Tutorial 6-7. Dynamic Programming and Greedy
|
|
- Christopher Stafford
- 5 years ago
- Views:
Transcription
1 Tutorial 6-7 Dynamic Programming and Greedy
2 Dynamic Programming Why DP? Natural Recursion may be expensive. For example, the Fibonacci: F(n)=F(n-1)+F(n-2) Recursive implementation memoryless : time= 1 5 n DP implementation: memorial time=o(n), with extra space O(n) or O(1) Basic Idea of DP O (( + ) ) 2 Computing each subproblem only once Find a reverse topological order for the subproblem graph Scheduling the subproblems according to the reverse topological order Record the subproblem solutions for later use
3 Examples Least Cost of Matrix Multiplication Optimal Binary Search Tree Separating Sequence of Word Coin change problem Longest Common Subsequence
4 Longest Common Subsequence(LCS) (CLRS: P350) Given two sequences x[1..m] and y[1..n], find a longest common subsequence. For example, x: A B C B D A B y: B D C A B A LCS(x,y)= BCBA or BCAB or BDAB
5 Brute-force Solution Check every subsequence of x[1..m] to see if it is also a subsequence of y[1..n]. Analysis Checking=O(n) time per subsequence 2 m subsequence of x Worst-case running time=o(n2 m ) =exponential time!
6 DP Solution Simplification Look at the length of a longest-common subsequence Extend the algorithm to find the LCS itself Strategy: Consider prefixes of x and y Define c[i,j]= LCS(x[1..i],y[1..j]) Then c[m,n]= LCS(x,y).
7 (1) DP Equation 0, if i=0 or j=0 ci [, j] = ci [ 1, j 1] + 1, if x[i]=y[j] max{ ci [ 1, j], ci [, j 1]} otherwise According to the equation, the problem can be solved by using recursive: LCS(x,y,i,j) If x[i]=y[j] then c[i,j] LCS(x,y,i-1,j-1)+1; else c[i,j] max{lcs(x,y,i-1,j),lcs(x,y,i,j-1)}; However, in the worst case, the time complexity of recursive implementation is also exponential!
8 (2) Memorization implementation After computing a solution to a subproblem, store it in a table. Subsequent calls check the table to avoid redoing work. LCS(x,y,i,j) if c[i,j]=nil If x[i]=y[j] then c[i,j] LCS(x,y,i-1,j-1)+1; else c[i,j] max{lcs(x,y,i-1,j),lcs(x,y,i,j-1)}; Time=Θ(mn), Space=Θ(mn)
9 (3)no-recursive implementation Idea: compute the table bottom-up LCS-LENGTH(X, Y) 1 m length[x] 2 n length[y] 3 for i 1 to m 4 do c[i, 0] 0 5 for j 0 to n 6 do c[0, j] 0 7 for i 1 to m 8 do for j 1 to n 9 do if xi = yj 10 then c[i, j] c[i -1, j -1] b[i, j] " " 12 else if c[i -1, j] c[i, j -1] 13 then c[i, j] c[i -1, j] 14 b[i, j] " " 15 else c[i, j] c[i, j -1] 16 b[i, j] 17 return c and b Time=Θ(mn), Space=Θ(mn)
10
11 Greedy Strategy Greedy-choice property A globally optimal solution can be arrived at by making a locally optimal (greedy) choice. In other words, when we are considering which choice to make, we make the choice that looks best in the current problem, without considering results from subproblems. Optimal substructure A problem exhibits optimal substructure if an optimal solution to the problem contains within it optimal solutions to subproblems. This property is a key ingredient of assessing the applicability of dynamic programming as well as greedy algorithms.
12 Steps: Cast the optimization problem as one in which we make a choice and are left with one subproblem to solve. Prove that there is always an optimal solution to the original problem that makes the greedy choice, so that the greedy choice is always safe. Demonstrate that, having made the greedy choice, what remains is a subproblem with the property that if we combine an optimal solution to the subproblem with the greedy choice we have made, we arrive at an optimal solution to the original problem.
13 Examples: MST Problem Prim s Algortithm Kruskal s Algorithm Single-Source Shortest Path Problem Dijstra s Algorithm Huffman Codes
14 Interval Scheduling Problem Also known as activity selection problem (CLRS P371) Problem description A resource, i.e. a lecture room, a computer A set of requests {1,2,,n}, the ith request corresponds to an interval of time starting at s(i), finishing at f(i) Two requests i and j are compatible if the request intervals do not overlap The goal is to accept as large a compatible subset as possible. Compatible sets of maximum size will be called optimal.
