Weighted Heuristic Anytime Search
|
|
|
- Arnold Palmer
- 6 years ago
- Views:
Transcription
1 Weighted Heuristic Anytime Search Flerova, Marinescu, and Dechter 1 1 University of South Carolina April 24, 2017
2 Basic Heuristic Best-first Search Blindly follows the heuristic Weighted A Search For w > 1 f (n) = д(n) + w h(n) Larger w yields greedier searches
3 Basic Heuristic Best-first Search Blindly follows the heuristic Weighted A Search For w > 1 f (n) = д(n) + w h(n) Larger w yields greedier searches
4 Graphical Definition (Graphical Model) A tuple M = X, D, F, where 1 X = {X 0,..., X n 1 } is a set of variables 2 D = {D 0,..., D n 1 } is a set of domains 3 F = { f 0 (X S0 ),..., f r 1 (X Sr 1 ) } is a set of scopes: X Si X i. f i : X Si R + 4 A combination operator {, } The model M represents the function C(X ) = r 1 i=0 f i (X Si )
5 Graphical Definition (Graphical Model) A tuple M = X, D, F, where 1 X = {X 0,..., X n 1 } is a set of variables 2 D = {D 0,..., D n 1 } is a set of domains 3 F = { f 0 (X S0 ),..., f r 1 (X Sr 1 ) } is a set of scopes: X Si X i. f i : X Si R + 4 A combination operator {, } The model M represents the function C(X ) = r 1 i=0 f i (X Si )
6 Graphical Given a model M = X, D, F,, the most common optimization task is either most probable explanation or maximum a posteriori MPE Find the optimal value C : C = C(x ) = max X r 1 i=0 f i (X Si ) MAP Find the optimizing configuration x : x = argmax X r 1 i=0 f i (X Si )
7 Graphical Given a model M = X, D, F,, the most common optimization task is either most probable explanation or maximum a posteriori MPE Find the optimal value C : C = C(x ) = max X r 1 i=0 f i (X Si ) MAP Find the optimizing configuration x : x = argmax X r 1 i=0 f i (X Si )
8 Graphical Given a model M = X, D, F,, the most common optimization task is either most probable explanation or maximum a posteriori MPE Find the optimal value C : C = C(x ) = max X r 1 i=0 f i (X Si ) MAP Find the optimizing configuration x : x = argmax X r 1 i=0 f i (X Si )
9 Graphical : MPE/MAP WCSP MPE C MPE = C(x ) = max X r 1 i=0 f i (X Si ) WCSP Weighted Constraint Satisfaction Problem (MPE in negative log-space) C WCSP C(x ) = min X r 1 i=0 f i (X Si )
10 Graphical : MPE/MAP WCSP MPE C MPE = C(x ) = max X r 1 i=0 f i (X Si ) WCSP Weighted Constraint Satisfaction Problem (MPE in negative log-space) C WCSP C(x ) = min X r 1 i=0 f i (X Si )
11 Primal Graph Definition The primal graph of a model is a graph where the vertices are the variables and edges connect variables within the same scope Scopes X S0 = {A, B} X S1 = {A,C} X S2 = {C, D} C D A B E F X S3 = {B, D} X S4 = {B, F } X S5 = {E, F } Figure: Primal
12 Primal Graph Definition The primal graph of a model is a graph where the vertices are the variables and edges connect variables within the same scope Scopes X S0 = {A, B} X S1 = {A,C} X S2 = {C, D} C D A B E F X S3 = {B, D} X S4 = {B, F } X S5 = {E, F } Figure: Primal
