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Recursive Back Tracking & Dynamic Programming Jeff Edmonds York University Lecture 7 COSC 3101 1 Techniques Problems Optimization Problems Best Path A Sequence of Decisions Printing Neatly The Little Bird & Friend Longest Common Subsequence Optimal Substructure Knapsack Problem Memoization The Event Scheduling Problem Set of Sub-Instances Parsing Tracing Dyn. Prog. Alg Satisfiability Reversing Code Speeding Up Running Time Multiple Opt Solutions Review Question for Little Bird Review & Don'ts 2 Dynamic Programming • A hard topic. • I try to provide a unified way to think of it and a fixed set of steps to follow. • Even if you don’t get the details of the algorithm correct, at least get the right structure. • I provide analogies (little bird) to make it hopefully more fun & easier to follow. 3 Optimization Problems • An important and practical class of computational problems. • For most of these, the best known algorithm runs in exponential time. • Industry would pay dearly to have faster algorithms. • Heuristics • Some have quick Greedy or Dynamic Programming algorithms • For the rest, Recursive Back Tracking is the best option. 4 Optimization Problems Ingredients: •Instances: The possible inputs to the problem. •Solutions for Instance: Each instance has an exponentially large set of solutions. •Cost of Solution: Each solution has an easy to compute cost or value. 5 Optimization Problems Specification of an Optimization Problem •<preCond>: The input is one instance. •<postCond>: The output is one of the valid solutions for this instance with optimal cost. (minimum or maximum) •The solution might not be unique. •Be clear about these ingredients! 6 Search Graph For Best Path We use it because it nicely demonstrates the concepts in a graphical way. 7 Search Graph For Best Path An instance (input) consists of <G,s,t>. G is a weighted directed layered graph s source node t sink node 4 5 3 2 3 2 5 2 2 1 4 3 8 Search Graph For Best Path An instance (input) consists of <G,s,t>. A solution for an instance is a path from s to t. The cost of a solution is the sum of the weights. The goal is to find a path with minimum total cost. 2+6+3+7=18 4+2+1+5=12 9 Brute Force Algorithm Try all paths, return the best. But there may be an exponential number of paths! 10 An Algorithm As A Sequence of Decisions I ask a question about the solution. “Which edge should we take first?” Some how I decide <s,v3>. My friend asks the next question. “Which edge do we take second?” Some how he decides <v3,v5>. His friend asks the next question. “Which edge do we take third?” Some how he decided <v5,v8>. 11 An Algorithm As A Sequence of Decisions I ask a question about the solution. “Which edge should we take first?” How do I decide? The greedy algorithm? Taking the best first edge. Does not work! 12 Local vs Global Considerations • We are able to make local observations and choices. – Eg. Which edge out of s is cheapest? • But it is hard to see the global consequences – Which path is the overall cheapest? • Sometimes a local initial sacrifice can globally lead to a better overall solution. 13 An Algorithm As A Sequence of Decisions I ask a question about the solution. “Which edge should we take first?” How do I decide? •In reality we will try all possible first edges. •But let's skip this part by pretending that we have a little bird to answer this little question. 14 "Little Bird" Abstraction (It is up to you whether or not to use it) Recall: Non-deterministic Finite Automata Non-deterministic Turing Machine 0 These have a higher power to tell them which way to go. The little bird is a little higher power, answering a little question about an optimal solution. 15 Little Bird & Friend Alg I ask a question about the solution. “Which edge should we take first?” The bird answers <s,v1>. My friend asks the next question. “Which edge do we take second?” The bird answers <v1,v4>. But we don’t want to worry about how our friend solves his problem. 16 Sub-Instance for Friend Our instance is <G,s,t>: Find best path from s to t. Our friend is recursion •i.e. he is a smaller version of ourselves •we can trust him to give us a correct answer •as long as we give him •a smaller instance •of the same problem. •What sub-instance do we give him? 17 Little Bird & Friend Alg The bird answers <s,v1>. If I trust the little bird, I take step along edge <s,v1> and ask my friend, “Which is the best path from v1 to t?” Friend answers <v1,v6,t> with weight 10. To get my solution I tack on the bird’s edge making the path <s,v1,v6,t> with weight 10+3=13. 18 Faulty Bird But what if we do not have a bird that we trust? This work is not wasted, because we have found •the best solution to our instance from amongst those consistent with this bird' s answer. •i.e. the best path from s to t from amongst those starting with <s,v1>. Define optS<I,k> to be: the optimum solution for instance I consistent with the kth bird' s answer. 19 Faulty Bird But what if we do not have a bird that we trust? This work is not wasted, because we have found •the best solution to our instance from amongst those consistent with this bird' s answer. •i.e. the best path from s to t from amongst those starting with <s,v1>. In reality we will try all possible first edges, giving ….. 20 Faulty Bird …the best path from amongst those starting with <s,v1>. 21 Faulty Bird … and the best path from amongst those starting with <s,v2>. 22 Faulty Bird … and the best path from amongst those starting with <s,v3>. 23 Faulty Bird … and the best path from amongst those starting with <s,v4>. 24 Faulty Bird At least one of these four paths must be an over all best path. I give the best of the best as the best path. 25 Bird/Friend - Best of the Best Consider our instance I. Consider the set of solutions A sequence of question to a little bird about a solution forms a tree of possible answers. 26 Bird/Friend - Best of the Best Consider our instance I. Consider the set of solutions k But we only care about the first bird answer. k The answers classifies Solutions consistent the possible solutions. with the kth bird' s answer. 27 Bird/Friend - Best of the Best optS<I,k> Consider our instance I. Consider the set of solutions Define optS<I,k> to be: the optimum solution k for instance I consistent with the kth bird' s answer. Do this for each k. Solutions consistent with the kth bird' s answer. 28 Bird/Friend - Best of the Best optS[I] optS<I,k> Consider our instance I. Consider the set of solutions Define optS<I,k> to be: the optimum solution kmax k for instance I consistent with the kth bird' s answer. Do this for each k. Let kmax be the bird' s answer giving the best optS<I,k>. optS[I] = optS<I,k max> = Bestk optS<I,k > 29 Bird/Friend - Best of the Best Constructing optS<I,k> : the optimum solution for instance I consistent with the kth bird' s answer. Given my instance I. I ask my little bird for an answer k. I ask my friend for his solution. I combine them. 30 Recursive backtracking code always has this same basic structure. 31 Be clear what are •the instances •it’s solution •the cost of a sol. 32 Loop through the bird answers. Be clear which is the current one being tried. 33 Give the bird & friend algorithm as a comment. (Unless it is in an earlier question.) 34 What is the bird asked? What does she answer? 35 Get help from friend Be clear what sub-instance you give him. Store the solution & cost he gives you. 36 How do you form your solution from the friend’s and from the bird’s? 37 How do you form your cost from the friend’s and from the bird’s? 38 optSolk is a best solution for our instance from amongst those consistent with the bird's kth answer. Take the best of the best 39 Return the solution and cost for the original instance. 