15 Designing a Greedy Algorithm (1) We can make greedy decision based on different rules, but they do not always give good solutions. (1) earliest starting time: counter example (a) (2) smallest interval : counter example (b) (3) fewest conflicts : counter example (c) (a) (b) (c)
16 Designing a Greedy Algorithm (2) A greedy rule does lead to the optimal solution: The smallest finish time Idea: we should accept the request that finished first, that is, the request i for which f(i) is as small as possible We insure that our resource becomes free as soon as possible while still satisfying one request, in this way, we can maximize the time left to satisfy other requests.
17 Proof the Optimality of Greedy Strategy Let A be the solution from the greedy strategy Let O be the optimal solution In order to prove A is optimal, we should simply show that A = O
18 Let A={i 1,i 2,,i k }, O={j 1,j 2,,j m }, our goal is to prove k=m Assume requests in A and O are ordered in the left-to-right order of the corresponding intervals Claim 1: For all indices r k, we have f(i r ) f(j r ). Proof by induction Base: for r=1, the statement is true; Hypothesis: assume the statement is true for r-1 Induction: in order to prove f(i r ) f(j r ), we show that f(i r ) f(j r ) is impossible: if so, like the following figure, j r will be finished before i r, according to the greedy rule, i r should be selected in A. i r-1 i r j r-1 j r Cliam 2: The greedy algorithm returns an optimal solution Proof by contradiction If A is not optimal, there must be m>k. According to claim 1, f(i k ) f(j k ). Since m>k, there is a request j k+1 in O, it starts after j k ends, hence after i k ends. So j k+1 is compatible to all elements in A and it can be added into A. -a contradiction
19 Analyzing the Greedy Algorithm The greedy algorithm runs in O(nlgn), which equals to the cost of sorting the n requests in order of finishing time.
20 Huffman Codes (CLRS P385) Character-coding problem Prefix Codes: codes in which no codeword is also a prefix of some other codeword
21 Constructing a Huffman Code Huffman invented a greedy algorithm that constructs an optimal prefix code called a Huffman code. HUFFMAN(C) 1 n C // C is a set of characters, each character c has frequency f(c) 2 Q C // Q ia a min-priority queue 3 for i=1 to n -1 4 do allocate a new node z 5 left[z] x EXTRACT-MIN (Q) 6 right[z] y EXTRACT-MIN (Q) 7 f [z] f [x] + f [y] 8 INSERT(Q, z) 9 return EXTRACT-MIN(Q) Return the root of the tree.
22
23 Analysis Assumes Q is implemented as a binary min-heap. For a set C of n characters, the initialization of Q in line 2 can be performed in O (n) time. The for loop in lines 3-8 is executed exactly n - 1 times, and since each heap operation requires time O (lg n), the loop contributes O (n lg n) to the running time. Thus, the total running time of HUFFMAN on a set of n characters is O (n lg n). Correctness of Huffman s algorithm (Please refer to CLRS P )
CMPS 2200 Fall Dynamic Programming. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk
CMPS 00 Fall 04 Dynamic Programming Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 9/30/4 CMPS 00 Intro. to Algorithms Dynamic programming Algorithm design technique
More informationCS473-Algorithms I. Lecture 11. Greedy Algorithms. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorithms I Lecture 11 Greedy Algorithms 1 Activity Selection Problem Input: a set S {1, 2,, n} of n activities s i =Start time of activity i, f i = Finish time of activity i Activity i takes place
More informationDesign and Analysis of Algorithms
CSE 101, Winter 018 D/Q Greed SP s DP LP, Flow B&B, Backtrack Metaheuristics P, NP Design and Analysis of Algorithms Lecture 8: Greed Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Optimization
More information16.Greedy algorithms
16.Greedy algorithms 16.1 An activity-selection problem Suppose we have a set S = {a 1, a 2,..., a n } of n proposed activities that with to use a resource. Each activity a i has a start time s i and a
More informationChapter 16: Greedy Algorithm
Chapter 16: Greedy Algorithm 1 About this lecture Introduce Greedy Algorithm Look at some problems solvable by Greedy Algorithm 2 Coin Changing Suppose that in a certain country, the coin dominations consist