13 A C D A B E F B C D E F (a) Primal (b) Induced Graph Figure: Induced graph over the natural ordering.
14 A C D A B E F B C D E F (a) Primal (b) Induced Graph Figure: Induced graph over the natural ordering.
15 A A C E C E B B D F D F (a) Orig + Ind. Edges (b) Pseudo-tree Figure: Pseudo-tree with edges chosen to respect the order
16 A A C E C E B B D F D F (a) Orig + Ind. Edges (b) Pseudo-tree Figure: Pseudo-tree with edges chosen to respect the order
17 OR AND OR AND OR A 0 1 B B C C C C E E E E.. Figure: Context-Minimal Graph For Pseudotree
18 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition The context-minimal search graph, denoted C T, is the search graph obtained after merging all identical subproblems. C T is exponential in the depth of T
19 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition The context-minimal search graph, denoted C T, is the search graph obtained after merging all identical subproblems. C T is exponential in the depth of T
20 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition A solution tree T of C T is a subtree satisfying the following conditions: 1 It contains the root of C T 2 If an internal AND node n is in T, then all the children of n are in T 3 if an internal OR node n is in T, then exactly one child of n is in T 4 Every leaf in T is a terminal node
21 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition A solution tree T of C T is a subtree satisfying the following conditions: 1 It contains the root of C T 2 If an internal AND node n is in T, then all the children of n are in T 3 if an internal OR node n is in T, then exactly one child of n is in T 4 Every leaf in T is a terminal node
22 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition A solution tree T of C T is a subtree satisfying the following conditions: 1 It contains the root of C T 2 If an internal AND node n is in T, then all the children of n are in T 3 if an internal OR node n is in T, then exactly one child of n is in T 4 Every leaf in T is a terminal node
23 Assume a graphical model M = X, D, F, with primal graph G, pseudotree T, and search tree S T Definition A solution tree T of C T is a subtree satisfying the following conditions: 1 It contains the root of C T 2 If an internal AND node n is in T, then all the children of n are in T 3 if an internal OR node n is in T, then exactly one child of n is in T 4 Every leaf in T is a terminal node
24 Best First Search State-of-the-art A for search space. Too complicated to fit on a slide Highlights Input: Graphical Model M = X, D, F, Σ Initial weight w 0 Pseudotree T rooted at X 1 heuristic h i (precalculated) Output: Optimal solution to M
25 Best First Search State-of-the-art A for search space. Too complicated to fit on a slide Highlights Input: Graphical Model M = X, D, F, Σ Initial weight w 0 Pseudotree T rooted at X 1 heuristic h i (precalculated) Output: Optimal solution to M
26 Best First Search State-of-the-art A for search space. Too complicated to fit on a slide Highlights Input: Graphical Model M = X, D, F, Σ Initial weight w 0 Pseudotree T rooted at X 1 heuristic h i (precalculated) Output: Optimal solution to M
27 Best First Search Let M = {X, D, F, Σ} where X = {A, B,C, D} D = {0, 1} s s X F is given by the following tables: A B f (A, B) B f (B) B C f (B,C) A B f (A, B)
28 Best First Search Scopes F = {f (A, B), f (B,C), f (A, D), f (B)} B A C D Figure: Primal Graph
29 Best First Search Scopes F = {f (A, B), f (B,C), f (A, D), f (B)} B A C D Figure: Primal Graph
30 Best First Search B h = 7 h = h = 3 A h = 1 h = 1 D h = h = 4 C h = h = 0 h = 0 D h = A h = 2 C h = h = 1 h = 1 h = 0 h = h = 0 h = h = 0 h = 0
31 Best First Search Algorithm Sketch Down Pass: Expand nodes and mark terminal nodes solved Up Pass: Update v(n) for each node according to the following rules: OR Nodes: v(n) = min w(n, k) + v(k) k succ(n) AND Nodes: v(n) = v(k) k succ(n)
32 Best First Search B v = 7 v = v = 3 A v = 1 v = 1 v = 4 C v = v = 0 v = 0 D v = v = 0 v = 0
COMPSCI 276 Fall 2007
Exact Inference lgorithms for Probabilistic Reasoning; OMPSI 276 Fall 2007 1 elief Updating Smoking lung ancer ronchitis X-ray Dyspnoea P lung cancer=yes smoking=no, dyspnoea=yes =? 2 Probabilistic Inference
Exact Inference Algorithms Bucket-elimination