40 Base Cases: Instances that are too small to have smaller instances to give to friends. What are these? What are their solutions and costs? 41 Optimal Substructure In order to be able to design a recursive backtracking algorithm for a computational problem, the problem needs to have a recursive structure, i.e. for a path from s to t to be optimal, the sub-path from vi to t must optimal. If shorter from vi to t. shorter to s to t. 42 Optimal Substructure In order to be able to design a recursive backtracking algorithm for a computational problem, the problem needs to have an optimal substructure, i.e. for a path from s to t to be optimal, the sub-path from vi to t must optimal. And finding such a sub-path is a sub-instance of the same computational problem. 43 Optimal Substructure • Optimal substructure means that – Every optimal solution to a problem contains... – ...optimal solutions to subproblems • Optimal substructure does not mean that – If you have optimal solutions to all subproblems... – ...then you can combine any of them to get an optimal solution to a larger problem. • Example: In Canadian coinage, – The optimal solution to 7¢ is 5¢ + 1¢ + 1¢, and – The optimal solution to 6¢ is 5¢ + 1¢, but – The optimal solution to 13¢ is not 5¢ + 1¢ + 1¢ + 5¢ + 1¢ • But there is some way of dividing up 13¢ into subsets with optimal solutions (say, 11¢ + 2¢) that will give an optimal solution for 13¢ – Hence, the making change problem exhibits optimal substructure. 44 Optimal Substructure Don’t all problems have this optimal substructure property? 45 Optimal Substructure Longest simple path B 1 2 3 1 4 • Consider the A C D following graph: • The longest simple path (path not containing a cycle) from A to D is A B C D • However, the subpath A B is not the longest simple path from A to B (A C B is longer) • The principle of optimality is not satisfied for this problem • Hence, the longest simple path problem cannot be solved by a dynamic programming approach NP-Complete 46 Same as Brute Force Algorithm I try each edge out of s. A friend tries each edge out of these. A friend tries each edge out of these. Time? Same as the brute force algorithm that tries each path. 47 Same as Brute Force Algorithm But there may be an exponential number of paths! 48 Speeding Up the Time Why do all this work with birds & friends? •How else would you iterate through all paths? •But sometimes we can exploit the structure to speed up the algorithm. 49 Speeding Up the Time Sometimes entire an branch can be pruned off. •Perhaps because these solutions are not valid or not highly valued. •Or because there is at least one optimal solution elsewhere in the tree. •A Greedy algorithm prunes off all branches except the one that looks best. 50 Speeding Up the Time Memoization •Remembers the solutions for the sub-instances so that if ever it needs to be solved again, the answer can be used. •This effectively prunes off this later branch of the classification tree. 51 Exponential Time Redoing Work “Which is the best path from v7 to t?” How many friends solve this sub-instance? 52 Exponential Time Redoing Work “Which is the best path from s to t?” 53 Exponential Time Redoing Work “Which is the best path from v1 to t?” 54 Exponential Time Redoing Work “Which is the best path from v4 to t?” 55 Exponential Time Redoing Work “Which is the best path from v7 to t?” There’s one. 56 Exponential Time Redoing Work “Which is the best path from s to t?” 57 Exponential Time Redoing Work “Which is the best path from v3 to t?” 58 Exponential Time Redoing Work “Which is the best path from v5 to t?” 59 Exponential Time Redoing Work “Which is the best path from v7 to t?” There’s another. 60 Exponential Time Redoing Work “Which is the best path from v7 to t?” How many friends solve this sub-instance? Once for each path to v7 Waste time redoing work Save time by only doing once. 61 Depth First Search Drop bread crumbs and don’t revisit. But we need shortest path 62 Dynamic Programming Having many friends solving this same sub-instance is a waste of time. We allocate one friend to the job. 63 Dynamic Programming It is my job to learn and remember the optSol to my sub-Instance i.e. the best path from v7 to t 64 Dynamic Programming When I need to find the best path from v4 to t I will ask you for the best path from v7 to t I will find my best path and tell you. 65 Dynamic Programming When I need to find the best path from v2 to t I will ask you for the best path from v7 to t I remember my best path and will tell you. 66 Dynamic Programming When I need to find the best path from v5 to t I will ask you for the best path from v7 to t I remember my best path and will tell you. 67 Dynamic Programming Avoid waiting. When I need to find the best path from v2 to t I will ask you for the best path from v7 to t I will find my best path and tell you. But I hate to wait for you. Recursion has a lot of overhead Why don’t you go first? 68 Dynamic Programming Before anyone asks me, I will find my best path and remember it. 69 Set of Sub-Instances But what sub-instance need to be solved and in which order? Given an instance I, Imagine running the recursive algorithm on it. Determine the complete set of sub-Instances ever given to you, your friends, their friends, … 70 Set of Sub-Instances Guess the complete set S of sub-Instances. “Best path from v7 to t?” Yes “Best path from v21 to t?” No v21 is not a part of our v21 original instance. 71 Set of Sub-Instances Guess the complete set S of sub-Instances. “Best path from v7 to t?” Yes “Best path from v21 to t?” No “Best path from v3 to v7?” No All paths considered end in t. 72 Set of Sub-Instances Guess the complete set S of sub-Instances. “Best path from v7 to t?” Yes “Best path from v21 to t?” No “Best path from v3 to v7?” No All paths considered end in t. 73 Set of Sub-Instances Guess the complete set S of sub-Instances. “Best path from v7 to t?” Yes “Best path from v21 to t?” No “Best path from v3 to v7?” No i “Best path from vi to t?” Yes 74 Set of Sub-Instances Guess the complete set S of sub-Instances is i “Best path from vi to t?” Assign one friend to each sub-Instance. 75 Set of Sub-Instances Guess the complete set S of sub-Instances is i “Best path from vi to t?” The set S of sub-Instances needs to: •include our given I 76 Set of Sub-Instances Guess the complete set S of sub-Instances is i “Best path from vi to t?” The set S of sub-Instances needs to: •include our given I •closed under “friend” operation Integers closed under addition x,y I x+y I sub-Instance S subsub-Instance S 77 Set of Sub-Instances Guess the complete set S of sub-Instances is i “Best path from vi to t?” The set S of sub-Instances needs to: •include our given I •closed under “friend” operation •each sub-Instance needs to be asked of some friend, friend, … 78 Set of Sub-Instances Guess the complete set S of sub-Instances is i “Best path from vi to t?” The set S of sub-Instances needs to: •include our given I •closed under “friend” operation •each sub-Instance needs to be asked of some friend, friend, … A fine set of sub-instances! 79 Order to complete The complete set S of sub-Instances is i “Best path from vi to t?” In what order should they go? •in an order such that no friend must wait. •from “smallest” to “largest” For this problem, the order relies on the graph being “leveled.” 80 Order to complete The complete set S of sub-Instances is i “Best path from vi to t?” In what order should they go? •in an order such that no friend must wait. •from “smallest” to “largest” First Base Case easy Last Instance to be solved. 81 Dynamic Programming "Which is the best path from t to t?" Base Case easy 82 Dynamic Programming "Which is the best path from v8 to t?" easy 83 Dynamic Programming "Which is the best path from v7 to t?" easy 84 Dynamic Programming "Which is the best path from v6 to t?" easy 85 Dynamic Programming "Which is the best path from v5 to t?" Harder 86 Dynamic Programming "Which is the best path from v5 to t?" Little bird suggests first edge <v5,v7> Friend gives best path <v7,t>. 