Find: a multiset M { 1,..., n } so that. i M w i W and. i M v i is maximized. Find: a set S { 1,..., n } so that. i S w i W and. i S v i is maximized.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Find: a multiset M { 1,..., n } so that. i M w i W and. i M v i is maximized. Find: a set S { 1,..., n } so that. i S w i W and. i S v i is maximized."

Transcription

1 Knapsack gain Slides for IS 675 PV hapter 6: ynamic Programming, Part 2 Jim Royer EES October 28, 2009 The Knapsack Problem (KP) knapsack with weight capacity W. Items 1,..., n where item i has weight w i and value v i.... with repetition Find: a multiset M 1,..., n } so that... without repetition Find: a set S 1,..., n } so that i S w i W and i S v i is maximized. Image from: IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21 Knapsack with repetition Knapsack without repetition, 1 Knapsack with repetition knapsack with capacity W. Items 1,..., n Item i has weight w i & value v i. Find: a multiset M 1,..., n } K(w) = max. value gained from a knapsack with cap. w = max i:w i w K(w w i) + v i (Why?) array K[0..W] K[0] 0 for w 1 to W do K[w] 0 for i 1 to n do if w i w then K[w] max(k[w], K[w w i ] + v i ) This runs in Θ(n W) time. Since we usually measure the size of W as W = the number of bits in the binary rep. of W. Hence, Θ(n W) = Θ(n 2 W ). So this is only useful for small values of W. Knapsack without repetition knapsack with capacity W. Items 1,..., n Item i has weight w i & value v i. Find: a set M 1,..., n } Problem K[w w n ] is not useful since it does not tell you whether item n was used in an optimal solution. Therefore, we refine things to: K[w, j] the best value obtainable with = capacity w using items from 1,..., j K[w, j 1], if w j > w; = max(k[w, j 1], K[w w j, j 1] + v j ), otherwise. (Why?) IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

2 Knapsack without repetition, 2 Knapsack w/o repetition knapsack with capacity W. Items 1,..., n Item i has weight w i & value v i. Find: a set M 1,..., n } Our recursive relation is: K[w, j] the best value obtainable with = capacity w using items from 1,..., j K[w, j 1], if w j > w; = max(k[w, j 1], K[w w j, j 1] + v j ), otherwise. array K[0..W, 0..n] // This is also Θ(n W) time. for w 0 to W do K[w, 0] 0 // Hence only useful K[0, j] 0 // when W is small. for w 1 to W do if w i > w then K[w, j] K[w, j 1] else K[w, j] max(k[w, j 1], K[w w i, j 1] + v i ) IS 675 (EES) ynamic Programming, part 2 October 28, / 21 hain matrix multiplication, 1 Recall Multiplying an d 0 d 1 matrix by an d 1 d 2 matrix results in a d 0 d 2 matrix and takes (d 0 d 1 d 2 )-many scalar multiplies. The hain Matrix Multiplication Problem (MMP) d 0,..., d n N + and matrices 1,..., n where dim( i ) = d i 1 d i. Find: The cheapest way to order the multiplications. Example: Suppose that is an matrix. is an 20 1 matrix. is an 1 10 matrix. is an matrix. Then: Parenthesization ost computation ost ( ( )) ,000 (( ) ) ,200 ( ) ( ) ,000 ( ( )) ,200 (( ) ) ,000 IS 675 (EES) ynamic Programming, part 2 October 28, / 21 hain matrix multiplication, 2 hain matrix multiplication, 3 Parenthesization ost computation ost ( ( )) ,000 (( ) ) ,200 ( ) ( ) ,000 ( ( )) ,200 (( ) ) ,000 The hain Matrix Multiplication Problem (MMP) d 0,..., d n N + and matrices 1,..., n where dim( i ) = d i 1 d i. Find: The cheapest way to order the multiplications. The (i, k) subproblem, 1 i k n Find: (i, k) = the min. cost of the i k. (i, i) = 0. (i, k) = min j=i,...,k 1 ((i, j) + (j + 1, k) + d i 1 d j d k ), where i < k. (1, n) is the minimal cost of the MMP for 1,..., n. Question: Why does optimal substructure hold? IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

3 hain matrix multiplication, 4 The (i, k) subproblem, 1 i k n Find: (i, k) = the min. cost of the i k. (i, i) = 0. (i, k) = min j=i,...,k 1 (1, n) is the minimal cost of the MMP for 1,..., n. ( (i, j) + (j + 1, k) + di 1 d j d k ), where i < k. for i 1 to n do [i, i] 0 for s 1 to n 1 do // ompute [1, 1 + s], [2, 2 + s],..., [n s, n] for i 1 to n s do k i + s; [i, k] + for j i to k 1 do [i, k] min([i, k], [i, j] + [j + 1, k] + d i 1 d j d k return [1, n] Run Time: Θ(n 3 ). Question: How do you reconstruct the order of multiplication from the [, ] table? IS 675 (EES) ynamic Programming, part 2 October 28, / 21 Shortest paths: ll-pairs, 1 Vaughn Truth or onseq. Tucumcari lbuquerque lamogordo Taos Socorro Silver ity Santa Rosa Santa Fe Ruidoso Roswell Reserve Red River Raton Los lamos Portales Lordsburg Las Vegas IS 675 (EES) ynamic Programming, part 2 October 28, / 21 Las ruces Hobbs Farmington Gallup eming lovis layton hama arlsbad rtesia Shortest paths: ll-pairs, 2 Shortest paths: ll-pairs, 3 ll-pairs Shortest Paths Problem (PSP) G = ( 1,..., n }, E) an undirected graph and len:e R +. onstruct: S[1..n, 1..n] so that S[i, j] = the length of a shortest G-path from i to j. ssumption: G is initially given by a matrix [1..n, 1..n] so that len(i, j), if (i, j) E; [i, j] = +, if (i, j) / E. PSP, restated: Given [1..n, 1..n], compute S[1..n, 1..n]. Question: What are good subproblems? is the approximation to S in which that paths have no vertices: ❶ ❸ ❶ ❷ ❸ llowing more and more vertices gives subproblems... vertices from 1,..., k } len(i, j), if (i, j) E; dist[i, j, 0] = [i, j] = +, if (i, j) / E. dist[i, j, n] = S[i, j] = the length of a shortest G-path from i to j IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