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J LECTURE 12 Dynamic programming Longest common subsequence Optimal substructure Overlapping subproblems Prof. Charles E. Leiserson Dynamic programming Design technique,
More informationDynamic Programming Assembly-Line Scheduling. Greedy Algorithms
Chapter 13 Greedy Algorithms Activity Selection Problem 0-1 Knapsack Problem Huffman Code Construction Dynamic Programming Assembly-Line Scheduling C-C Tsai P.1 Greedy Algorithms A greedy algorithm always
More informationG205 Fundamentals of Computer Engineering. CLASS 21, Mon. Nov Stefano Basagni Fall 2004 M-W, 1:30pm-3:10pm
G205 Fundamentals of Computer Engineering CLASS 21, Mon. Nov. 22 2004 Stefano Basagni Fall 2004 M-W, 1:30pm-3:10pm Greedy Algorithms, 1 Algorithms for Optimization Problems Sequence of steps Choices at
More informationAlgorithms Dr. Haim Levkowitz
91.503 Algorithms Dr. Haim Levkowitz Fall 2007 Lecture 4 Tuesday, 25 Sep 2007 Design Patterns for Optimization Problems Greedy Algorithms 1 Greedy Algorithms 2 What is Greedy Algorithm? Similar to dynamic
More informationCMPS 102 Solutions to Homework 7
CMPS 102 Solutions to Homework 7 Kuzmin, Cormen, Brown, lbrown@soe.ucsc.edu November 17, 2005 Problem 1. 15.4-1 p.355 LCS Determine an LCS of x = (1, 0, 0, 1, 0, 1, 0, 1) and y = (0, 1, 0, 1, 1, 0, 1,
More informationLecture: Analysis of Algorithms (CS )
Lecture: Analysis of Algorithms (CS483-001) Amarda Shehu Spring 2017 1 The Fractional Knapsack Problem Huffman Coding 2 Sample Problems to Illustrate The Fractional Knapsack Problem Variable-length (Huffman)
More informationMain approach: always make the choice that looks best at the moment. - Doesn t always result in globally optimal solution, but for many problems does
Greedy algorithms Main approach: always make the choice that looks best at the moment. - More efficient than dynamic programming - Doesn t always result in globally optimal solution, but for many problems
More informationCOMP251: Greedy algorithms
COMP251: Greedy algorithms Jérôme Waldispühl School of Computer Science McGill University Based on (Cormen et al., 2002) Based on slides from D. Plaisted (UNC) & (goodrich & Tamassia, 2009) Disjoint sets
More informationCS60020: Foundations of Algorithm Design and Machine Learning. Sourangshu Bhattacharya
CS60020: Foundations of Algorithm Design and Machine Learning Sourangshu Bhattacharya Dynamic programming Design technique, like divide-and-conquer. Example: Longest Common Subsequence (LCS) Given two
More informationECE608 - Chapter 16 answers
¼ À ÈÌ Ê ½ ÈÊÇ Ä ÅË ½µ ½ º½¹ ¾µ ½ º½¹ µ ½ º¾¹½ µ ½ º¾¹¾ µ ½ º¾¹ µ ½ º ¹ µ ½ º ¹ µ ½ ¹½ ½ ECE68 - Chapter 6 answers () CLR 6.-4 Let S be the set of n activities. The obvious solution of using Greedy-Activity-
More informationIN101: Algorithmic techniques Vladimir-Alexandru Paun ENSTA ParisTech
IN101: Algorithmic techniques Vladimir-Alexandru Paun ENSTA ParisTech License CC BY-NC-SA 2.0 http://creativecommons.org/licenses/by-nc-sa/2.0/fr/ Outline Previously on IN101 Python s anatomy Functions,
More informationDynamic Programming II
June 9, 214 DP: Longest common subsequence biologists often need to find out how similar are 2 DNA sequences DNA sequences are strings of bases: A, C, T and G how to define similarity? DP: Longest common
More informationIntroduction to Algorithms
Introduction to Algorithms, Lecture 1 /1/200 Introduction to Algorithms.04J/1.401J LECTURE 11 Graphs, MST, Greedy, Prim Graph representation Minimum spanning trees Greedy algorithms hallmarks. Greedy choice
More informationGreedy Algorithms. CLRS Chapters Introduction to greedy algorithms. Design of data-compression (Huffman) codes
Greedy Algorithms CLRS Chapters 16.1 16.3 Introduction to greedy algorithms Activity-selection problem Design of data-compression (Huffman) codes (Minimum spanning tree problem) (Shortest-path problem)
More informationGreedy Algorithms CSE 6331
Greedy Algorithms CSE 6331 Reading: Sections 16.1, 16.2, 16.3, Chapter 23. 1 Introduction Optimization Problem: Construct a sequence or a set of elements {x 1,..., x k } that satisfies given constraints
More informationEfficient Sequential Algorithms, Comp309. Motivation. Longest Common Subsequence. Part 3. String Algorithms
Efficient Sequential Algorithms, Comp39 Part 3. String Algorithms University of Liverpool References: T. H. Cormen, C. E. Leiserson, R. L. Rivest Introduction to Algorithms, Second Edition. MIT Press (21).