87 Dynamic Programming "Which is the best path from v5 to t?" Little bird suggests first edge <v5,v8> Friend gives best path <v8,t>. 88 Dynamic Programming "Which is the best path from v5 to t?" Take best of best 89 Dynamic Programming "Which is the best path from v4 to t?" 90 Dynamic Programming "Which is the best path from v4 to t?" Little bird suggests first edge <v4,v6> Friend gives best path <v7,t>. 91 Dynamic Programming "Which is the best path from v4 to t?" Little bird suggests first edge <v4,t> Friend gives best path <t,t>. 92 Dynamic Programming "Which is the best path from v4 to t?" Little bird suggests first edge <v4,v7> Friend gives best path <v7,t>. 93 Dynamic Programming "Which is the best path from v4 to t?" Take best of best 94 Dynamic Programming "Which is the best path from v3 to t?" 95 Dynamic Programming "Which is the best path from v3 to t?" Little bird suggests first edge <v3,v5> Friend gives best path <v5,t>. 96 Dynamic Programming "Which is the best path from v3 to t?" Little bird suggests first edge <v3,v8> Friend gives best path <v8,t>. 97 Dynamic Programming "Which is the best path from v3 to t?" Take best of best 98 Dynamic Programming "Which is the best path from v2 to t?" 99 Dynamic Programming "Which is the best path from v2 to t?" Little bird suggests first edge <v2,v4> Friend gives best path <v4,t>. 100 Dynamic Programming "Which is the best path from v2 to t?" Little bird suggests first edge <v2,v7> Friend gives best path <v7,t>. 101 Dynamic Programming "Which is the best path from v2 to t?" Take best of best 102 Dynamic Programming "Which is the best path from v1 to t?" 103 Dynamic Programming "Which is the best path from v1 to t?" Little bird suggests first edge <v1,v3> Friend gives best path <v3,t>. 104 Dynamic Programming "Which is the best path from v1 to t?" Little bird suggests first edge <v1,v4> Friend gives best path <v4,t>. 105 Dynamic Programming "Which is the best path from v1 to t?" Little bird suggests first edge <v1,v5> Friend gives best path <v5,t>. 106 Dynamic Programming "Which is the best path from v1 to t?" Take best of best 107 Dynamic Programming "Which is the best path from s to t?" Original Problem 108 Dynamic Programming "Which is the best path from s to t?" Little bird suggests first edge <s,v1> Friend gives best path <v1,t>. 109 Dynamic Programming "Which is the best path from s to t?" Little bird suggests first edge <s,v2> Friend gives best path <v2,t>. 110 Dynamic Programming "Which is the best path from s to t?" Little bird suggests first edge <s,v3> Friend gives best path <v3,t>. 111 Dynamic Programming "Which is the best path from s to t?" Little bird suggests first edge <s,v4> Friend gives best path <v4,t>. 112 Dynamic Programming "Which is the best path from s to t?" Take best of best DONE 113 Dynamic Programming Construct a table •for storing an optimal solution & cost •for each sub-instance. Map Sub-Instances i “Best path from vi to t?” Indexes i ϵ [n], i.e. for each node vi Cell of table “Which is the best path from vi to t?” t, v , v , v , v , …., i s 114 Dynamic Programming Fill out a table containing an optimal solution for each sub-instance. “Which is the best path from vi to t?” t, v8, v7, v6, v5, …., s Base case Original 115 Communication Friend k gives friend i a best path from vk to t. Recursive BackTracking i <optSubSol,optSubCost> = LeveledGraph(<G,vk,t>) i k return(optSolmin,optCostmin) , Dynamic Programming k k optSol[k] = optSolmin i optSubSol = optSol[k] optSolk = <vi,vk> + optSol[k] 116 Dynamic Programming code always has this same basic structure. 117 Be clear what are •the instances •it’s solution •the cost of a solution. 118 Dynamic Programs do not recurse making the instance smaller and smaller. Instead, it up front determines the set S of all sub-instances that ever need to be solved. Be clear what sub-instances are. 119 Be clear what sub-instances are. How are they indexed? Tables indexed by these sub-instances store an optimal solution and it’s cost. 120 The set S of sub-instances are solved from smallest to largest so that no body waits. Base Cases: Instances that are too small to have smaller instances to give to friends. They get solved first and their solutions stored. 121 Then we iterate through the remaining sub-instances. From smallest to largest. Each gets solved and their solutions stored. 122 Consider yourself to be a friend working on one of these. Be clear which sub-instance is yours. Solve this as you did before. 123 Loop through the bird answers. Be clear which is the current one being tried. 124 Give the bird & friend algorithm as a comment. (Unless it is in an earlier question.) 125 What is the bird asked? What does she answer? 126 k k k k Get help from friend Be clear what sub-instance you give your friend. 127 k k k k Get help from friend Instead of recursing, we simply look in the table for the solution. Because his instance is smaller, he has already solved it and stored sol in the table. 128 How do you form your solution from the friend’s and from the bird’s? 129 How do you form your cost from the friend’s and from the bird’s? 130 optSol<i,k> is a best solution for our instance subI[i] from amongst those consistent with the bird's kth answer. Take the best of the best 131 Store the solution to our instance subI[i] in the table. 132 Base Cases: Instances that are too small to have smaller instances to give to friends. Is this code correct? 133 Dynamic Programs do not recurse making the instance smaller and smaller. Hence, lets not worry about our instance I being a base case. 134 But there is a table of subinstances that must be solved. Some of these will be base cases and their solutions must be stored in the table. 135 But there is a table n t= of subinstances n n n that must be solved. Some of these will be base cases and their solutions must be stored in the table. 136 But there is a table of subinstances that must be solved. Then we solve the rest. 137 0 s= 0 n 0 Return the solution and cost for the original instance.138 Order Feels Backwards Path from s to t. Path from t to t. 139 An Esthetically Better Path from s to s. Path from s to t. 140 Reversing 141 Reversing Determine the complete set of sub-instances “Which is the best path from s to vi?” i 142 Reversing Fill out a table containing an optimal solution for each sub-instance. “Which is the best path from s to vi?” s, v1, v2, v3, v4, …., t Base case Original 143 144 Running Time Time = # of Sub-Instances × # of Bird Answers × q(1) ? =n×d 145 Communication Time optSolk = <optSol[k],vi> Friend k gives best path from s to vk to friend i, who adds the edge <vk,vi>. Time = q(1) ? k Size of path = q(n). Time = q(n) i 146 Running Time Time = # of Sub-Instances × # of Bird Answers × size of solution = q(n × d × n) Store path costs, not paths Space = # of Sub-Instances × q(1) = q(n) 147 Store Path Costs, not Paths "What is cost of the best path from s to v7?" 148 Store Path Costs, not Paths "What is cost of the best path from s to v7?" Little bird suggests last 8 edge <v4,v7> with weight 2. Friend gives cost 8 of best path <s,v4>. Best cost via <v4,v7> is 8+2=10. 149 Store Path Costs, not Paths "What is cost of the best path from s to v7?" 2 Little bird suggests last edge <v2,v7> with weight 7. Friend gives cost 2 of best path <s,v2>. Best cost via <v2,v7> is 2+7=9. 150 Store Path Costs, not Paths "What is cost of the best path from s to v7?" Little bird suggests last 6 edge <v5,v7> with weight 5. Friend gives cost 6 of best path <s,v5>. Best cost via <v5,v7> is 6+5=11. 151 Store Path Costs, not Paths "What is cost of the best path from s to v7?" 