Exact Inference Algorithms Bucket-elimination COMPSCI 276, Spring 2011 Class 5: Rina Dechter (Reading: class notes chapter 4, Darwiche chapter 6) 1 Belief Updating Smoking lung Cancer Bronchitis X-ray
Exact Inference Algorithms Bucket-elimination
Exact Inference Algorithms Bucket-elimination COMPSCI 276, Spring 2013 Class 5: Rina Dechter (Reading: class notes chapter 4, Darwiche chapter 6) 1 Belief Updating Smoking lung Cancer Bronchitis X-ray
Variational algorithms for marginal MAP
Variational algorithms for marginal MAP Alexander Ihler UC Irvine CIOG Workshop November 2011 Variational algorithms for marginal MAP Alexander Ihler UC Irvine CIOG Workshop November 2011 Work with Qiang
A new Evaluation of Forward Checking and its Consequences on Efficiency of Tools for Decomposition of CSPs
2008 20th IEEE International Conference on Tools with Artificial Intelligence A new Evaluation of Forward Checking and its Consequences on Efficiency of Tools for Decomposition of CSPs Philippe Jégou Samba
Bayesian Networks: Representation, Variable Elimination
Bayesian Networks: Representation, Variable Elimination CS 6375: Machine Learning Class Notes Instructor: Vibhav Gogate The University of Texas at Dallas We can view a Bayesian network as a compact representation
CS 395T Computational Learning Theory. Scribe: Mike Halcrow. x 4. x 2. x 6
CS 395T Computational Learning Theory Lecture 3: September 0, 2007 Lecturer: Adam Klivans Scribe: Mike Halcrow 3. Decision List Recap In the last class, we determined that, when learning a t-decision list,
Reasoning with Deterministic and Probabilistic graphical models Class 2: Inference in Constraint Networks Rina Dechter
Reasoning with Deterministic and Probabilistic graphical models Class 2: Inference in Constraint Networks Rina Dechter Road Map n Graphical models n Constraint networks Model n Inference n Search n Probabilistic
Section Notes 8. Integer Programming II. Applied Math 121. Week of April 5, expand your knowledge of big M s and logical constraints.
Section Notes 8 Integer Programming II Applied Math 121 Week of April 5, 2010 Goals for the week understand IP relaxations be able to determine the relative strength of formulations understand the branch
Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passing
Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passing Huayan Wang huayanw@cs.stanford.edu Computer Science Department, Stanford University, Palo Alto, CA 90 USA Daphne Koller koller@cs.stanford.edu
4.8 Huffman Codes. These lecture slides are supplied by Mathijs de Weerd
4.8 Huffman Codes These lecture slides are supplied by Mathijs de Weerd Data Compression Q. Given a text that uses 32 symbols (26 different letters, space, and some punctuation characters), how can we
Tree Decompositions and Tree-Width
Tree Decompositions and Tree-Width CS 511 Iowa State University December 6, 2010 CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, 2010 1 / 15 Tree Decompositions Definition
where X is the feasible region, i.e., the set of the feasible solutions.
3.5 Branch and Bound Consider a generic Discrete Optimization problem (P) z = max{c(x) : x X }, where X is the feasible region, i.e., the set of the feasible solutions. Branch and Bound is a general semi-enumerative
Compiling Knowledge into Decomposable Negation Normal Form
Compiling Knowledge into Decomposable Negation Normal Form Adnan Darwiche Cognitive Systems Laboratory Department of Computer Science University of California Los Angeles, CA 90024 darwiche@cs. ucla. edu
Computing Possibly Optimal Solutions for Multi-Objective Constraint Optimisation with Tradeoffs
Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 215) Computing Possibly Optimal Solutions for Multi-Objective Constraint Optimisation with Tradeoffs Nic
Each internal node v with d(v) children stores d 1 keys. k i 1 < key in i-th sub-tree k i, where we use k 0 = and k d =.
7.5 (a, b)-trees 7.5 (a, b)-trees Definition For b a an (a, b)-tree is a search tree with the following properties. all leaves have the same distance to the root. every internal non-root vertex v has at
Learning Decision Trees
Learning Decision Trees Machine Learning Fall 2018 Some slides from Tom Mitchell, Dan Roth and others 1 Key issues in machine learning Modeling How to formulate your problem as a machine learning problem?