More informationMain approach: always make the choice that looks best at the moment.
Greedy algorithms Main approach: always make the choice that looks best at the moment. - More efficient than dynamic programming - Always make the choice that looks best at the moment (just one choice;
More informationCS Algorithms. Dynamic programming and memoization. (Based on slides by Luebke, Lim, Wenbin)
CS 7 - lgorithms Dynamic programming and memoization (ased on slides by Luebke, Lim, Wenbin) When to Use Dynamic Programming Usually used to solve Optimization problems Usually the problem can be formulated
More information4.1 Interval Scheduling
41 Interval Scheduling Interval Scheduling Interval scheduling Job j starts at s j and finishes at f j Two jobs compatible if they don't overlap Goal: find maximum subset of mutually compatible jobs a
More informationElements of Dynamic Programming. COSC 3101A - Design and Analysis of Algorithms 8. Discovering Optimal Substructure. Optimal Substructure - Examples
Elements of Dynamic Programming COSC 3A - Design and Analysis of Algorithms 8 Elements of DP Memoization Longest Common Subsequence Greedy Algorithms Many of these slides are taken from Monica Nicolescu,
More informationESO207A: Data Structures and Algorithms End-semester exam
ESO207A: Data Structures and Algorithms End-semester exam Max marks: 120 Time: 180 mins. 17-July-2017 1. Answer all 7 questions. Questions 1 to 3 are from module 3 and questions 4 to 7 are from module
More informationWe ve done. Now. Next
We ve done Fast Fourier Transform Polynomial Multiplication Now Introduction to the greedy method Activity selection problem How to prove that a greedy algorithm works Huffman coding Matroid theory Next
More informationTU/e Algorithms (2IL15) Lecture 2. Algorithms (2IL15) Lecture 2 THE GREEDY METHOD
Algorithms (2IL15) Lecture 2 THE GREEDY METHOD x y v w 1 Optimization problems for each instance there are (possibly) multiple valid solutions goal is to find an optimal solution minimization problem:
More informationGreedy Algorithms CSE 780
Greedy Algorithms CSE 780 Reading: Sections 16.1, 16.2, 16.3, Chapter 23. 1 Introduction Optimization Problem: Construct a sequence or a set of elements {x 1,..., x k } that satisfies given constraints
More informationCS 758/858: Algorithms
CS 758/858: Algorithms http://www.cs.unh.edu/~ruml/cs758 Wheeler Ruml (UNH) Class 12, CS 758 1 / 22 Scheduling Rules Algorithm Proof Opt. Substructure Break Algorithms Wheeler Ruml (UNH) Class 12, CS 758
More informationCS473-Algorithms I. Lecture 10. Dynamic Programming. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorithms I Lecture 1 Dynamic Programming 1 Introduction An algorithm design paradigm like divide-and-conquer Programming : A tabular method (not writing computer code) Divide-and-Conquer (DAC):
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J LECTURE 13 Graph algorithms Graph representation Minimum spanning trees Greedy algorithms Optimal substructure Greedy choice Prim s greedy MST algorithm Prof.
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J LECTURE 16 Greedy Algorithms (and Graphs) Graph representation Minimum spanning trees Optimal substructure Greedy choice Prim s greedy MST algorithm Prof. Charles
More informationGreedy Algorithms. Alexandra Stefan
Greedy Algorithms Alexandra Stefan 1 Greedy Method for Optimization Problems Greedy: take the action that is best now (out of the current options) it may cause you to miss the optimal solution You build
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms CSE 5311 Lecture 16 Greedy algorithms Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 Overview A greedy
More informationDynamic Programmming: Activity Selection
Dynamic Programmming: Activity Selection Select the maximum number of non-overlapping activities from a set of n activities A = {a 1,, a n } (sorted by finish times). Identify easier subproblems to solve.