9 2 Take best of best: 10, 9, 11 We also learn the wise little bird’s advice. We will store this in the table too. 152 Running Time birdsAdvice[i] = kmin 153 Leave these lines as comments for extra clarity for the reader 154 Find Optimal Path Previous algorithm gives: • Cost of the best path from s to vi, i. • Bird’s advice of last edge to vi. We run the bird-friend algorithm again, but with a reliable bird. 155 Find Optimal Path The bird gives that the last edge of the best path from s to t is <v8,t>. 156 Find Optimal Path The bird gives that the last edge of the best path from s to v8 is <v5,v8>. 157 Find Optimal Path The bird gives that the last edge of the best path from s to v5 is <v3,v5>. 158 Find Optimal Path The bird gives that the last edge of the best path from s to v3 is <s,v3>. 159 Find Optimal Path Done! 160 Find Optimal Path This could be done iteratively. As an exercise, design it. 161 Multiple Optimal Solutions I ask the bird: “Which is the last edge?” She could give either answer. By giving this edge she says “There exists an optimal solution consistent with this answer.” Similar to greedy proof. 6 162 Multiple Optimal Solutions I ask the bird: “Which is the last edge?” We try all the bird answers. When we try this bird answer, we find this best solution. When we try this bird answer, we find this best solution. When we take best of best, we choose between them. 6 163 Review Designing Recursive Back Tracking Algorithm •What are instances, solutions, and costs? •Given an instance I, •What question do you ask the little bird? •Given a bird answer k [K], •What instance sub-Instance do your give your friend? •Assume he gives you optSubSol for subI. •How do you produce an optSol for I from •the bird’s k and •the friend’s optSubSol? •How do you determine the cost of optSol from •the bird’s k and •the cost of the friend’s optSubSol? •Try all bird’s answers and take best of best. 164 Review Recursive Back Tracking Algorithm Dynamic Programming Algorithm •Given an instance I, •Imagine running the recursive alg on it. • Determine the complete set of sub-Instances ever given to you, your friends, their friends, … • Build a table indexed by these sub-Instances •Fill in the table in order so that nobody waits. •the cost of its optimal solution •advice given by the bird •Run the recursive alg with bird’s advice to find the solution to your instance. 165 The Question For the Little Bird Purpose of Little Bird: •An abstraction from which it is easier to focus on the difficult issues. •Her answers give us a list of things to try. •Temporarily trusting the bird, helps us focus on the remaining question helping us formulate sub-instance for friend. •Coming up with which question is one of the main creative steps. •Hint: Ask about a local property •There are only so many question that you might ask so just try them all. 166 The Question For the Little Bird An instance: Graph, s, and t A solution: a path t s I ask the bird: “What is the first edge in the path?” The Dynamic Programming reverses the recursive backtracking algorithm. Hence, to end up with a “forward order”, we first reverse the recursive backtracking algorithm.167 The Question For the Little Bird An instance: Graph, s, and t A solution: a path t s I ask the bird: “What is the last edge in the path?” The Dynamic Programming reverses the recursive backtracking algorithm. Hence, to end up with a “forward order”, we first reverse the recursive backtracking algorithm.168 The Question For the Little Bird An instance: Graph, s, and t A solution: a path t s I ask the bird: “What is the last edge in the path?” A good question for the bird leaves you with a good recursive sub-instance to ask your friend. “What is the rest of the path?” 169 The Question For the Little Bird An instance: Graph, s, and t A solution: a path t s I ask the bird: “What is the last edge in the path?” Giving a good follow up question for your friend to ask the bird. “What is the second last edge in the path?” 170 The Question For the Little Bird • You can only ask the bird a little question. – Together with your question, you provide the little bird with a list A1, A2, …, AK of possible answers. – The little bird answers, k [1..K]. – For an efficient algorithm, K must be small. •Eg. “What is best last edge?” – K = number of edges into node t. t s K 171 The Question For the Little Bird • You can only ask the bird a little question. – Together with your question, you provide the little bird with a list A1, A2, …, AK of possible answers. – The little bird answers, k [1..K]. – For an efficient algorithm, K must be small. •Eg. “What is an optimal solution?” – K = # of solutions. Trying all is the Brute Force algorithm. 172 The Question For the Little Bird An instance: Graph, s, and t A solution: a path t s I ask the bird: “How many edges are in the path?” Bad Question: •it is not a local property •How does this help us solve the problem? •What is a good follow up question for the friend to ask? 173 The Question For the Little Bird An instance: ??? A solution: a sequence of objects Z=abcd I ask the bird: “What is the last object in the sequence?” # of answers K = # of possible last objects. I ask my friend: “What is the rest of the solution?” 174 The Question For the Little Bird An instance: a sequence of objects X =asbefchda X= a s b e f c h d a A solution: a subset of these objects Z=abcd I ask the bird: “What is the last object in the sequence?” # of answers K = # of possible last objects. Is there a smaller question that we could ask? 175 The Question For the Little Bird An instance: a sequence of objects X=asbefchda A solution: a subset of these objects Z=abcd I ask the bird: “Is the last object of the instance included in the optimal solution?” # of answers K = 2, Yes or No 176 The Question For the Little Bird An instance: ??? A solution: a binary tree of objects 38 25 51 17 31 42 63 I ask the bird: 4 21 28 35 40 49 55 71 “What object is at the root?” I ask my friend: I ask a second friend: “What is the left sub-tree?”“What is the right sub-tree?” Previous problems had one friend given a bird ans.177 The Question For the Little Bird An instance: ??? A solution: a binary tree of objects 38 25 51 17 31 42 63 I ask the bird: 4 21 28 35 40 49 55 71 “What object is at a leaf?” Bad Question: •How does this help us solve the problem? •What is a good follow up question for the friend to ask? 178 Printing Neatly 179 Printing Neatly An instance: text to print neatly & # chars per line “Love life man while there as we be”, 11 A solution: # of words to put on each line. The cost: a measure of how neat, few blanks on the end of each line. 3 = 8 Love.life.. 2 small punishment 3 = 8 man.while.. 2 3 = 216 big punishment there...... 6 3 = 27 as.we.be... 3 11 259 The goal is to to print it as “neatly” as possible. 180 Brute Force Algorithm Try all ways to print, return the best. But there may be an exponential number of ways to! love.life.. man........ love....... life.man... love....... life.man... love.life.. man........ 181 Bird & Friend Algorithm An instance:“Love life man while there as we be”, 11 I ask the bird: “How many words on the last line?” She may answer 3. I ask the friend: I combine “Which is the best way to print bird’s and the remaining n-3 words?” friend’s answers. 182 Bird & Friend Algorithm An instance:“Love life man while there as we be”, 11 Even if the bird was wrong, this work is not wasted. This is best way to print from amongst those ending in 3 words. We try the bird answers 12words, 34 5 and take best of best. 183 Same as Brute Force Algorithm I try each # words on last line. A friend tries # on next. A friend tries # on next. Time? Same as the brute force algorithm that tries each path. 184 Memoization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 185 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, “Love life man while there as we be”, 11 •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … 186 Set of Sub-Instances Guess the complete set of sub-Instances. “Love life man while there as we be”, 11 “Love life man while there”, 11 Yes “Hi there”, 81 No “man while”, 11 No This may appear on a line, but it will never be a sub-Instance for a friend. 