4 length of the shortest path from i to j in which only nodes 1, 2,..., k} can be used as interme diates. Initially, dist(i, j, 0) is the length of the direct edge between i and j, if it exists, and is Shortest paths: ll-pairs, 3 otherwise. a,.h. Papadimitriou, and U.V. Vazirani 187 Shortest paths: ll-pairs, 4 What happens when we expand the set to include an extra node k? We mus reexamine all pairs Question: i, j and check How do whether we go using from dist[, k as, an k 1] to dist[,, k], point where: gives us a shorter shortest paths path from i to j. ut this is easy: a shortest path from i to j that uses k along with possibly want to find the shortest path not just between s and t but between other all lower-numbered pairs nodes goes through k just once (why? because we assume One approach would be to execute our general shortest-path algorithm that there from are no negative cycles). nd we have already vertices from calculated 1,... the, k } length of the shortes.1 (since there may be the negative length of edges) a shortest V times, G-path once from for i to each j using starting path node. from The i to k and from k to j using only lower-numbered vertices: ng time would then be O( V 2 E ). vertices We ll now from see 1, a.. better., k } alternative, the O( V 3 ) k ogramming-based Floyd-Warshall len(i, algorithm. j), if (i, j) E; dist(i, k, k 1) is a good dist[i, subproblem j, 0] = [i, for j] = computing distances between all pairs of vertices in a +, if (i, j) / E. i dist(k, j, k 1) ply solving the problem for more and more pairs or starting points is unhelpful, leads right dist[i, back j, n] to= the S[i, O( V j] = 2 the E ) length algorithm. of a shortest G-path from i to j a comes to mind: the shortest path u w 1 w l v between u and v dist(i, j, k 1) j number of nodes possibly none. Suppose we disallow ether. Then Question: we can How solve do we all-pairs go from shortest dist[,, k paths 1] to at dist[, once:, k]? the shortest Thus, pathusing from k gives us a shorter path from i to j if and only if mply the direct edge (u, v), if it exists. What if we now gradually expand the set dist[i, j, k 1] = dist[i, j, k] means: there is a shortest path from i to j using dist(i, k, k vertices 1) + dist(k, from j, 1, k..., k 1) } that < dist(i, does not j, use k k. 1), ble nodes? We can do this one node at a time, updating the shortest s at each stage. Eventually this set grows to all of V, at which pointin allwhich vertices case dist(i, j, k) should be updated accordingly. to be on all paths, and we have found the true shortest paths between vertices Here isofthe Floyd-Warshall algorithm and as you can see, it takes O( V 3 ) time. IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21 ncretely, number the vertices in V as 1, 2,..., n}, and let dist(i, j, k) denote forthe i = 1 to n: e shortest path from i to j in which only nodes 1, 2,..., k} can be used as intermeially, for j = 1 to n: dist(i, j, 0) is the length of the direct edge between i and j, if it exists, and is dist(i, j, 0) = e. Shortest paths: ll-pairs, 5 appens when we expand the set to include an extra node k? We must Shortest paths: ll-pairs, 6 all pairs Question: i, j and check How do whether we go using from dist[, k as, an k 1] to dist[,, k], point where: gives us a shorter i to j. ut this is easy: a shortest path from i to j that uses k along with possibly Question: How do we go from dist[,, k 1] to dist[,, k]? -numbered nodes goes through k just once (why? because we assume are no negative dist[i, cycles). j, k] = nd we have already vertices from calculated 1,... the, k } length of the shortest vertices from 1,..., k } to k and from k to j using only lower-numbered vertices: = min(dist[i, j, k 1], dist[i, k, k 1] + dist[k, j, k 1]). dist(i, k, k 1) k for i 1 to n do // Initialization i dist(k, j, k 1) dist[i, j, 0] [i, j] for i 1 to n do // Main iteration dist(i, j, k 1) j for k 1 to n do g k gives us a shorter path from i to j if and only if dist[i, j, k] min(dist[i, j, k 1], dist[i, j, k 1] = dist[i, j, k] means: shortest paths from i to j using dist[i, k, k 1] + dist[k, j, k 1]) for i dist(i, k, k vertices 1) + dist(k, from j, 1, k..., k 1) } must < dist(i, use k. j, k 1), 1 to n do // Output dist[i, k, k 1] + dist[k, j, k 1]. (Why?) S[i, j] dist[i, j, n] se dist(i, j, k) should be updated accordingly. the Floyd-Warshall algorithm and as you can see, it takes O( V 3 return S ) time. i = 1 to n: IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