Statistical Methods for NLP
Statistical Methods for NLP Stochastic Grammars Joakim Nivre Uppsala University Department of Linguistics and Philology joakim.nivre@lingfil.uu.se Statistical Methods for NLP 1(22) Structured Classification
BMI/CS 776 Lecture 4. Colin Dewey
BMI/CS 776 Lecture 4 Colin Dewey 2007.02.01 Outline Common nucleotide substitution models Directed graphical models Ancestral sequence inference Poisson process continuous Markov process X t0 X t1 X t2
Disconnecting Networks via Node Deletions
1 / 27 Disconnecting Networks via Node Deletions Exact Interdiction Models and Algorithms Siqian Shen 1 J. Cole Smith 2 R. Goli 2 1 IOE, University of Michigan 2 ISE, University of Florida 2012 INFORMS
Learning in State-Space Reinforcement Learning CIS 32
Learning in State-Space Reinforcement Learning CIS 32 Functionalia Syllabus Updated: MIDTERM and REVIEW moved up one day. MIDTERM: Everything through Evolutionary Agents. HW 2 Out - DUE Sunday before the
Nearest Neighbor Search with Keywords
Nearest Neighbor Search with Keywords Yufei Tao KAIST June 3, 2013 In recent years, many search engines have started to support queries that combine keyword search with geography-related predicates (e.g.,
Math Models of OR: Branch-and-Bound
Math Models of OR: Branch-and-Bound John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA November 2018 Mitchell Branch-and-Bound 1 / 15 Branch-and-Bound Outline 1 Branch-and-Bound
Introduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
Evolutionary Tree Analysis. Overview
CSI/BINF 5330 Evolutionary Tree Analysis Young-Rae Cho Associate Professor Department of Computer Science Baylor University Overview Backgrounds Distance-Based Evolutionary Tree Reconstruction Character-Based
Design and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 5: Divide and Conquer (Part 2) Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ A Lower Bound on Convex Hull Lecture 4 Task: sort the
HW #4. (mostly by) Salim Sarımurat. 1) Insert 6 2) Insert 8 3) Insert 30. 4) Insert S a.
HW #4 (mostly by) Salim Sarımurat 04.12.2009 S. 1. 1. a. 1) Insert 6 2) Insert 8 3) Insert 30 4) Insert 40 2 5) Insert 50 6) Insert 61 7) Insert 70 1. b. 1) Insert 12 2) Insert 29 3) Insert 30 4) Insert
Digital search trees JASS
Digital search trees Analysis of different digital trees with Rice s integrals. JASS Nicolai v. Hoyningen-Huene 28.3.2004 28.3.2004 JASS 04 - Digital search trees 1 content Tree Digital search tree: Definition
arxiv: v1 [stat.ml] 26 Feb 2013
![arxiv: v1 [stat.ml] 26 Feb 2013](/thumbs/90/103662159.jpg "arxiv: v1 [stat.ml] 26 Feb 2013")
Variational Algorithms for Marginal MAP Qiang Liu Donald Bren School of Information and Computer Sciences University of California, Irvine Irvine, CA, 92697-3425, USA qliu1@uci.edu arxiv:1302.6584v1 [stat.ml]
Binary Decision Diagrams. Graphs. Boolean Functions
Binary Decision Diagrams Graphs Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant
AND/OR Importance Sampling
D/ Importance Sampling Vibhav Gogate and Rina Dechter Department of Information and Computer Science, University of California, Irvine, C 9697, {vgogate,dechter}@ics.uci.edu bstract The paper introduces
On the Relationship between Sum-Product Networks and Bayesian Networks
On the Relationship between Sum-Product Networks and Bayesian Networks International Conference on Machine Learning, 2015 Han Zhao Mazen Melibari Pascal Poupart University of Waterloo, Waterloo, ON, Canada
Query Optimization: Exercise
Query Optimization: Exercise Session 6 Bernhard Radke November 27, 2017 Maximum Value Precedence (MVP) [1] Weighted Directed Join Graph (WDJG) Weighted Directed Join Graph (WDJG) 1000 0.05 R 1 0.005 R
Optimization Bounds from Binary Decision Diagrams
Optimization Bounds from Binary Decision Diagrams J. N. Hooker Joint work with David Bergman, André Ciré, Willem van Hoeve Carnegie Mellon University ICS 203 Binary Decision Diagrams BDDs historically
Search and Lookahead. Bernhard Nebel, Julien Hué, and Stefan Wölfl. June 4/6, 2012
Search and Lookahead Bernhard Nebel, Julien Hué, and Stefan Wölfl Albert-Ludwigs-Universität Freiburg June 4/6, 2012 Search and Lookahead Enforcing consistency is one way of solving constraint networks:
Sum-Product Networks: A New Deep Architecture
Sum-Product Networks: A New Deep Architecture Pedro Domingos Dept. Computer Science & Eng. University of Washington Joint work with Hoifung Poon 1 Graphical Models: Challenges Bayesian Network Markov Network
Assignment 5: Solutions
Comp 21: Algorithms and Data Structures Assignment : Solutions 1. Heaps. (a) First we remove the minimum key 1 (which we know is located at the root of the heap). We then replace it by the key in the position
SOLVING INTEGER LINEAR PROGRAMS. 1. Solving the LP relaxation. 2. How to deal with fractional solutions?