187 Set of Sub-Instances The set of sub-Instances is the set of prefixes. “Love life man while there as we be”, 11 “Love life man while there as we”, 11 “Love life man while there as”, 11 “Love life man while there”, 11 “Love life man while”, 11 “Love life man”, 11 “Love life”, 11 “Love”, 11 The set S of sub-Instances needs to: “”, 11 •include our given I •closed under “friend” operation sub-Instance S 188 Set of Sub-Instances The set of sub-Instances is the set of prefixes. “Love life man while there as we be”, 11 “Love life man while there as we”, 11 “Love life man while there as”, 11 “Love life man while there”, 11 “Love life man while”, 11 “Love life man”, 11 “Love life”, 11 “Love”, 11 The set S of sub-Instances needs to: “”, 11 •include our given I •closed under “friend” operation The bird sub-Instance S answers 12words, subsub-Instance i 34 5 189 Set of Sub-Instances The set of sub-Instances is the set of prefixes. “Love life man while there as we be”, 11 “Love life man while there as we”, 11 “Love life man while there as”, 11 “Love life man while there”, 11 “Love life man while”, 11 “Love life man”, 11 “Love life”, 11 “Love”, 11 The set S of sub-Instances needs to: “”, 11 •include our given I •closed under “friend” operation •each sub-Instance needs to be asked of some friend, friend, … 190 Set of Sub-Instances The set of sub-Instances is the set of prefixes. “Love life man while there as we be”, 11 “Love life man while there as we”, 11 “Love life man while there as”, 11 “Love life man while there”, 11 “Love life man while”, 11 The set S of sub-Instances needs to: •include our given I •closed under “friend” operation The bird •each sub-Instance needs to be answers 1. asked of some friend, friend, … A fine set of sub-instances! 191 Set of Sub-Instances The set of sub-Instances is the set of prefixes. “Love life man while there as we be”, 11 “Love life man while there as we”, 11 “Love life man while there as”, 11 “Love life man while there”, 11 “Love life man while”, 11 “Love life man”, 11 “Love life”, 11 “Love”, 11 In what order should they go? “”, 11 •in an order such that no friend must wait. •from “smallest” to “largest” First Base Case easy Last Instance to be solved. 192 The Table Construct a table •for storing the cost of opt sol and bird’s advice. •for each sub-instance. Map Sub-Instances The set of prefixes of words. Indexes i ϵ [n], i.e. for each word. Cell of table “Which is the best printing of first i words?” i 193 Dynamic Programming Fill out a table containing an optimal solution for each sub-instance. “Which is the best printing of first i words?” Base case Original 194 “Love life man while there as we be”, 11 The 5th sub-instance is “Love life man while there”, 11 5 words with 4, 4, 3, 5, 5 letters. 195 “Love life man while there as we be”, 11 The 5th sub-instance is Love.life.. “Love life man while there”, 11 man.while.. Its solution is there...... with 2,2,1 words on each line. The bird’s advice is 1 word on last. Solution’s cost is 23 + 23 +63 = 232 196 “Love life man while there as we be”, 11 Assume the table is filled in so far. We will work to fill in the last line 197 “Love life man while there as we be”, 11 Love.life.. man.while.. there.as.we be......... 198 “Love life man while there as we be”, 11 Love.life.. man.while.. there.as... we.be...... 199 “Love life man while there as we be”, 11 Love.life.. man.while.. there...... as.we.be... 200 “Love life man while there as we be”, 11 there.as.we.be 201 “Love life man while there as we be”, 11 Tried all bird answers. Choose best of the best. 202 “Love life man while there as we be”, 11 Choose best of the best. 203 Dynamic Programming code always has this same basic structure. Amusingly, when formatting this code, I had to fight with line breaks to get the height/width ratio Printing Neatly. 204 Be clear what are •the instances •it’s solution •the cost of a solution. 205 Dynamic Programs do not recurse making the instance smaller and smaller. Instead, it up front determines the set S of all sub-instances that ever need to be solved. Be clear what sub-instances are. 206 Be clear what sub-instances are. How are they indexed? Tables indexed by these sub-instances store an optimal solution and it’s cost. 207 The set S of sub-instances are solved from smallest to largest so that no body waits. Base Cases: Instances that are too small to have smaller instances to give to friends. They get solved first and their solutions stored. 208 Then we iterate through the remaining sub-instances. From smallest to largest. Each gets solved and their solutions stored. Actually, we store the bird’s advice instead of the solution. 209 Consider yourself to be a friend working on one of these. Be clear which sub-instance is yours. Solve this as you did before. 210 Loop through the bird answers. Be clear which is the current one being tried. 211 Give the bird & friend algorithm as a comment. (Unless it is in an earlier question.) 212 What is the bird asked? What does she answer? 213 i-k i-k i-k Get help from friend Be clear what sub-instance you give your friend. 214 i-k i-k i-k Instead of recursing, we simply look in the table for the solution. Because his instance is smaller, he has already solved it and stored sol in the table. 215 How do you form your solution from the friend’s and from the bird’s? 216 How do you form your cost from the friend’s and from the bird’s? 217 optSol<i,k> is a best solution for our instance subI[i] from amongst those consistent with the bird's kth answer. Take the best of the best 218 Store the solution to our instance subI[i] in the table. Actually, we store the bird’s advice instead of the solution. 219 Base Cases: Instances that are too small to have smaller instances to give to friends. Is this code correct? 220 Dynamic Programs do not recurse making the instance smaller and smaller. Hence, lets not worry about our instance I being a base case. 221 But there is a table of subinstances that must be solved. Some of these will be base cases and their solutions must be stored in the table. 222 But there is a table of subinstances that must be solved. Some of these will be base cases and their solutions must be stored in the table. 223 But there is a table of subinstances that must be solved. Then we solve the rest. 224 Return the solution and cost for the original instance. But actually, we don’t have the solution. We must rerun it, this time with advice from the bird. 225 Time = # of Sub-Instances × # of Bird Answers = q(n × n) Space = # of Sub-Instances = q(n) 226 Find Optimal Path Previous algorithm gives cost and bird’s advice. We run the bird-friend algorithm again, but with a reliable bird. 227 “Love life man while there as we be”, 11 Love.life.. <2 ,2 ,1,3> man.while.. there…… as we be… 228 Longest Common Subsequence problem X=asbefchda Y=rtwabgjcktfd 229 Longest Common Subsequence problem An instance: Two strings X X=asbefchdat Y Y=rtwabgjcktfd A solution: A common subsequence. Z=abcd The cost: The length of Z. The goal is to find a longest common subsequence. 