More informationDynamic Programming (Part #2)
Dynamic Programming (Part #) Introduction to Algorithms MIT Press (Chapter 5) Matrix-Chain Multiplication Problem: given a sequence A, A,, A n, compute the product: A A A n Matrix compatibility: C = A
More informationCSE 431/531: Algorithm Analysis and Design (Spring 2018) Greedy Algorithms. Lecturer: Shi Li
CSE 431/531: Algorithm Analysis and Design (Spring 2018) Greedy Algorithms Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Main Goal of Algorithm Design Design fast
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms Instructor: SharmaThankachan Lecture 10: Greedy Algorithm Slides modified from Dr. Hon, with permission 1 About this lecture Introduce Greedy Algorithm Look at some problems
More informationCS141: Intermediate Data Structures and Algorithms Greedy Algorithms
CS141: Intermediate Data Structures and Algorithms Greedy Algorithms Amr Magdy Activity Selection Problem Given a set of activities S = {a 1, a 2,, a n } where each activity i has a start time s i and
More informationAnalysis of Algorithms Prof. Karen Daniels
UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010 Lecture 2 Tuesday, 2/2/10 Design Patterns for Optimization Problems Greedy Algorithms Algorithmic Paradigm Context
More informationCSC 373 Lecture # 3 Instructor: Milad Eftekhar
Huffman encoding: Assume a context is available (a document, a signal, etc.). These contexts are formed by some symbols (words in a document, discrete samples from a signal, etc). Each symbols s i is occurred
More informationCSE 421 Applications of DFS(?) Topological sort
CSE 421 Applications of DFS(?) Topological sort Yin Tat Lee 1 Precedence Constraints In a directed graph, an edge (i, j) means task i must occur before task j. Applications Course prerequisite: course
More informationAnalysis of Algorithms - Greedy algorithms -
Analysis of Algorithms - Greedy algorithms - Andreas Ermedahl MRTC (Mälardalens Real-Time Reseach Center) andreas.ermedahl@mdh.se Autumn 2003 Greedy Algorithms Another paradigm for designing algorithms
More informationHuffman Coding. Version of October 13, Version of October 13, 2014 Huffman Coding 1 / 27
Huffman Coding Version of October 13, 2014 Version of October 13, 2014 Huffman Coding 1 / 27 Outline Outline Coding and Decoding The optimal source coding problem Huffman coding: A greedy algorithm Correctness
More informationSubsequence Definition. CS 461, Lecture 8. Today s Outline. Example. Assume given sequence X = x 1, x 2,..., x m. Jared Saia University of New Mexico
Subsequence Definition CS 461, Lecture 8 Jared Saia University of New Mexico Assume given sequence X = x 1, x 2,..., x m Let Z = z 1, z 2,..., z l Then Z is a subsequence of X if there exists a strictly
More informationGreedy Algorithms. Textbook reading. Chapter 4 Chapter 5. CSci 3110 Greedy Algorithms 1/63
CSci 3110 Greedy Algorithms 1/63 Greedy Algorithms Textbook reading Chapter 4 Chapter 5 CSci 3110 Greedy Algorithms 2/63 Overview Design principle: Make progress towards a solution based on local criteria
More informationDesign and Analysis of Algorithms 演算法設計與分析. Lecture 7 April 6, 2016 洪國寶
Design and Analysis of Algorithms 演算法設計與分析 Lecture 7 April 6, 2016 洪國寶 1 Course information (5/5) Grading (Tentative) Homework 25% (You may collaborate when solving the homework, however when writing up
More informationDynamic Programming CS 445. Example: Floyd Warshll Algorithm: Computing all pairs shortest paths
CS 44 Dynamic Programming Some of the slides are courtesy of Charles Leiserson with small changes by Carola Wenk Example: Floyd Warshll lgorithm: Computing all pairs shortest paths Given G(V,E), with weight
More informationCSE 431/531: Analysis of Algorithms. Greedy Algorithms. Lecturer: Shi Li. Department of Computer Science and Engineering University at Buffalo
CSE 431/531: Analysis of Algorithms Greedy Algorithms Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Main Goal of Algorithm Design Design fast algorithms to solve
More informationCSC 373: Algorithm Design and Analysis Lecture 4
CSC 373: Algorithm Design and Analysis Lecture 4 Allan Borodin January 14, 2013 1 / 16 Lecture 4: Outline (for this lecture and next lecture) Some concluding comments on optimality of EST Greedy Interval
More informationWe will go over the basic scenarios, in which it is appropriate to apply this technique, and a few concrete applications.