5 Shortest paths: ll-pairs, 7 The traveling salesman problem for i 1 to n do // Initialization dist[i, j, 0] [i, j] for i 1 to n do // Main iteration for k 1 to n do dist[i, j, k] min(dist[i, j, k 1], dist[i, k, k 1] + dist[k, j, k 1]) for i 1 to n do // Output S[i, j] dist[i, j, n] return S Time complexity: Θ( V 3 ). (Why?) Space complexity: lso Θ( V 3 ), but this is easy to improve to Θ( V 2 ). The traveling salesman problem G a complete graph on verts 1,..., n and d(i, j) = (the distance of i to j) < Find: minimal cost of a complete tour of G. Subproblems For S 1,..., n } with 1 S, and j S: the minimal cost of a path from [S, j] = 1 to j using just nodes in S. [S, 1] =, when S > 1. [S, j] = min [S j }, i] + dist(i, j). i (S j } Note: This is a set up for hapter There are at most 2 n n-many subproblems, and each one takes O(n) time to solve.!!! This takes O(n 2 2 n ) time!!! 2 IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21 Independent sets in trees, 1 Independent sets in trees, 2 efinition Suppose G = (V, E) is an undirected graph. (a) u and v V are independent when (u, v) / E. (b) U V is an independent set when every pair of elements from U are independent. The Independent Set Problem (ISP) G an undirected graph. Find: max-sized independent set for G. The Independent Set Problem for Trees T a tree. Find: max-sized independent set for T. Example In the graph below: 1, 4, 5 } is not an independent set. 1, 5 } and 1, 6 } are independent sets 2, 3, 6 } and 1, 4, 6 } are max-sized independent sets efinition Suppose G = (V, E) is an undirected graph. (a) u and v V are independent when (u, v) / E. (b) U V is an independent set when every pair of elements from U are independent. The Indep. Set Problem for Trees T a tree. Find: max-sized independent set for T. Strategy Pick some vertex of T as the root, r. Now each vertex of T is the root of a subtree. For each v in T, define I(v) = the size of a largest independent set in v s subtree. I(v) = 1 when v is a leaf. I(r) = the size of a largest indep. set in T. So, what is the recursion? IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

6 Independent sets in trees, 3 Strategy Pick some vertex of T as the root, r. For each v in T: I(v) = the size of a largest indep. set in v s subtree. I(v) = 1 when v is a leaf and I(r) = the size of a largest indep. set in T. Wwhat is the recursion for I(u)? ase: u is in some maximal sized indep. set for u s subtree Then I(u) = 1 + I(v) : v is a grandchild of u }. ase: u is in no maximal sized indep. set for u s subtree Then I(u) = I(v) : v is a child of u }. Independent sets in trees, 4 Strategy Pick some vertex of T as the root, r. For each u in T compute: I(u) = the size of a largest indep. set in u s subtree = max ( 1 + v is a grandchild of u I(v), v is a child of u I(v) ) This can be done in Θ( V ) time. For the general graph case, O(2 V ) is the best known time. Note: This is another set up for hapter 8. I(u) = max ( 1 + v is a grandchild of u I(v), v is a child of u I(v) ) IS 675 (EES) ynamic Programming, part 2 October 28, / 21 IS 675 (EES) ynamic Programming, part 2 October 28, / 21

Dynamic Programming: Shortest Paths and DFA to Reg Exps

Dynamic Programming: Shortest Paths and DFA to Reg Exps CS 374: Algorithms & Models of Computation, Fall 205 Dynamic Programming: Shortest Paths and DFA to Reg Exps Lecture 7 October 22, 205 Chandra & Manoj (UIUC) CS374 Fall 205 / 54 Part I Shortest Paths with

More information

Dynamic Programming: Shortest Paths and DFA to Reg Exps

Dynamic Programming: Shortest Paths and DFA to Reg Exps CS 374: Algorithms & Models of Computation, Spring 207 Dynamic Programming: Shortest Paths and DFA to Reg Exps Lecture 8 March 28, 207 Chandra Chekuri (UIUC) CS374 Spring 207 / 56 Part I Shortest Paths

More information

CS483 Design and Analysis of Algorithms

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

More information

Chapter 8 Dynamic Programming

Chapter 8 Dynamic Programming Chapter 8 Dynamic Programming Copyright 007 Pearson Addison-Wesley. All rights reserved. Dynamic Programming Dynamic Programming is a general algorithm design technique for solving problems defined by

More information

Chapter 8 Dynamic Programming

Chapter 8 Dynamic Programming Chapter 8 Dynamic Programming Copyright 2007 Pearson Addison-Wesley. All rights reserved. Dynamic Programming Dynamic Programming is a general algorithm design technique for solving problems defined by

More information

Slides for CIS 675. Huffman Encoding, 1. Huffman Encoding, 2. Huffman Encoding, 3. Encoding 1. DPV Chapter 5, Part 2. Encoding 2

Slides for CIS 675. Huffman Encoding, 1. Huffman Encoding, 2. Huffman Encoding, 3. Encoding 1. DPV Chapter 5, Part 2. Encoding 2 Huffman Encoding, 1 EECS Slides for CIS 675 DPV Chapter 5, Part 2 Jim Royer October 13, 2009 A toy example: Suppose our alphabet is { A, B, C, D }. Suppose T is a text of 130 million characters. What is

More information

All-Pairs Shortest Paths

All-Pairs Shortest Paths All-Pairs Shortest Paths Version of October 28, 2016 Version of October 28, 2016 All-Pairs Shortest Paths 1 / 26 Outline Another example of dynamic programming Will see two different dynamic programming

More information

Computer Science & Engineering 423/823 Design and Analysis of Algorithms

Computer Science & Engineering 423/823 Design and Analysis of Algorithms Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 03 Dynamic Programming (Chapter 15) Stephen Scott and Vinodchandran N. Variyam sscott@cse.unl.edu 1/44 Introduction Dynamic

More information

CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms

CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms Professor Henry Carter Fall 2016 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic

More information

Data Structures in Java

Data Structures in Java Data Structures in Java Lecture 21: Introduction to NP-Completeness 12/9/2015 Daniel Bauer Algorithms and Problem Solving Purpose of algorithms: find solutions to problems. Data Structures provide ways

More information

CS173 Lecture B, November 3, 2015

CS173 Lecture B, November 3, 2015 CS173 Lecture B, November 3, 2015 Tandy Warnow November 3, 2015 CS 173, Lecture B November 3, 2015 Tandy Warnow Announcements Examlet 7 is a take-home exam, and is due November 10, 11:05 AM, in class.