230 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers one of : •Last of X is not included •Last of Y is not included •Last of X is included •Last of Y is included •Neither are included •Both are included 231 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is not included I ask my friend: The instance: X=asbetchd Y=rtwabgjcktfd 232 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is not included I combine My friend answers: bird’s and The instance:c d Z=ab friend’s answers X=asbetchd and give the same Z. Y=rtwabgjcktfd 233 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of Y is not included I ask my friend: The instance: X=asbetchda Y=rtwabgjcktf 234 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: I combine •Last of Y is not included bird’s and My friend answers: friend’s answers Z=abc The instance: and give the same Z. X=asbe chda X = a s b e tt c h d a Not as good as last Y= wabg ck Y = rr tt w a b g jj c k tt ff but we need to try.235 Bird & Friend Algorithm An instance: X=asbetchd Last chars equal Y=rtwabgjckdfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X and last of Y are both included I ask my friend: The instance: X=asbetch Y=rtwabgjckdf 236 Bird & Friend Algorithm An instance: X=asbetchd Last chars equal Y=rtwabgjckdfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X and last of Y are both included I combine My friend answers: bird’s and Z=abc The instance: friend’s answers X=asbetch and give Y=rtwabgjckdf Zd = abcd. 237 Bird & Friend Algorithm An instance: X=asbetchda Last chars not equal Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X and last of Y are both included I politely tell her that she is wrong. 238 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is included I ask my friend: The instance: X=asbetchd Y=rtwabgjcktfd 239 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is included My friend answers: I combine The instance:c d Z=ab bird’s and Wrong X=asbetchd friend’s answers Y=rtwabgjcktfd and give 240 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is included I ask my friend: The instance: X=asbetchd Y=rtw 241 Bird & Friend Algorithm An instance: X=asbetchda Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers: •Last of X is included I combine My friend answers: bird’s and Z=t The instance: friend’s answers X=asbetchd and give Y=rtw Za = ta. 242 Bird & Friend Algorithm An instance: X=asbetchda Last chars not equal Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers one of : Can we eliminate •Last of X is not included some of her answers? •Last of Y is not included Given any optSol •Last of X is included ? she needs to have •Last of Y is included ? a valid answer. •Neither are included ? •Both are included 243 Bird & Friend Algorithm An instance: X=asbetchda Last chars not equal Y=rtwabgjcktfd I ask the bird: “Is the last character of either X or Y included in Z?” She answers one of : •Last of X is not included •Last of Y is not included •Last of X is included •Last of Y is included •Neither are included # of answers K = 3 •Both are included 244 Same as Brute Force Algorithm I try each of 3 bird ans. My friends try 3 His friends try 3 Time? Same as the brute force algorithm that tries each solution. 245 Memorization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 246 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, X=asbetchda Y=rtwabgjcktfd •Imagine running the recursive alg on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends… Is this a sub-Instance? X’ = a s b e t c Yes Y’ = r t w a b g j c k 247 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, X=asbetchda Y=rtwabgjcktfd •Imagine running the recursive alg on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … Is this a sub-Instance? X’ = b e t No Y’ = a b g j c k 248 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, X=asbetchda Y=rtwabgjcktfd •Imagine running the recursive alg on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … Is this a sub-Instance? Yes X’ = x1,…xi Y’ = y1,…,yj i [0..|X|] j [0..|Y|] |X| × |Y| of these. 249 Set of Sub-Instances Guess the complete set S of sub-Instances. Xi = x1,…xi Yj = y1,…,yj i [0..|X|] j [0..|Y|] The set S of sub-Instances needs to: •include our given I Yes: i = |X| & j = |Y| 250 Set of Sub-Instances Guess the complete set S of sub-Instances. Xi = x1,…xi Yj = y1,…,yj i [0..|X|] j [0..|Y|] The set S of sub-Instances needs to: •include our given I •closed under “friend” operation sub-Instance S subsub-Instance S Xi = x1,…xi Xi-1 = x1,…xi-1 Yj = y1,…,yj S Xi = x1,…xi Yj-1 = y1,…,yj-1 Yj = y1,…,yj Xi-1 = x1,…xi-1 Yj-1 = y1,…,yj-1 S 251 Set of Sub-Instances Guess the complete set S of sub-Instances. Xi = x1,…xi Yj = y1,…,yj i [0..|X|] j [0..|Y|] The set S of sub-Instances needs to: •include our given I •closed under “friend” operation sub-Instance S subsub-Instance S •each sub-Instance needs to be asked of some friend, friend, … We showed this. This is a fine set of sub-Instances! 252 The Table Construct a table •for storing the cost of opt sol and bird’s advice. •for each sub-instance. Map Sub-Instances Xi = x1,…xi Yj = y1,…,yj Indexes i [0..|X|] j [0..|Y|] Cell of table “LCS of x1,…xi and y1,…,yj ?” j i 253 The Table 254 Table Y Original instance I = <X,Y> X 255 Table Yj j= sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Optimal Solution = Longest Common i= Subsequence Cost = length of LCS. 256 Table Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Optimal Solution = Longest Common Subsequence Bird’s Advice •delete xi 257 Table Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Optimal Solution = Longest Common Subsequence Bird’s Advice •delete xi •take both xi and yj 258 Table Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Optimal Solution = Longest Common Subsequence Bird’s Advice •delete xi •delete yj •take both xi and yj 259 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •delete xi Friend’s sub-Instance Our cost 5 = friend’s cost 260 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •delete yj Friend’s sub-Instance Our cost 5 = friend’s cost 261 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •take both xi and yj Friend’s sub-Instance Our cost 6 = friend’s cost +1 262 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •Take best of best 6 263 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •delete xi Friend’s sub-Instance Our cost 4 = friend’s cost 264 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •delete yj Friend’s sub-Instance Our cost 3 = friend’s cost 265 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •take both xi and yj Sorry bird is wrong. - Our cost = - 266 Fill in Box Yj sub-Instancei,j = Xi = x1,…xi Xi Yj = y1,…,yj Fill in box •Try all bird’s ans. •Take best of best 4 267 Fill in Box 268 Fill in Box 269 Order to Fill in Table Order to fill table: •so that nobody waits This guy waits for 270 Order to Fill in Table (later) 271 Base Cases Base Cases: •general algorithm does not work •This guy’s friends are 272 Base Cases Base Cases: •general algorithm does not work 273 Base Cases 274 With Advice 275 With Advice Done 276 Knapsack Problem Get as much value as you can into the knapsack 277 Knapsack Problem Ingredients: •Instances: The volume V of the knapsack. The volume and price of n objects <<v1,p1>,<v2,p2>,… ,<vn,pn>>. •Solutions: A set of objects that fit in the knapsack. •i.e. i S vi V •Cost of Solution: The total value of objects in set. •i.e. i S pi •Goal: Get as much value as you can into the knapsack. 278 Greedy Algorithm Greedy Criteria: Most valuable pi v=4,p=4 v=7,p=5 v=4,p=4 v=4,p=4 v=4,p=4 V=8 V=8 Greedy give 5 Optimal gives 8 279 Greedy Algorithm Greedy Criteria: Most dense in value pi vi v=7,p=5 v=4,p=4 v=4,p=4 V=8 v=7,p=5 V=7 V=7 Greedy give 4 Optimal gives 5 280 Greedy Algorithm Greedy Criteria: Most dense in value pi vi If fractional solutions are allowed. Works Often an Integer solution is MUCH v=4,p=4 harder to find. ¾ of v=4,p=4 v=7,p=5 V=7 V=7 Greedy give 4 + ¾ × 4 = 7 = Optimal gives 5 281 Bird & Friend Algorithm My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 A solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>>. I ask the bird: “What is the last object to take?” # of answers K = n 282 Bird & Friend Algorithm My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 A solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>>. I ask the bird: “Do we keep the last object?” # of answers K = 2 Yes & No 283 Bird & Friend Algorithm My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 Bird says, Yes keep the last object. Trust her and put it into your knapsack. I ask my friend: To fill the rest of the knapsack. But what instance do I give him? 284 Bird & Friend Algorithm His instance: <V-vn:<v1,p1>,<v2,p2>,.........