Chapter 16 The Greedy Method We have looked at the divide n conquer and dynamic programming techniques, and will now discuss another general technique, the greedy method, on designing algorithms. We will
More informationPresentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Dynamic Programming
Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 25 Dynamic Programming Terrible Fibonacci Computation Fibonacci sequence: f = f(n) 2
More informationChapter 9. Greedy Technique. Copyright 2007 Pearson Addison-Wesley. All rights reserved.
Chapter 9 Greedy Technique Copyright 2007 Pearson Addison-Wesley. All rights reserved. Greedy Technique Constructs a solution to an optimization problem piece by piece through a sequence of choices that
More information/463 Algorithms - Fall 2013 Solution to Assignment 3
600.363/463 Algorithms - Fall 2013 Solution to Assignment 3 (120 points) I (30 points) (Hint: This problem is similar to parenthesization in matrix-chain multiplication, except the special treatment on
More informationSankalchand Patel College of Engineering - Visnagar Department of Computer Engineering and Information Technology. Assignment
Class: V - CE Sankalchand Patel College of Engineering - Visnagar Department of Computer Engineering and Information Technology Sub: Design and Analysis of Algorithms Analysis of Algorithm: Assignment
More informationAlgorithms for Data Science
Algorithms for Data Science CSOR W4246 Eleni Drinea Computer Science Department Columbia University Thursday, October 1, 2015 Outline 1 Recap 2 Shortest paths in graphs with non-negative edge weights (Dijkstra
More informationGreedy Algorithms CHAPTER 16
CHAPTER 16 Greedy Algorithms In dynamic programming, the optimal solution is described in a recursive manner, and then is computed ``bottom up''. Dynamic programming is a powerful technique, but it often
More informationData Structures and Algorithms Week 8
Data Structures and Algorithms Week 8 Dynamic programming Fibonacci numbers Optimization problems Matrix multiplication optimization Principles of dynamic programming Longest Common Subsequence Algorithm
More informationGreedy Algorithms and Matroids. Andreas Klappenecker
Greedy Algorithms and Matroids Andreas Klappenecker Giving Change Coin Changing Suppose we have n types of coins with values v[1] > v[2] > > v[n] > 0 Given an amount C, a positive integer, the following
More informationCSED233: Data Structures (2017F) Lecture12: Strings and Dynamic Programming
(2017F) Lecture12: Strings and Dynamic Programming Daijin Kim CSE, POSTECH dkim@postech.ac.kr Strings A string is a sequence of characters Examples of strings: Python program HTML document DNA sequence
More informationCS60020: Foundations of Algorithm Design and Machine Learning. Sourangshu Bhattacharya
CS60020: Foundations of Algorithm Design and Machine Learning Sourangshu Bhattacharya Graphs (review) Definition. A directed graph (digraph) G = (V, E) is an ordered pair consisting of a set V of vertices
More information15.Dynamic Programming
15.Dynamic Programming Dynamic Programming is an algorithm design technique for optimization problems: often minimizing or maximizing. Like divide and conquer, DP solves problems by combining solutions
More informationApplied Algorithm Design Lecture 3
Applied Algorithm Design Lecture 3 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 3 1 / 75 PART I : GREEDY ALGORITHMS Pietro Michiardi (Eurecom) Applied Algorithm
More informationGreedy algorithms is another useful way for solving optimization problems.
Greedy Algorithms Greedy algorithms is another useful way for solving optimization problems. Optimization Problems For the given input, we are seeking solutions that must satisfy certain conditions. These
More information16 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 informationCSci 231 Final Review
CSci 231 Final Review Here is a list of topics for the final. Generally you are responsible for anything discussed in class (except topics that appear italicized), and anything appearing on the homeworks.