SOLVING INTEGER LINEAR PROGRAMS 1. Solving the LP relaxation. 2. How to deal with fractional solutions? Integer Linear Program: Example max x 1 2x 2 0.5x 3 0.2x 4 x 5 +0.6x 6 s.t. x 1 +2x 2 1 x 1 + x 2
Lecture 16. Error-free variable length schemes (contd.): Shannon-Fano-Elias code, Huffman code
Lecture 16 Agenda for the lecture Error-free variable length schemes (contd.): Shannon-Fano-Elias code, Huffman code Variable-length source codes with error 16.1 Error-free coding schemes 16.1.1 The Shannon-Fano-Elias
Even More on Dynamic Programming
Algorithms & Models of Computation CS/ECE 374, Fall 2017 Even More on Dynamic Programming Lecture 15 Thursday, October 19, 2017 Sariel Har-Peled (UIUC) CS374 1 Fall 2017 1 / 26 Part I Longest Common Subsequence
AVL Trees. Manolis Koubarakis. Data Structures and Programming Techniques
AVL Trees Manolis Koubarakis 1 AVL Trees We will now introduce AVL trees that have the property that they are kept almost balanced but not completely balanced. In this way we have O(log n) search time
Machine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 18 The Junction Tree Algorithm Collect & Distribute Algorithmic Complexity ArgMax Junction Tree Algorithm Review: Junction Tree Algorithm end message
Chapter 4: Computation tree logic
INFOF412 Formal verification of computer systems Chapter 4: Computation tree logic Mickael Randour Formal Methods and Verification group Computer Science Department, ULB March 2017 1 CTL: a specification
On the Average Path Length of Complete m-ary Trees
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 17 2014, Article 14.6.3 On the Average Path Length of Complete m-ary Trees M. A. Nyblom School of Mathematics and Geospatial Science RMIT University
Binary Decision Diagrams
Binary Decision Diagrams Literature Some pointers: H.R. Andersen, An Introduction to Binary Decision Diagrams, Lecture notes, Department of Information Technology, IT University of Copenhagen Tools: URL:
IE418 Integer Programming
IE418: Integer Programming Department of Industrial and Systems Engineering Lehigh University 2nd February 2005 Boring Stuff Extra Linux Class: 8AM 11AM, Wednesday February 9. Room??? Accounts and Passwords
Properties of Context-Free Languages
Properties of Context-Free Languages Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
maxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2
ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving
The Necessity of Bounded Treewidth for Efficient Inference in Bayesian Networks
The Necessity of Bounded Treewidth for Efficient Inference in Bayesian Networks Johan H.P. Kwisthout and Hans L. Bodlaender and L.C. van der Gaag 1 Abstract. Algorithms for probabilistic inference in Bayesian
CSE 202 Homework 4 Matthias Springer, A
CSE 202 Homework 4 Matthias Springer, A99500782 1 Problem 2 Basic Idea PERFECT ASSEMBLY N P: a permutation P of s i S is a certificate that can be checked in polynomial time by ensuring that P = S, and
Binary Decision Diagrams
Binary Decision Diagrams Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant in 1986.