More information

Algorithm Design CS 515 Fall 2015 Sample Final Exam Solutions

Algorithm Design CS 515 Fall 2015 Sample Final Exam Solutions Algorithm Design CS 515 Fall 2015 Sample Final Exam Solutions Copyright c 2015 Andrew Klapper. All rights reserved. 1. For the functions satisfying the following three recurrences, determine which is the

More information

Lecture 7: Shortest Paths in Graphs with Negative Arc Lengths. Reading: AM&O Chapter 5

Lecture 7: Shortest Paths in Graphs with Negative Arc Lengths. Reading: AM&O Chapter 5 Lecture 7: Shortest Paths in Graphs with Negative Arc Lengths Reading: AM&O Chapter Label Correcting Methods Assume the network G is allowed to have negative arc lengths but no directed negativelyweighted

More information

Dynamic Programming (CLRS )

Dynamic Programming (CLRS ) Dynamic Programming (CLRS.-.) Today we discuss a technique called dynamic programming. It is neither especially dynamic nor especially programming related. We will discuss dynamic programming by looking

More information

Computer Science 385 Analysis of Algorithms Siena College Spring Topic Notes: Limitations of Algorithms

Computer Science 385 Analysis of Algorithms Siena College Spring Topic Notes: Limitations of Algorithms Computer Science 385 Analysis of Algorithms Siena College Spring 2011 Topic Notes: Limitations of Algorithms We conclude with a discussion of the limitations of the power of algorithms. That is, what kinds

More information

CMPSCI 611 Advanced Algorithms Midterm Exam Fall 2015

CMPSCI 611 Advanced Algorithms Midterm Exam Fall 2015 NAME: CMPSCI 611 Advanced Algorithms Midterm Exam Fall 015 A. McGregor 1 October 015 DIRECTIONS: Do not turn over the page until you are told to do so. This is a closed book exam. No communicating with

More information

CS60007 Algorithm Design and Analysis 2018 Assignment 1

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

More information

Computer Science & Engineering 423/823 Design and Analysis of Algorithms

Computer Science & Engineering 423/823 Design and Analysis of Algorithms Computer Science & Engineering 423/823 Design and Analysis of s Lecture 09 Dynamic Programming (Chapter 15) Stephen Scott (Adapted from Vinodchandran N. Variyam) 1 / 41 Spring 2010 Dynamic programming

More information

Design and Analysis of Algorithms

Design and Analysis of Algorithms Design and Analysis of Algorithms CSE 5311 Lecture 22 All-Pairs Shortest Paths Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 All Pairs

More information

Dynamic Programming. Prof. S.J. Soni

Dynamic Programming. Prof. S.J. Soni Dynamic Programming Prof. S.J. Soni Idea is Very Simple.. Introduction void calculating the same thing twice, usually by keeping a table of known results that fills up as subinstances are solved. Dynamic

More information

CS 470/570 Dynamic Programming. Format of Dynamic Programming algorithms:

CS 470/570 Dynamic Programming. Format of Dynamic Programming algorithms: CS 470/570 Dynamic Programming Format of Dynamic Programming algorithms: Use a recursive formula But recursion can be inefficient because it might repeat the same computations multiple times So use an

More information

Dynamic Programming( Weighted Interval Scheduling)

Dynamic Programming( Weighted Interval Scheduling) Dynamic Programming( Weighted Interval Scheduling) 17 November, 2016 Dynamic Programming 1 Dynamic programming algorithms are used for optimization (for example, finding the shortest path between two points,

More information

1 Assembly Line Scheduling in Manufacturing Sector

1 Assembly Line Scheduling in Manufacturing Sector Indian Institute of Information Technology Design and Manufacturing, Kancheepuram, Chennai 600 27, India An Autonomous Institute under MHRD, Govt of India http://www.iiitdm.ac.in COM 209T Design and Analysis

More information

Dynamic Programming. p. 1/43

Dynamic Programming. p. 1/43 Dynamic Programming Formalized by Richard Bellman Programming relates to planning/use of tables, rather than computer programming. Solve smaller problems first, record solutions in a table; use solutions

More information

Maximum sum contiguous subsequence Longest common subsequence Matrix chain multiplication All pair shortest path Kna. Dynamic Programming

Maximum sum contiguous subsequence Longest common subsequence Matrix chain multiplication All pair shortest path Kna. Dynamic Programming Dynamic Programming Arijit Bishnu arijit@isical.ac.in Indian Statistical Institute, India. August 31, 2015 Outline 1 Maximum sum contiguous subsequence 2 Longest common subsequence 3 Matrix chain multiplication

More information

Introduction. I Dynamic programming is a technique for solving optimization problems. I Key element: Decompose a problem into subproblems, solve them

Introduction. I Dynamic programming is a technique for solving optimization problems. I Key element: Decompose a problem into subproblems, solve them ntroduction Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 03 Dynamic Programming (Chapter 15) Stephen Scott and Vinodchandran N. Variyam Dynamic programming is a technique

More information

Dynamic Programming ACM Seminar in Algorithmics

Dynamic Programming ACM Seminar in Algorithmics Dynamic Programming ACM Seminar in Algorithmics Marko Genyk-Berezovskyj berezovs@fel.cvut.cz Tomáš Tunys tunystom@fel.cvut.cz CVUT FEL, K13133 February 27, 2013 Problem: Longest Increasing Subsequence

More information

CS Algorithms and Complexity

CS Algorithms and Complexity CS 50 - Algorithms and Complexity Linear Programming, the Simplex Method, and Hard Problems Sean Anderson 2/15/18 Portland State University Table of contents 1. The Simplex Method 2. The Graph Problem

More information

1. (a) Explain the asymptotic notations used in algorithm analysis. (b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)).