More informationGreedy algorithms part 2, and Huffman code
Greedy algorithms part 2, and Huffman code Two main properties: 1. Greedy choice property: At each decision point, make the choice that is best at the moment. We typically show that if we make a greedy
More informationGreedy Algorithms CLRS Laura Toma, csci2200, Bowdoin College
Greedy Algorithms CLRS 16.1-16.2 Laura Toma, csci2200, Bowdoin College Overview. Sometimes we can solve optimization problems with a technique called greedy. A greedy algorithm picks the option that looks
More informationName: Lirong TAN 1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G.
1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G. A shortest s-t path is a path from vertex to vertex, whose sum of edge weights is minimized. (b) Give the pseudocode
More informationGreedy Algorithms Huffman Coding
Greedy Algorithms Huffman Coding Huffman Coding Problem Example: Release 29.1 of 15-Feb-2005 of TrEMBL Protein Database contains 1,614,107 sequence entries, comprising 505,947,503 amino acids. There are
More informationCSE331 Introduction to Algorithms Lecture 15 Minimum Spanning Trees
CSE1 Introduction to Algorithms Lecture 1 Minimum Spanning Trees Antoine Vigneron antoine@unist.ac.kr Ulsan National Institute of Science and Technology July 11, 201 Antoine Vigneron (UNIST) CSE1 Lecture
More informationCS Final - Review material
CS4800 Algorithms and Data Professor Fell Fall 2009 October 28, 2009 Old stuff CS 4800 - Final - Review material Big-O notation Though you won t be quizzed directly on Big-O notation, you should be able
More informationString Patterns and Algorithms on Strings
String Patterns and Algorithms on Strings Lecture delivered by: Venkatanatha Sarma Y Assistant Professor MSRSAS-Bangalore 11 Objectives To introduce the pattern matching problem and the important of algorithms
More informationPART IV. Given 2 sorted arrays, What is the time complexity of merging them together?
General Questions: PART IV Given 2 sorted arrays, What is the time complexity of merging them together? Array 1: Array 2: Sorted Array: Pointer to 1 st element of the 2 sorted arrays Pointer to the 1 st
More informationReview of course COMP-251B winter 2010
Review of course COMP-251B winter 2010 Lecture 1. Book Section 15.2 : Chained matrix product Matrix product is associative Computing all possible ways of parenthesizing Recursive solution Worst-case running-time
More informationComputer Science & Engineering 423/823 Design and Analysis of Algorithms
Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 05 Minimum-Weight Spanning Trees (Chapter 23) Stephen Scott (Adapted from Vinodchandran N. Variyam) sscott@cse.unl.edu Introduction
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 16 Dynamic Programming Least Common Subsequence Saving space Adam Smith Least Common Subsequence A.k.a. sequence alignment edit distance Longest Common Subsequence
More informationMinimum Spanning Trees
Minimum Spanning Trees Problem A town has a set of houses and a set of roads. A road connects 2 and only 2 houses. A road connecting houses u and v has a repair cost w(u, v). Goal: Repair enough (and no
More information5. For recursive algorithms, what do we need to define and solve to do the runtime analysis?
CS430 Lecture 1 Activities Opening Questions 1. Algorithm A takes 5 seconds to sort 1000 records, and Algorithm B takes 10 seconds to sort 1000 records. You have the code for both algorithms. When deciding
More informationCSC 1700 Analysis of Algorithms: Minimum Spanning Tree
CSC 1700 Analysis of Algorithms: Minimum Spanning Tree Professor Henry Carter Fall 2016 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic programming
More informationGreedy Algorithms. At each step in the algorithm, one of several choices can be made.