Dublin City Schools Mathematics Graded Course of Study Algebra I Philosophy
Philosophy The Dublin City Schools Mathematics Program is designed to set clear and consistent expectations in order to help support children with the development of mathematical understanding. We believe
Fibonacci Heaps These lecture slides are adapted from CLRS, Chapter 20.
Fibonacci Heaps These lecture slides are adapted from CLRS, Chapter 20. Princeton University COS 4 Theory of Algorithms Spring 2002 Kevin Wayne Priority Queues Operation mae-heap insert find- delete- union
Bayesian Networks Factor Graphs the Case-Factor Algorithm and the Junction Tree Algorithm
Bayesian Networks Factor Graphs the Case-Factor Algorithm and the Junction Tree Algorithm 1 Bayesian Networks We will use capital letters for random variables and lower case letters for values of those
Fault-Tolerant Consensus
Fault-Tolerant Consensus CS556 - Panagiota Fatourou 1 Assumptions Consensus Denote by f the maximum number of processes that may fail. We call the system f-resilient Description of the Problem Each process
Propositional Calculus - Semantics (3/3) Moonzoo Kim CS Dept. KAIST
Propositional Calculus - Semantics (3/3) Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Overview 2.1 Boolean operators 2.2 Propositional formulas 2.3 Interpretations 2.4 Logical Equivalence and substitution
4 : Exact Inference: Variable Elimination
10-708: Probabilistic Graphical Models 10-708, Spring 2014 4 : Exact Inference: Variable Elimination Lecturer: Eric P. ing Scribes: Soumya Batra, Pradeep Dasigi, Manzil Zaheer 1 Probabilistic Inference
Integer Programming. Wolfram Wiesemann. December 6, 2007
Integer Programming Wolfram Wiesemann December 6, 2007 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration
Basing Decisions on Sentences in Decision Diagrams
Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence Basing Decisions on Sentences in Decision Diagrams Yexiang Xue Department of Computer Science Cornell University yexiang@cs.cornell.edu
Lesson 10: Comparing Functions and their features
Lesson 10: Comparing Functions and their features Standards: MAFS.912.F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
The Separation Problem for Binary Decision Diagrams
The Separation Problem for Binary Decision Diagrams J. N. Hooker Joint work with André Ciré Carnegie Mellon University ISAIM 2014 Separation Problem in Optimization Given a relaxation of an optimization
On the Isomorphism Problem for Decision Trees and Decision Lists
On the Isomorphism Problem for Decision Trees and Decision Lists V. Arvind 1, Johannes Köbler 2, Sebastian Kuhnert 2, Gaurav Rattan 1, and Yadu Vasudev 1 1 The Institute of Mathematical Sciences, Chennai,
CS60007 Algorithm Design and Analysis 2018 Assignment 1
CS60007 Algorithm Design and Analysis 2018 Assignment 1 Palash Dey and Swagato Sanyal Indian Institute of Technology, Kharagpur Please submit the solutions of the problems 6, 11, 12 and 13 (written in
Min-Max Message Passing and Local Consistency in Constraint Networks
Min-Max Message Passing and Local Consistency in Constraint Networks Hong Xu, T. K. Satish Kumar, and Sven Koenig University of Southern California, Los Angeles, CA 90089, USA hongx@usc.edu tkskwork@gmail.com
Binary Search Trees. Motivation
Binary Search Trees Motivation Searching for a particular record in an unordered list takes O(n), too slow for large lists (databases) If the list is ordered, can use an array implementation and use binary
APPROXIMATION COMPLEXITY OF MAP INFERENCE IN SUM-PRODUCT NETWORKS
APPROXIMATION COMPLEXITY OF MAP INFERENCE IN SUM-PRODUCT NETWORKS Diarmaid Conaty Queen s University Belfast, UK Denis D. Mauá Universidade de São Paulo, Brazil Cassio P. de Campos Queen s University Belfast,
COMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING +
Mathematical Programming 19 (1980) 213-219. North-Holland Publishing Company COMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING + Katta G. MURTY The University of Michigan, Ann Arbor, MI, U.S.A.