1. (a) Explain the asymptotic notations used in algorithm analysis. (b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)). Code No: R05220502 Set No. 1 1. (a) Explain the asymptotic notations used in algorithm analysis. (b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)). 2. (a) List some of the relative advantages

More information

CS/COE

CS/COE CS/COE 1501 www.cs.pitt.edu/~nlf4/cs1501/ P vs NP But first, something completely different... Some computational problems are unsolvable No algorithm can be written that will always produce the correct

More information

COL351: Analysis and Design of Algorithms (CSE, IITD, Semester-I ) Name: Entry number:

COL351: Analysis and Design of Algorithms (CSE, IITD, Semester-I ) Name: Entry number: Name: Entry number: There are 5 questions for a total of 75 points. 1. (5 points) You are given n items and a sack that can hold at most W units of weight. The weight of the i th item is denoted by w(i)

More information

General Methods for Algorithm Design

General Methods for Algorithm Design General Methods for Algorithm Design 1. Dynamic Programming Multiplication of matrices Elements of the dynamic programming Optimal triangulation of polygons Longest common subsequence 2. Greedy Methods

More information

Lecture 13. More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees.

Lecture 13. More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees. Lecture 13 More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees. Announcements HW5 due Friday! HW6 released Friday! Last time Not coding in an action

More information

Quiz 1 Solutions. Problem 2. Asymptotics & Recurrences [20 points] (3 parts)

Quiz 1 Solutions. Problem 2. Asymptotics & Recurrences [20 points] (3 parts) Introduction to Algorithms October 13, 2010 Massachusetts Institute of Technology 6.006 Fall 2010 Professors Konstantinos Daskalakis and Patrick Jaillet Quiz 1 Solutions Quiz 1 Solutions Problem 1. We

More information

BKLUP PRELIMINARY SUMMARY REPORT JULY 2016

BKLUP PRELIMINARY SUMMARY REPORT JULY 2016 PARTICIPATED LAST MONTH TOTAL BLKUP HOURS OTHER Alamogordo PD 2 Otero 6 0 0 0 0 0 4 3 N/P Albuquerque PD 3 Bernalillo N/P Anthony PD 1 Dona Ana 4 0 0 0 0 0 3 0 Artesia PD 2 Eddy 16 0 0 0 0 3 9 0 N/P Aztec

More information

Exercises NP-completeness

Exercises NP-completeness Exercises NP-completeness Exercise 1 Knapsack problem Consider the Knapsack problem. We have n items, each with weight a j (j = 1,..., n) and value c j (j = 1,..., n) and an integer B. All a j and c j

More information

Advanced Analysis of Algorithms - Midterm (Solutions)

Advanced Analysis of Algorithms - Midterm (Solutions) Advanced Analysis of Algorithms - Midterm (Solutions) K. Subramani LCSEE, West Virginia University, Morgantown, WV {ksmani@csee.wvu.edu} 1 Problems 1. Solve the following recurrence using substitution:

More information

Chapter 7 Network Flow Problems, I

Chapter 7 Network Flow Problems, I Chapter 7 Network Flow Problems, I Network flow problems are the most frequently solved linear programming problems. They include as special cases, the assignment, transportation, maximum flow, and shortest

More information

Lecture 6 September 21, 2016

Lecture 6 September 21, 2016 ICS 643: Advanced Parallel Algorithms Fall 2016 Lecture 6 September 21, 2016 Prof. Nodari Sitchinava Scribe: Tiffany Eulalio 1 Overview In the last lecture, we wrote a non-recursive summation program and

More information

CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices

CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices This lab consists of exercises on real-valued vectors and matrices. Most of the exercises will required pencil and paper. Put

More information

Week 5: Quicksort, Lower bound, Greedy

Week 5: Quicksort, Lower bound, Greedy Week 5: Quicksort, Lower bound, Greedy Agenda: Quicksort: Average case Lower bound for sorting Greedy method 1 Week 5: Quicksort Recall Quicksort: The ideas: Pick one key Compare to others: partition into

More information

Branch-and-Bound. Leo Liberti. LIX, École Polytechnique, France. INF , Lecture p. 1

Branch-and-Bound. Leo Liberti. LIX, École Polytechnique, France. INF , Lecture p. 1 Branch-and-Bound Leo Liberti LIX, École Polytechnique, France INF431 2011, Lecture p. 1 Reminders INF431 2011, Lecture p. 2 Problems Decision problem: a question admitting a YES/NO answer Example HAMILTONIAN

More information

CS Analysis of Recursive Algorithms and Brute Force

CS Analysis of Recursive Algorithms and Brute Force CS483-05 Analysis of Recursive Algorithms and Brute Force Instructor: Fei Li Room 443 ST II Office hours: Tue. & Thur. 4:30pm - 5:30pm or by appointments lifei@cs.gmu.edu with subject: CS483 http://www.cs.gmu.edu/

More information

Practice Final Solutions. 1. Consider the following algorithm. Assume that n 1. line code 1 alg(n) { 2 j = 0 3 if (n = 0) { 4 return j

Practice Final Solutions. 1. Consider the following algorithm. Assume that n 1. line code 1 alg(n) { 2 j = 0 3 if (n = 0) { 4 return j Practice Final Solutions 1. Consider the following algorithm. Assume that n 1. line code 1 alg(n) 2 j = 0 3 if (n = 0) 4 return j } 5 else 6 j = 2n+ alg(n 1) 7 return j } } Set up a recurrence relation

More information

Dynamic Programming. Cormen et. al. IV 15

Dynamic Programming. Cormen et. al. IV 15 Dynamic Programming Cormen et. al. IV 5 Dynamic Programming Applications Areas. Bioinformatics. Control theory. Operations research. Some famous dynamic programming algorithms. Unix diff for comparing

