Integer Programming ISE 418. Lecture 7. Dr. Ted Ralphs

Integer Programming ISE 418. Lecture 7. Dr. Ted Ralphs Integer Programming ISE 418 Lecture 7 Dr. Ted Ralphs ISE 418 Lecture 7 1 Reading for This Lecture Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Wolsey Chapter 7 CCZ Chapter 1 Constraint

More information

Fundamentals of Database Systems Prof. Arnab Bhattacharya Department of Computer Science and Engineering Indian Institute of Technology, Kanpur

Fundamentals of Database Systems Prof. Arnab Bhattacharya Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Fundamentals of Database Systems Prof. Arnab Bhattacharya Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture - 18 Database Indexing: Hashing We will start on

More information

Hash-Based Indexing 1

Hash-Based Indexing 1 Hash-Based Indexing 1 Tree Indexing Summary Static and dynamic data structures ISAM and B+ trees Speed up both range and equality searches B+ trees very widely used in practice ISAM trees can be useful

More information

56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998

56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation

More information

Chapter 18. Indexing Structures for Files. Chapter Outline. Indexes as Access Paths. Primary Indexes Clustering Indexes Secondary Indexes

Chapter 18. Indexing Structures for Files. Chapter Outline. Indexes as Access Paths. Primary Indexes Clustering Indexes Secondary Indexes Chapter 18 Indexing Structures for Files Chapter Outline Types of Single-level Ordered Indexes Primary Indexes Clustering Indexes Secondary Indexes Multilevel Indexes Dynamic Multilevel Indexes Using B-Trees

More information

Search Algorithms for Discrete Optimization Problems

Search Algorithms for Discrete Optimization Problems Search Algorithms for Discrete Optimization Problems Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar To accompany the text ``Introduction to Parallel Computing'', Addison Wesley, 2003. 1 Topic

More information

15.083J Integer Programming and Combinatorial Optimization Fall Enumerative Methods

15.083J Integer Programming and Combinatorial Optimization Fall Enumerative Methods 5.8J Integer Programming and Combinatorial Optimization Fall 9 A knapsack problem Enumerative Methods Let s focus on maximization integer linear programs with only binary variables For example: a knapsack

More information

Database files Organizations Indexing B-tree and B+ tree. Copyright 2011 Ramez Elmasri and Shamkant Navathe

Database files Organizations Indexing B-tree and B+ tree. Copyright 2011 Ramez Elmasri and Shamkant Navathe Database files Organizations Indexing B-tree and B+ tree Outline Type of Single-Level Ordered Indexes Multilevel Indexes Dynamic Multilevel Indexes Using B-Trees and B + -Trees Indexes on Multiple Keys

More information

Rasterization: Geometric Primitives

Rasterization: Geometric Primitives Rasterization: Geometric Primitives Outline Rasterizing lines Rasterizing polygons 1 Rasterization: What is it? How to go from real numbers of geometric primitives vertices to integer coordinates of pixels

More information

Toward the joint design of electronic and optical layer protection

Toward the joint design of electronic and optical layer protection Toward the joint design of electronic and optical layer protection Massachusetts Institute of Technology Slide 1 Slide 2 CHALLENGES: - SEAMLESS CONNECTIVITY - MULTI-MEDIA (FIBER,SATCOM,WIRELESS) - HETEROGENEOUS

More information

Table of Contents. Course Minutiae. Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis 2 / 32

Table of Contents. Course Minutiae. Course Overview Algorithm Design Strategies Algorithm Correctness Asymptotic Analysis 2 / 32 Intro Lecture CS 584/684: Algorithm Design and Analysis Daniel Leblanc1 1 Senior Adjunct Instructor Portland State University Maseeh College of Engineering and Computer Science Spring 2018 1 / 32 2 / 32

More information

Planning and Control: Markov Decision Processes

Planning and Control: Markov Decision Processes CSE-571 AI-based Mobile Robotics Planning and Control: Markov Decision Processes Planning Static vs. Dynamic Predictable vs. Unpredictable Fully vs. Partially Observable Perfect vs. Noisy Environment What

More information

Recursive Sequences. Lecture 24 Section 5.6. Robb T. Koether. Hampden-Sydney College. Wed, Feb 27, 2013