Design and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 4: Divide and Conquer (I) Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Divide and Conquer ( DQ ) First paradigm or framework DQ(S)
Graph structure in polynomial systems: chordal networks
Graph structure in polynomial systems: chordal networks Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
Introduction to Bin Packing Problems
Introduction to Bin Packing Problems Fabio Furini March 13, 2015 Outline Origins and applications Applications: Definition: Bin Packing Problem (BPP) Solution techniques for the BPP Heuristic Algorithms
Reduced Ordered Binary Decision Diagrams
Reduced Ordered Binary Decision Diagrams Lecture #12 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de December 13, 2016 c JPK
Inference and Representation
Inference and Representation David Sontag New York University Lecture 5, Sept. 30, 2014 David Sontag (NYU) Inference and Representation Lecture 5, Sept. 30, 2014 1 / 16 Today s lecture 1 Running-time of
Zebo Peng Embedded Systems Laboratory IDA, Linköping University
TDTS 01 Lecture 8 Optimization Heuristics for Synthesis Zebo Peng Embedded Systems Laboratory IDA, Linköping University Lecture 8 Optimization problems Heuristic techniques Simulated annealing Genetic
Integer Programming ISE 418. Lecture 8. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer
Observations Homework Checkpoint quizzes Chapter assessments (Possibly Projects) Blocks of Algebra
September The Building Blocks of Algebra Rates, Patterns and Problem Solving Variables and Expressions The Commutative and Associative Properties The Distributive Property Equivalent Expressions Seeing
Scout and NegaScout. Tsan-sheng Hsu.
Scout and NegaScout Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Abstract It looks like alpha-beta pruning is the best we can do for an exact generic searching procedure.
Solving the Maximum Agreement Subtree and Maximum Comp. Tree problems on bounded degree trees. Sylvain Guillemot, François Nicolas.
Solving the Maximum Agreement Subtree and Maximum Compatible Tree problems on bounded degree trees LIRMM, Montpellier France 4th July 2006 Introduction The Mast and Mct problems: given a set of evolutionary
Bayesian Networks. Exact Inference by Variable Elimination. Emma Rollon and Javier Larrosa Q
Bayesian Networks Exact Inference by Variable Elimination Emma Rollon and Javier Larrosa Q1-2015-2016 Emma Rollon and Javier Larrosa Bayesian Networks Q1-2015-2016 1 / 25 Recall the most usual queries
Data Structures and Algorithms " Search Trees!!
Data Structures and Algorithms " Search Trees!! Outline" Binary Search Trees! AVL Trees! (2,4) Trees! 2 Binary Search Trees! "! < 6 2 > 1 4 = 8 9 Ordered Dictionaries" Keys are assumed to come from a total
CS483 Design and Analysis of Algorithms
CS483 Design and Analysis of Algorithms Lectures 15-16 Dynamic Programming Instructor: Fei Li lifei@cs.gmu.edu with subject: CS483 Office hours: STII, Room 443, Friday 4:00pm - 6:00pm or by appointments
Computational Models - Lecture 5 1
Computational Models - Lecture 5 1 Handout Mode Iftach Haitner. Tel Aviv University. November 28, 2016 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice Herlihy, Brown University.
min3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.
ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe
Mini-Buckets: A General Scheme for Bounded Inference
Mini-Buckets: A General Scheme for Bounded Inference RINA DECHTER University of California, Irvine, Irvine, California AND IRINA RISH IBM T. J. Watson Research Center, Hawthorne, New York Abstract. This
Data Mining Classification: Basic Concepts and Techniques. Lecture Notes for Chapter 3. Introduction to Data Mining, 2nd Edition
Data Mining Classification: Basic Concepts and Techniques Lecture Notes for Chapter 3 by Tan, Steinbach, Karpatne, Kumar 1 Classification: Definition Given a collection of records (training set ) Each
Advanced Implementations of Tables: Balanced Search Trees and Hashing
Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the
arxiv: v1 [cs.ds] 9 Apr 2018
![arxiv: v1 [cs.ds] 9 Apr 2018](/thumbs/91/107706762.jpg "arxiv: v1 [cs.ds] 9 Apr 2018")
From Regular Expression Matching to Parsing Philip Bille Technical University of Denmark phbi@dtu.dk Inge Li Gørtz Technical University of Denmark inge@dtu.dk arxiv:1804.02906v1 [cs.ds] 9 Apr 2018 Abstract
Absolute Value Equations and Inequalities. Use the distance definition of absolute value.