More information

Algorithm Design Strategies V

Algorithm Design Strategies V Algorithm Design Strategies V Joaquim Madeira Version 0.0 October 2016 U. Aveiro, October 2016 1 Overview The 0-1 Knapsack Problem Revisited The Fractional Knapsack Problem Greedy Algorithms Example Coin

More information

Randomized Sorting Algorithms Quick sort can be converted to a randomized algorithm by picking the pivot element randomly. In this case we can show th

Randomized Sorting Algorithms Quick sort can be converted to a randomized algorithm by picking the pivot element randomly. In this case we can show th CSE 3500 Algorithms and Complexity Fall 2016 Lecture 10: September 29, 2016 Quick sort: Average Run Time In the last lecture we started analyzing the expected run time of quick sort. Let X = k 1, k 2,...,

More information

Limitations of Algorithm Power

Limitations of Algorithm Power Limitations of Algorithm Power Objectives We now move into the third and final major theme for this course. 1. Tools for analyzing algorithms. 2. Design strategies for designing algorithms. 3. Identifying

More information

CSE 202 Homework 4 Matthias Springer, A

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

More information

Discrete Optimization 2010 Lecture 2 Matroids & Shortest Paths

Discrete Optimization 2010 Lecture 2 Matroids & Shortest Paths Matroids Shortest Paths Discrete Optimization 2010 Lecture 2 Matroids & Shortest Paths Marc Uetz University of Twente m.uetz@utwente.nl Lecture 2: sheet 1 / 25 Marc Uetz Discrete Optimization Matroids

More information

Analysis of Algorithms I: All-Pairs Shortest Paths

Analysis of Algorithms I: All-Pairs Shortest Paths Analysis of Algorithms I: All-Pairs Shortest Paths Xi Chen Columbia University The All-Pairs Shortest Paths Problem. Input: A directed weighted graph G = (V, E) with an edge-weight function w : E R. Output:

More information

Methods for solving recurrences

Methods for solving recurrences Methods for solving recurrences Analyzing the complexity of mergesort The merge function Consider the following implementation: 1 int merge ( int v1, int n1, int v, int n ) { 3 int r = malloc ( ( n1+n

More information

Lecture 11. Single-Source Shortest Paths All-Pairs Shortest Paths

Lecture 11. Single-Source Shortest Paths All-Pairs Shortest Paths Lecture. Single-Source Shortest Paths All-Pairs Shortest Paths T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to, rd Edition, MIT Press, 009 Sungkyunkwan University Hyunseung Choo choo@skku.edu

More information

Activity selection. Goal: Select the largest possible set of nonoverlapping (mutually compatible) activities.

Activity selection. Goal: Select the largest possible set of nonoverlapping (mutually compatible) activities. Greedy Algorithm 1 Introduction Similar to dynamic programming. Used for optimization problems. Not always yield an optimal solution. Make choice for the one looks best right now. Make a locally optimal

More information

Intro to Contemporary Math

Intro to Contemporary Math Intro to Contemporary Math Hamiltonian Circuits and Nearest Neighbor Algorithm Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Hamiltonian Circuits and the Traveling Salesman

More information

Dominating Set Counting in Graph Classes

Dominating Set Counting in Graph Classes Dominating Set Counting in Graph Classes Shuji Kijima 1, Yoshio Okamoto 2, and Takeaki Uno 3 1 Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan kijima@inf.kyushu-u.ac.jp

More information

CS325: Analysis of Algorithms, Fall Final Exam

CS325: Analysis of Algorithms, Fall Final Exam CS: Analysis of Algorithms, Fall 0 Final Exam I don t know policy: you may write I don t know and nothing else to answer a question and receive percent of the total points for that problem whereas a completely

More information

CS 6901 (Applied Algorithms) Lecture 3

CS 6901 (Applied Algorithms) Lecture 3 CS 6901 (Applied Algorithms) Lecture 3 Antonina Kolokolova September 16, 2014 1 Representative problems: brief overview In this lecture we will look at several problems which, although look somewhat similar

More information

Data Structures and Algorithms (CSCI 340)

Data Structures and Algorithms (CSCI 340) University of Wisconsin Parkside Fall Semester 2008 Department of Computer Science Prof. Dr. F. Seutter Data Structures and Algorithms (CSCI 340) Homework Assignments The numbering of the problems refers

More information

Randomized Algorithms III Min Cut

Randomized Algorithms III Min Cut Chapter 11 Randomized Algorithms III Min Cut CS 57: Algorithms, Fall 01 October 1, 01 11.1 Min Cut 11.1.1 Problem Definition 11. Min cut 11..0.1 Min cut G = V, E): undirected graph, n vertices, m edges.

More information

Combinatorial Optimization

Combinatorial Optimization Combinatorial Optimization Problem set 8: solutions 1. Fix constants a R and b > 1. For n N, let f(n) = n a and g(n) = b n. Prove that f(n) = o ( g(n) ). Solution. First we observe that g(n) 0 for all

More information

Lecture 13. More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees.