Greedy Algorithms At each step in the algorithm, one of several choices can be made. Greedy Strategy: make the choice that is the best at the moment. After making a choice, we are left with one subproblem
More informationGreedy Algorithms and Matroids. Andreas Klappenecker
Greedy Algorithms and Matroids Andreas Klappenecker Greedy Algorithms A greedy algorithm solves an optimization problem by working in several phases. In each phase, a decision is made that is locally optimal
More informationLecturers: Sanjam Garg and Prasad Raghavendra March 20, Midterm 2 Solutions
U.C. Berkeley CS70 : Algorithms Midterm 2 Solutions Lecturers: Sanjam Garg and Prasad aghavra March 20, 207 Midterm 2 Solutions. (0 points) True/False Clearly put your answers in the answer box in front
More informationGreedy Algorithms. Previous Examples: Huffman coding, Minimum Spanning Tree Algorithms
Greedy Algorithms A greedy algorithm is one where you take the step that seems the best at the time while executing the algorithm. Previous Examples: Huffman coding, Minimum Spanning Tree Algorithms Coin
More informationGreedy Algorithms. Subhash Suri. October 9, 2017
Greedy Algorithms Subhash Suri October 9, 2017 1 Introduction Greedy algorithms are a commonly used paradigm for combinatorial algorithms. Combinatorial problems intuitively are those for which feasible
More informationAlgorithm Design Techniques part I
Algorithm Design Techniques part I Divide-and-Conquer. Dynamic Programming DSA - lecture 8 - T.U.Cluj-Napoca - M. Joldos 1 Some Algorithm Design Techniques Top-Down Algorithms: Divide-and-Conquer Bottom-Up
More informationCSE 101, Winter Design and Analysis of Algorithms. Lecture 11: Dynamic Programming, Part 2
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 11: Dynamic Programming, Part 2 Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Goal: continue with DP (Knapsack, All-Pairs SPs, )
More informationDijkstra s algorithm for shortest paths when no edges have negative weight.
Lecture 14 Graph Algorithms II 14.1 Overview In this lecture we begin with one more algorithm for the shortest path problem, Dijkstra s algorithm. We then will see how the basic approach of this algorithm
More informationCMSC Introduction to Algorithms Spring 2012 Lecture 16
CMSC 351 - Introduction to Algorithms Spring 2012 Lecture 16 Instructor: MohammadTaghi Hajiaghayi Scribe: Rajesh Chitnis 1 Introduction In this lecture we will look at Greedy Algorithms and Dynamic Programming.
More informationWeek 11: Minimum Spanning trees
Week 11: Minimum Spanning trees Agenda: Minimum Spanning Trees Prim s Algorithm Reading: Textbook : 61-7 1 Week 11: Minimum Spanning trees Minimum spanning tree (MST) problem: Input: edge-weighted (simple,
More informationGreedy Algorithms and Matroids. Andreas Klappenecker
Greedy Algorithms and Matroids Andreas Klappenecker Greedy Algorithms A greedy algorithm solves an optimization problem by working in several phases. In each phase, a decision is made that is locally optimal
More informationSo far... Finished looking at lower bounds and linear sorts.
So far... Finished looking at lower bounds and linear sorts. Next: Memoization -- Optimization problems - Dynamic programming A scheduling problem Matrix multiplication optimization Longest Common Subsequence
More informationEST Solutions. Ans 1(a): KMP Algorithm for Preprocessing: KMP Algorithm for Searching:
EST Solutions Ans 1(a): KMP Algorithm for Preprocessing: KMP Algorithm for Searching: Ans 1(b): A priority queue is a data structure for maintaining a set S of elements, each with an associated value called
More informationCS521 \ Notes for the Final Exam
CS521 \ Notes for final exam 1 Ariel Stolerman Asymptotic Notations: CS521 \ Notes for the Final Exam Notation Definition Limit Big-O ( ) Small-o ( ) Big- ( ) Small- ( ) Big- ( ) Notes: ( ) ( ) ( ) ( )
More informationSolving NP-hard Problems on Special Instances
Solving NP-hard Problems on Special Instances Solve it in poly- time I can t You can assume the input is xxxxx No Problem, here is a poly-time algorithm 1 Solving NP-hard Problems on Special Instances
More information1 Non greedy algorithms (which we should have covered
1 Non greedy algorithms (which we should have covered earlier) 1.1 Floyd Warshall algorithm This algorithm solves the all-pairs shortest paths problem, which is a problem where we want to find the shortest
More informationDepartment of Computer Science and Engineering Analysis and Design of Algorithm (CS-4004) Subject Notes
Page no: Department of Computer Science and Engineering Analysis and Design of Algorithm (CS-00) Subject Notes Unit- Greedy Technique. Introduction: Greedy is the most straight forward design technique.
More informationlooking ahead to see the optimum
! Make choice based on immediate rewards rather than looking ahead to see the optimum! In many cases this is effective as the look ahead variation can require exponential time as the number of possible
More informationGreedy Algorithms. Informal Definition A greedy algorithm makes its next step based only on the current state and simple calculations on the input.
Greedy Algorithms Informal Definition A greedy algorithm makes its next step based only on the current state and simple calculations on the input. easy to design not always correct challenge is to identify
More information