Chapter 2 Section 7 2.7 Absolute Value Equations and Inequalities Objectives 1 2 3 4 5 6 Use the distance definition of absolute value. Solve equations of the form ax + b = k, for k > 0. Solve inequalities
Lecture 9: PGM Learning
13 Oct 2014 Intro. to Stats. Machine Learning COMP SCI 4401/7401 Table of Contents I Learning parameters in MRFs 1 Learning parameters in MRFs Inference and Learning Given parameters (of potentials) and
Analysis of Algorithms. Outline 1 Introduction Basic Definitions Ordered Trees. Fibonacci Heaps. Andres Mendez-Vazquez. October 29, Notes.
Analysis of Algorithms Fibonacci Heaps Andres Mendez-Vazquez October 29, 2015 1 / 119 Outline 1 Introduction Basic Definitions Ordered Trees 2 Binomial Trees Example 3 Fibonacci Heap Operations Fibonacci
Graph structure in polynomial systems: chordal networks
Graph structure in polynomial systems: chordal networks Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort
Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort Xi Chen Columbia University We continue with two more asymptotic notation: o( ) and ω( ). Let f (n) and g(n) are functions that map
Discrete Mathematics
Discrete Mathematics Trees H. Turgut Uyar Ayşegül Gençata Yayımlı Emre Harmancı 2001-2016 License You are free to: c 2001-2016 T. Uyar, A. Yayımlı, E. Harmancı Share copy and redistribute the material
Lessons from MIP Search. John Hooker Carnegie Mellon University November 2009
Lessons from MIP Search John Hooker Carnegie Mellon University November 2009 Outline MIP search The main ideas Duality and nogoods From MIP to AI (and back) Binary decision diagrams From MIP to constraint
Multiple Choice Tries and Distributed Hash Tables
Multiple Choice Tries and Distributed Hash Tables Luc Devroye and Gabor Lugosi and Gahyun Park and W. Szpankowski January 3, 2007 McGill University, Montreal, Canada U. Pompeu Fabra, Barcelona, Spain U.
Bayesian Inference. Definitions from Probability: Naive Bayes Classifiers: Advantages and Disadvantages of Naive Bayes Classifiers:
Bayesian Inference The purpose of this document is to review belief networks and naive Bayes classifiers. Definitions from Probability: Belief networks: Naive Bayes Classifiers: Advantages and Disadvantages
Realization Plans for Extensive Form Games without Perfect Recall
Realization Plans for Extensive Form Games without Perfect Recall Richard E. Stearns Department of Computer Science University at Albany - SUNY Albany, NY 12222 April 13, 2015 Abstract Given a game in
Computing the acceptability semantics. London SW7 2BZ, UK, Nicosia P.O. Box 537, Cyprus,
Computing the acceptability semantics Francesca Toni 1 and Antonios C. Kakas 2 1 Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK, ft@doc.ic.ac.uk 2 Department of Computer
Appendix of Computational Protein Design Using AND/OR Branch and Bound Search
Appendix of Computational Protein Design Using AND/OR Branch and Bound Search Yichao Zhou 1, Yuexin Wu 1, and Jianyang Zeng 1,2, 1 Institute for Interdisciplinary Information Sciences, Tsinghua University,
Variable Elimination (VE) Barak Sternberg
Variable Elimination (VE) Barak Sternberg Basic Ideas in VE Example 1: Let G be a Chain Bayesian Graph: X 1 X 2 X n 1 X n How would one compute P X n = k? Using the CPDs: P X 2 = x = x Val X1 P X 1 = x
Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers
Discrete Mathematics and Theoretical Computer Science DMTCS vol. subm., by the authors, 1 1 Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers Christopher Eagle 1 and