Lecture 13. More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees. Lecture 13 More dynamic programming! Longest Common Subsequences, Knapsack, and (if time) independent sets in trees. Announcements HW5 due Friday! HW6 released Friday! Last time Not coding in an action

More information

CS281A/Stat241A Lecture 19

CS281A/Stat241A Lecture 19 CS281A/Stat241A Lecture 19 p. 1/4 CS281A/Stat241A Lecture 19 Junction Tree Algorithm Peter Bartlett CS281A/Stat241A Lecture 19 p. 2/4 Announcements My office hours: Tuesday Nov 3 (today), 1-2pm, in 723

More information

July 18, Approximation Algorithms (Travelling Salesman Problem)

July 18, Approximation Algorithms (Travelling Salesman Problem) Approximation Algorithms (Travelling Salesman Problem) July 18, 2014 The travelling-salesman problem Problem: given complete, undirected graph G = (V, E) with non-negative integer cost c(u, v) for each

More information

Notes on the Matrix-Tree theorem and Cayley s tree enumerator

Notes on the Matrix-Tree theorem and Cayley s tree enumerator Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will

More information

Algorithms and Data Structures Final Lesson

Algorithms and Data Structures Final Lesson Algorithms and Data Structures Final Lesson Michael Schwarzkopf https://www.uni weimar.de/de/medien/professuren/medieninformatik/grafische datenverarbeitung Bauhaus University Weimar July 11, 2018 (Corrected

More information

Applications of Binary Search

Applications of Binary Search Applications of Binary Search The basic idea of a binary search can be used in many different places. In particular, any time you are searching for an answer in a search space that is somehow sorted, you

More information

IS 2610: Data Structures

IS 2610: Data Structures IS 2610: Data Structures Graph April 12, 2004 Graph Weighted graph call it networks Shortest path between nodes s and t in a network Directed simple path from s to t with the property that no other such

More information

Query Optimization: Exercise

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

More information

CSI 4105 MIDTERM SOLUTION

CSI 4105 MIDTERM SOLUTION University of Ottawa CSI 4105 MIDTERM SOLUTION Instructor: Lucia Moura Feb 6, 2010 10:00 am Duration: 1:50 hs Closed book Last name: First name: Student number: There are 4 questions and 100 marks total.

More information

Dynamic Programming: Matrix chain multiplication (CLRS 15.2)

Dynamic Programming: Matrix chain multiplication (CLRS 15.2) Dynamic Programming: Matrix chain multiplication (CLRS.) The problem Given a sequence of matrices A, A, A,..., A n, find the best way (using the minimal number of multiplications) to compute their product.

More information

Partha Sarathi Mandal

Partha Sarathi Mandal MA 252: Data Structures and Algorithms Lecture 32 http://www.iitg.ernet.in/psm/indexing_ma252/y12/index.html Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati The All-Pairs Shortest Paths Problem

More information

Solutions to Final Practice Problems 1-15

Solutions to Final Practice Problems 1-15 Solutions to Final Practice Problems 1-15 1. Given a sequence of n real numbers A 1... A n, determine in linear time a contiguous subsequence A i... A j (i j) for which the sum of elements in the subsequence

More information

Lecture 17: Trees and Merge Sort 10:00 AM, Oct 15, 2018

Lecture 17: Trees and Merge Sort 10:00 AM, Oct 15, 2018 CS17 Integrated Introduction to Computer Science Klein Contents Lecture 17: Trees and Merge Sort 10:00 AM, Oct 15, 2018 1 Tree definitions 1 2 Analysis of mergesort using a binary tree 1 3 Analysis of

More information

University of Toronto Department of Electrical and Computer Engineering. Final Examination. ECE 345 Algorithms and Data Structures Fall 2016

University of Toronto Department of Electrical and Computer Engineering. Final Examination. ECE 345 Algorithms and Data Structures Fall 2016 University of Toronto Department of Electrical and Computer Engineering Final Examination ECE 345 Algorithms and Data Structures Fall 2016 Print your first name, last name, UTORid, and student number neatly

More information

Divide and Conquer Algorithms. CSE 101: Design and Analysis of Algorithms Lecture 14

Divide and Conquer Algorithms. CSE 101: Design and Analysis of Algorithms Lecture 14 Divide and Conquer Algorithms CSE 101: Design and Analysis of Algorithms Lecture 14 CSE 101: Design and analysis of algorithms Divide and conquer algorithms Reading: Sections 2.3 and 2.4 Homework 6 will

More information

Single Source Shortest Paths

Single Source Shortest Paths CMPS 00 Fall 017 Single Source Shortest Paths Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk Paths in graphs Consider a digraph G = (V, E) with an edge-weight

More information

13 Dynamic Programming (3) Optimal Binary Search Trees Subset Sums & Knapsacks

13 Dynamic Programming (3) Optimal Binary Search Trees Subset Sums & Knapsacks 13 Dynamic Programming (3) Optimal Binary Search Trees Subset Sums & Knapsacks Average-case analysis Average-case analysis of algorithms and data structures: Input is generated according to a known probability

More information

Santa Claus Schedules Jobs on Unrelated Machines

Santa Claus Schedules Jobs on Unrelated Machines Santa Claus Schedules Jobs on Unrelated Machines Ola Svensson (osven@kth.se) Royal Institute of Technology - KTH Stockholm, Sweden March 22, 2011 arxiv:1011.1168v2 [cs.ds] 21 Mar 2011 Abstract One of the

More information

Do not turn this page until you have received the signal to start. Please fill out the identification section above. Good Luck!

Do not turn this page until you have received the signal to start. Please fill out the identification section above. Good Luck! CSC 373H5 F 2017 Midterm Duration 1 hour and 50 minutes Last Name: Lecture Section: 1 Student Number: First Name: Instructor: Vincent Maccio Do not turn this page until you have received the signal to

More information

Dynamic Programming. Reading: CLRS Chapter 15 & Section CSE 6331: Algorithms Steve Lai

Dynamic Programming. Reading: CLRS Chapter 15 & Section CSE 6331: Algorithms Steve Lai Dynamic Programming Reading: CLRS Chapter 5 & Section 25.2 CSE 633: Algorithms Steve Lai Optimization Problems Problems that can be solved by dynamic programming are typically optimization problems. Optimization

More information

Week 7 Solution. The two implementations are 1. Approach 1. int fib(int n) { if (n <= 1) return n; return fib(n 1) + fib(n 2); } 2.