Recursive Sequences. Lecture 24 Section 5.6. Robb T. Koether. Hampden-Sydney College. Wed, Feb 27, 2013 Recursive Sequences Lecture 24 Section 5.6 Robb T. Koether Hampden-Sydney College Wed, Feb 27, 2013 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 1 / 21 1 Recursive Sequences

More information

Register Allocation 1

Register Allocation 1 Register Allocation 1 Lecture Outline Memory Hierarchy Management Register Allocation Register interference graph Graph coloring heuristics Spilling Cache Management The Memory Hierarchy Registers < 1

More information

UNIT III BALANCED SEARCH TREES AND INDEXING

UNIT III BALANCED SEARCH TREES AND INDEXING UNIT III BALANCED SEARCH TREES AND INDEXING OBJECTIVE The implementation of hash tables is frequently called hashing. Hashing is a technique used for performing insertions, deletions and finds in constant

More information

Princeton University Computer Science COS226: Data Structures and Algorithms. Midterm, Fall 2013

Princeton University Computer Science COS226: Data Structures and Algorithms. Midterm, Fall 2013 Princeton University Computer Science COS226: Data Structures and Algorithms Midterm, Fall 2013 This test has 8 questions worth a total of 57 points. The exam is closed book, except that you are allowed

More information

Homework #6 (Constraint Satisfaction, Non-Deterministic Uncertainty and Adversarial Search) Out: 2/21/11 Due: 2/29/11 (at noon)

Homework #6 (Constraint Satisfaction, Non-Deterministic Uncertainty and Adversarial Search) Out: 2/21/11 Due: 2/29/11 (at noon) CS121 Introduction to Artificial Intelligence Winter 2011 Homework #6 (Constraint Satisfaction, Non-Deterministic Uncertainty and Adversarial Search) Out: 2/21/11 Due: 2/29/11 (at noon) How to complete

More information

Lecture 24: Image Retrieval: Part II. Visual Computing Systems CMU , Fall 2013

Lecture 24: Image Retrieval: Part II. Visual Computing Systems CMU , Fall 2013 Lecture 24: Image Retrieval: Part II Visual Computing Systems Review: K-D tree Spatial partitioning hierarchy K = dimensionality of space (below: K = 2) 3 2 1 3 3 4 2 Counts of points in leaf nodes Nearest

More information

MCA SEM-II Data Structure

MCA SEM-II Data Structure MCA SEM-II Data Structure Timings: 30 Minutes] Objective Questions Keys [ Marks - 40 1. The memory address of the first element of an array is called a. Floor address b. Foundation address c. First address

More information

Crew Scheduling Problem: A Column Generation Approach Improved by a Genetic Algorithm. Santos and Mateus (2007)

Crew Scheduling Problem: A Column Generation Approach Improved by a Genetic Algorithm. Santos and Mateus (2007) In the name of God Crew Scheduling Problem: A Column Generation Approach Improved by a Genetic Algorithm Spring 2009 Instructor: Dr. Masoud Yaghini Outlines Problem Definition Modeling As A Set Partitioning

More information

Recursive Sequences. Lecture 24 Section 5.6. Robb T. Koether. Hampden-Sydney College. Wed, Feb 26, 2014

Recursive Sequences. Lecture 24 Section 5.6. Robb T. Koether. Hampden-Sydney College. Wed, Feb 26, 2014 Recursive Sequences Lecture 24 Section 5.6 Robb T. Koether Hampden-Sydney College Wed, Feb 26, 2014 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 26, 2014 1 / 26 1 Recursive Sequences

More information

CS 380: Artificial Intelligence Lecture #3

CS 380: Artificial Intelligence Lecture #3 CS 380: Artificial Intelligence Lecture #3 William Regli Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms 1 Problem-solving agents Example: Romania

More information

CS 2150 Final Exam, Spring 2018 Page 1 of 10 UVa userid:

CS 2150 Final Exam, Spring 2018 Page 1 of 10 UVa userid: CS 2150 Final Exam, Spring 2018 Page 1 of 10 UVa userid: CS 2150 Final Exam Name You MUST write your e-mail ID on EACH page and put your name on the top of this page, too. If you are still writing when

More information

Basic Search Algorithms

Basic Search Algorithms Basic Search Algorithms Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Abstract The complexities of various search algorithms are considered in terms of time, space, and cost

More information