<vn-1,pn-1>,<vn,pn>>. V=9-4 v=7,p=5 v=4,p=4 v=4,p=4 His solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>>. My solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>,<vn,pn>> My cost: same + pn 285 Bird & Friend Algorithm My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 If we trust the bird and friend, this is valid and optimal. My solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>,<vn,pn>> My cost: same +pn 286 Bird & Friend Algorithm My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 Bird says, No do not keep the last object. Trust her and delete it. I ask my friend: To fill the knapsack with the rest. What instance do I give him? 287 Bird & Friend Algorithm His instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. V=9 v=7,p=5 v=4,p=4 v=4,p=4 His solution: <<v5,p5>,<v9,p9>,...........,<v82,p82>>. My solution: same If we trust the bird and friend, My cost: same this is valid and optimal. 288 Same as Brute Force Algorithm I try each of 2 bird ans. My friends tries 2 His friends tries 2 Time? Same as the brute force algorithm that tries each solution. 289 Memoization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 290 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, <V:<v1,p1>,<v2,p2>,<v3,p3>,<v4,p4>,<v5,p5>,<v6,p6>>. •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … Is this a sub-Instance? <V:<v1,p1>,<v2,p2>,<v3,p3>>. Yes, if the bird keeps saying “No”. 291 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, <V:<v1,p1>,<v2,p2>,<v3,p3>,<v4,p4>,<v5,p5>,<v6,p6>>. •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … Is this a sub-Instance? <V:<v1,p1>,<v2,p2>,<v3,p3>,<v4,p4>,<v5,p5>,<v6,p6>>. No, the set of objects is always a prefix of the original set. 292 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, <V:<v1,p1>,<v2,p2>,<v3,p3>,<v4,p4>,<v5,p5>,<v6,p6>>. •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … Quite possibly, if V’ V. It is easier to solve than to Is this a sub-Instance? determine if it is a sub-instance. <V’:<v1,p1>,<v2,p2>,<v3,p3>>. 293 Set of Sub-Instances My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. Guess the complete set S of sub-Instances. V’ [0..V] <V’:<v1,p1>,<v2,p2>,......,<vi,pi>>. i [0..n] The set S of sub-Instances needs to: •include our given I Yes: V’=V & i = n •closed under “friend” operation sub-Instance S subsub-Instance S <V’:<v1,p1>,<v2,p2>,......,<vi,pi>> S No <V’ :<v1,p1>,<v2,p2>,...,<vi-1,pi-1>> S Yes <V’-vi:<v1,p1>,<v2,p2>,...,<vi-1,pi-1>> 294 The Table Construct a table •for storing the cost of opt sol and bird’s advice. •for each sub-instance. Map Sub-Instances <V’:<v1,p1>,<v2,p2>,......,<vi,pi>>. Indexes V’ [0..V] i [0..n] “Which of first i objects Cell of table to put in a knapsack of size v’?” v’ i 295 The Table The complete set S of sub-Instances. V’ [0..V] <V’:<v1,p1>,<v2,p2>,......,<vi,pi>>. i [0..n] 0 1 2 V’-vi V’ V OptSol Cost 0 & 1 Bird’s Advice for this 2 sub-Instance same + pi No i-1 same Yes i Take best Our cost? n of best. 296 The Table The complete set S of sub-Instances. V’ [0..V] <V’:<v1,p1>,<v2,p2>,......,<vi,pi>>. i [0..n] 0 1 2 V’-vi V’ V OptSol Cost 0 & 1 Bird’s Advice for this 2 sub-Instance i-1 Order to fill i so nobody waits? n 297 The Code 298 299 300 Running Time My instance: <V:<v1,p1>,<v2,p2>,..........................,<vn,pn>>. The complete set S of sub-Instances is V’ [0..V] <V’:<v1,p1>,<v2,p2>,......,<vi,pi>>. i [0..n] No Yes Running time = ( # of sub-instances × # bird answers ) = ( Vn × 2 ) = ( 2#bits in V × n ) Polynomial? Exponential in “size” in instance! 301 The Knapsack Problem •Dynamic Programming Running time = ( V × n ) = ( 2#bits in V × n ) •Poly time if size of knapsack is small •Exponential time if size is an arbitrary integer. 302 The Knapsack Problem •Dynamic Programming Running time = ( V × n ) = ( 2#bits in V × n ) •NP-Complete If there is a poly-time algorithm for the Knapsack Problem For EVERY optimization problem there is a poly-time algorithm. 303 The Knapsack Problem •Dynamic Programming Running time = ( V × n ) = ( 2#bits in V × n ) •NP-Complete Likely there is not a poly-time algorithm for the Knapsack Problem. Likely there is not a poly-time algorithm for EVERY optimization problem. 304 The Knapsack Problem •Dynamic Programming Running time = ( V × n ) = ( 2#bits in V × n ) •NP-Complete •Approximate Algorithm •In poly-time, solution can be found that is (1+) as good as optimal. done 305 The Job/Event Scheduling Problem Schedule as many events in your room as possible 306 The Job/Event Scheduling Problem Ingredients: •Instances: Events with starting and finishing times <<s1,f1>,<s2,f2>,… ,<sn,fn>>. •Solutions: A set of events that do not overlap. •Cost of Solution: The number of events scheduled. •Goal: Given a set of events, schedule as many as possible. 307 Greedy Algorithm Greedy Criteria: Earliest Finishing Time Motivation: Schedule the event which will free up your room for someone else as soon as possible. 308 Weighted Event Scheduling Ingredients: •Instances: Events with starting and finishing times and weights <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. •Solutions: A set of events that do not overlap. •Cost of Solution: Total weight of events scheduled. •Goal: Given a set of events, schedule max weight 309 Greedy Algorithm 1 100 1 Greedy Criteria: Earliest Finishing Time Motivation: Schedule the event which will free up your room for someone else as soon as possible. 310 Bird & Friend Algorithm An instance: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. A solution: <<s5,f5,w5>,<s9,f9,w9>,… ,<s82,f82,w82>>. I ask the bird: “What is the last event to take?” # of answers K = n 311 Bird & Friend Algorithm An instance: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. A solution: <<s5,f5,w5>,<s9,f9,w9>,… ,<s82,f82,w82>>. I ask the bird: “Do we keep the last event?” # of answers K = 2 Yes & No 312 Bird & Friend Algorithm An instance: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. I ask the bird: “Do we keep the last event?” She answers: No I ask my friend: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. His solution: <<s5,f5,w5>,<s9,f9,w9>,… ,<s82,f82,w82>>. My solution: same My cost: same 313 Bird & Friend Algorithm An instance: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. I ask the bird: “Do we keep the last event?” She answers: Yes I ask my friend: <<s1,f1,w1>,<s2,f2,w2>,… ,<sn,fn,wn>>. His solution: <<s5,f5,w5>,<s9,f9,w9>,… ,<s82,f82,w82>>. My solution: same + <sn,fn,wn>. My cost: same +wn 314 Bird & Friend Algorithm My instance: last event Bird answers: “Yes keep the last event.” Cant keep any events that overlap with it. 315 Bird & Friend Algorithm My instance: last event Bird answers: “Yes keep the last event.” I ask my friend: 316 Bird & Friend Algorithm His instance: His solution My solution Bird answers: “Yes keep the last event.” I ask my friend: My solution: same + <sn,fn,wn>. Valid? 317 Bird & Friend Algorithm My instance: My solution Bird answers: “Yes keep the last event.” I ask my friend: My solution: same + <sn,fn,wn>. Valid? Yes 318 Bird & Friend Algorithm My instance: My solution Bird answers: “Yes keep the last event.” I ask my friend: My solution: same + <sn,fn,wn>. My cost: same +wn 319 Same as Brute Force Algorithm I try each of 2 bird ans. My friends tries 2 His friends tries 2 Time? Same as the brute force algorithm that tries each solution. 