Week 7 Solution. The two implementations are 1. Approach 1. int fib(int n) { if (n <= 1) return n; return fib(n 1) + fib(n 2); } 2. Week 7 Solution 1.You are given two implementations for finding the nth Fibonacci number(f Fibonacci numbers are defined by F(n = F(n 1 + F(n 2 with F(0 = 0 and F(1 = 1 The two implementations are 1. Approach

More information

NP-Complete Problems

NP-Complete Problems NP-Complete Problems Max Bisection Is NP-Complete max cut becomes max bisection if we require that S = V S. We shall reduce the more general max cut to max bisection. Add V isolated nodes to G to yield

More information

Weighted Activity Selection

Weighted Activity Selection Weighted Activity Selection Problem This problem is a generalization of the activity selection problem that we solvd with a greedy algorithm. Given a set of activities A = {[l, r ], [l, r ],..., [l n,

More information

DAA Unit- II Greedy and Dynamic Programming. By Mrs. B.A. Khivsara Asst. Professor Department of Computer Engineering SNJB s KBJ COE, Chandwad

DAA Unit- II Greedy and Dynamic Programming. By Mrs. B.A. Khivsara Asst. Professor Department of Computer Engineering SNJB s KBJ COE, Chandwad DAA Unit- II Greedy and Dynamic Programming By Mrs. B.A. Khivsara Asst. Professor Department of Computer Engineering SNJB s KBJ COE, Chandwad 1 Greedy Method 2 Greedy Method Greedy Principal: are typically

More information

4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle

4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle Directed Hamiltonian Cycle Chapter 8 NP and Computational Intractability Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle Γ. Then G' has an undirected Hamiltonian

More information

Mergesort and Recurrences (CLRS 2.3, 4.4)

Mergesort and Recurrences (CLRS 2.3, 4.4) Mergesort and Recurrences (CLRS 2.3, 4.4) We saw a couple of O(n 2 ) algorithms for sorting. Today we ll see a different approach that runs in O(n lg n) and uses one of the most powerful techniques for

More information

Algorithms and Data Structures 2016 Week 5 solutions (Tues 9th - Fri 12th February)

Algorithms and Data Structures 2016 Week 5 solutions (Tues 9th - Fri 12th February) Algorithms and Data Structures 016 Week 5 solutions (Tues 9th - Fri 1th February) 1. Draw the decision tree (under the assumption of all-distinct inputs) Quicksort for n = 3. answer: (of course you should

More information

Lecture 18: More NP-Complete Problems

Lecture 18: More NP-Complete Problems 6.045 Lecture 18: More NP-Complete Problems 1 The Clique Problem a d f c b e g Given a graph G and positive k, does G contain a complete subgraph on k nodes? CLIQUE = { (G,k) G is an undirected graph with

More information

Dynamic Programming. Reading: CLRS Chapter 15 & Section CSE 2331 Algorithms Steve Lai

Dynamic Programming. Reading: CLRS Chapter 15 & Section CSE 2331 Algorithms Steve Lai Dynamic Programming Reading: CLRS Chapter 5 & Section 25.2 CSE 233 Algorithms Steve Lai Optimization Problems Problems that can be solved by dynamic programming are typically optimization problems. Optimization

More information

Even More on Dynamic Programming

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

More information

Clustering using Mixture Models

Clustering using Mixture Models Clustering using Mixture Models The full posterior of the Gaussian Mixture Model is p(x, Z, µ,, ) =p(x Z, µ, )p(z )p( )p(µ, ) data likelihood (Gaussian) correspondence prob. (Multinomial) mixture prior

More information

Question Paper Code :

Question Paper Code : www.vidyarthiplus.com Reg. No. : B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011. Time : Three hours Fourth Semester Computer Science and Engineering CS 2251 DESIGN AND ANALYSIS OF ALGORITHMS (Regulation

More information

Lecture 15 - NP Completeness 1

Lecture 15 - NP Completeness 1 CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanford.edu) February 29, 2018 Lecture 15 - NP Completeness 1 In the last lecture we discussed how to provide

More information

CSE 431/531: Analysis of Algorithms. Dynamic Programming. Lecturer: Shi Li. Department of Computer Science and Engineering University at Buffalo

CSE 431/531: Analysis of Algorithms. Dynamic Programming. Lecturer: Shi Li. Department of Computer Science and Engineering University at Buffalo CSE 431/531: Analysis of Algorithms Dynamic Programming Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Paradigms for Designing Algorithms Greedy algorithm Make a

More information

Dynamic programming. Curs 2015

Dynamic programming. Curs 2015 Dynamic programming. Curs 2015 Fibonacci Recurrence. n-th Fibonacci Term INPUT: n nat QUESTION: Compute F n = F n 1 + F n 2 Recursive Fibonacci (n) if n = 0 then return 0 else if n = 1 then return 1 else

More information

CS Lunch. Dynamic Programming Recipe. 2 Midterm 2! Slides17 - Segmented Least Squares.key - November 16, 2016

CS Lunch. Dynamic Programming Recipe. 2 Midterm 2! Slides17 - Segmented Least Squares.key - November 16, 2016 CS Lunch 1 Michelle Oraa Ali Tien Dao Vladislava Paskova Nan Zhuang Surabhi Sharma Wednesday, 12:15 PM Kendade 307 2 Midterm 2! Monday, November 21 In class Covers Greedy Algorithms Closed book Dynamic

More information

Computational Complexity and Intractability: An Introduction to the Theory of NP. Chapter 9

Computational Complexity and Intractability: An Introduction to the Theory of NP. Chapter 9 1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives Classify problems as tractable or intractable Define decision problems Define the class P Define

More information

CS325: Analysis of Algorithms, Fall Midterm

CS325: Analysis of Algorithms, Fall Midterm CS325: Analysis of Algorithms, Fall 2017 Midterm I don t know policy: you may write I don t know and nothing else to answer a question and receive 25 percent of the total points for that problem whereas

More information