320 Memorization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 321 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … 322 Every subset of {1,…,9} is a possible sub-Instance. I.e. could be an exponential number of them. Hence, running time is exponential. Greedy algorithm sorted jobs by finishing time. Let us do that too. 323 Each sub-Instance is prefix. I.e. only n of them. Hence, running time is polynomial! 324 Set of Sub-Instances My instance: <<s1,f1,w1>,<s2,f2,w2>,................… ,<sn,fn,wn>>. Guess the complete set S of sub-Instances. <<s1,f1,w1>,<s2,f2,w2>,… ,<si,fi,wi>> i [0..n] The set S of sub-Instances needs to: •include our given I Yes: i = n •closed under “friend” operation sub-Instance S subsub-Instance S ? •each sub-Instance needs to be asked of some friend, friend… Only n sub-Instances Good enough. 325 Set of Sub-Instances Show closed under “friend” operation sub-Instance S subsub-Instance S sub-Instance = last event <<s1,f1,w1>,<s2,f2,w2>,................................,<si,fi,wi>> Events sorted by finishing time. 326 Set of Sub-Instances Show closed under “friend” operation sub-Instance S subsub-Instance S sub-Instance = last event <<s1,f1,w1>,<s2,f2,w2>,................................,<si,fi,wi>> Bird answers: “Yes keep the last event.” Delete overlapping events for friend. 327 Set of Sub-Instances Show closed under “friend” operation sub-Instance S subsub-Instance S subsub-Instance = <<s1,f1,w1>,<s2,f2,w2>,.....,<sj,fj,wj>> Bird answers: “Yes keep the last event.” Delete overlapping events for friend. 328 Set of Sub-Instances Show closed under “friend” operation sub-Instance S subsub-Instance S subsub-Instance = typical kept job typical deleted job <<s1,f1,w1>,<s2,f2,w2>,.....,<sj,fj,wj>> Event j is kept fj si set of kept jobs is a prefix of events. subsub-Instance S 329 Set of Sub-Instances My instance: <<s1,f1,w1>,<s2,f2,w2>,................… ,<sn,fn,wn>>. The complete set S of sub-Instances is <<s1,f1,w1>,<s2,f2,w2>,… ,<si,fi,wi>> i [0..n] Table: 0, 1, 2, 3, 4, …. n Base case Original 330 Set of Sub-Instances My instance: <<s1,f1,w1>,<s2,f2,w2>,................… ,<sn,fn,wn>>. The complete set S of sub-Instances is <<s1,f1,w1>,<s2,f2,w2>,… ,<si,fi,wi>> i [0..n] Running time = # of sub-instances × # bird answers = n × 2 But to find your friend’s “yes” sub-instance you must know how many events overlap with your last event. This takes time: O(logn) using binary search Done for a total of O(nlogn) time. 331 Parsing Input: s=6*8+((2+42)*(5+12)+987*7*123+15*54) Output: 332 Parsing Recursive Alg: •GetExp calls GetTerm •GetTerm calls GetFact •GetFact calls GetExp 333 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Context Free Grammar (Not look ahead one) For ease, we will assume every non-terminal either goes to two non-terminals or to one terminal 334 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T start non-terminal = T C A string to parse = a1a2a3 ..... an A C B T = baeaadbda A A C B C A Output: A parsing B T A C A B b a e a a d b b d a335 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T C A Recursive Algorithm: GetT does not know A C B T whether to call GetA, GetC, or GetT. A A C B C A B T A C A B b a e a a d b b d a336 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask Little Bird: C A •For first rule Ask Friend A C B T •Parse left A A C B C A Ask Another Friend B T A C •Parse right. A B b a e a a d b b d a337 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask Little Bird: C A •For first rule Instance to give Friend •? b a e a a d b b d a338 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask Little Bird: C A •For first rule Want from Friend: A C •Left sub-parse tree. A A C B Instance to give him: •C baeaadb B T A C A B b a e a a d b b d a339 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask Little Bird: C A •For first rule How can we know split? A C B T •Ask the Bird! A A C B C A B T A C A B b a e a a d b b d a340 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask Little Bird: C A •For first rule # of ans K = mT = # of rules for T. •For the split. # of ans K = n = # chars in string. Total # of ans K = mT × n. b a e a a d b b d a341 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask left friend: C A •Instance: C baeaadb •Solution: Left parsing A C A A C B B T A C A B b a e a a d b b d a342 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Ask right friend: C A •Instance: A bda B T •Solution: Right parsing C A b a e a a d b b d a343 Parsing T AB A AA B TA C CB CA BT BC AC TT a b c e d Input: T a1a2a3 ..... an T Combine: C A •Instance: A C B T •Bird’s Answer •Left Friend’s Answer A A C B C A •Right Friend’s Answer B T A C A B b a e a a d b b d a344 345 Same as Brute Force Algorithm I try each of 2 bird ans. My friends tries 2 His friends tries 2 Time? Same as the brute force algorithm that tries each solution. 346 Memoization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 347 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … 348 Set of Sub-Instances Determine the complete set of sub-Instances. My instance I: T a1a2a3 ..... an T gives: C A My left sub-Instance. A C gives: His right sub-Instance. b a e a a d b b d a349 Set of Sub-Instances Determine the complete set of sub-Instances. My instance I: T a1a2a3 ..... an sub-Instances: T’ aiai+1 ..... aj • non-terminals T’ T’= C • i,j [1,n] # of sub-Instances = # of non-terminals × n2 a1...ai-1 aiai+1...aj aj+1...an a d b 350 The Table Construct a table •for storing the cost of opt sol and bird’s advice. •for each sub-instance. Map Sub-Instances T’ aiai+1 ..... aj Indexes T’ i,j [1,n] T’ i Cell of table j 351 352 Running Time Running time = ( # of sub-instances × # bird answers ) = ( # of non-terminals × n2 × # of rules · n ) sub-Instances: T’ aiai+1 ..... aj non-terminals T’ gives: First rule and split & i,j [1,n] Done 353 Find a Satisfying Assignment An instance (input) consists of a circuit: c = (x3 or x5 or x6) and (x2 or x5 or x7) and (x3 or x4) true true false true true true true true A solution is an assignment of the variables. x1 = 0, x2 = 1, x3 = 0, x4 = 0, x5 = 1, x6 = 0, x7 = 1 The cost of a solution is • 1 if the assignment satisfies the circuit. • 0 if not. The goal is to find satisfying assignment. 354 Find a Satisfying Assignment Instance: c = (x3 or x5 or x6) and (x2 or x5 or x7) and (x3 or x4) Ask the little bird Value of x1 in an optimal solution or even better Value of x3 in an optimal solution We will have to try both x3 = 0 and x3 = 1. For now, suppose she answered x3 = 0. 355 Find a Satisfying Assignment Instance: c = (x3 or x5 or x6) and (x2 or x5 or x7) and (x3 or x4) true false true Commit to x3 = 0 and simplify Sub-Instance: c= (x2 or x5 or x7) and x4 Friend gives Sub-Solution: x1 = 0, x2 = 1, x4 = 0, x5 = 1, x6 = 0, x7 = 1 Our Solution: x1 = 0, x2 = 1, x3 = 0, x4 = 0, x5 = 1, x6 = 0, x7 = 1 356 Speeding Up the Time x3 0 1 x2 x1 0 1 0 1 x1 x1 x2 x2 0 1 0 1 0 1 0 1 In the end, some friend looks at each of the 2n assignments, 357 Memoization Assign one friend to each sub-instance. “Which is the best path from vi to t?” i 358 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, •Imagine running the recursive algorithm on it. •Determine the complete set of sub-Instances ever given to you, your friends, their friends, … 359 Set of Sub-Instances Determine the complete set of sub-Instances. Given an instance I, c = (x1 or y1) and (x2 or y2) and (x3 or y3) and (x4 or y4) Is this a sub-Instance? c = (x1) and (x3) Commit to y1=0, y2=1, y3=0, y4=1, and simplify Yes True for any subset of the xi. could be an exponential # of different sub-Instances. running time is exponential. 360 Speeding Up the Time x3 0 1 x2 x1 0 1 0 1 x1 x1 x2 x2 0 1 0 1 0 1 0 1 In the end, some friend looks at each of the 2n assignments, 361 Speeding Up the Time x3 0 1 x2 x1 0 1 0 1 x1 x1 x2 x2 0 1 0 1 0 1 0 1 But sometimes we can prune off branches. 362 Find a Satisfying Assignment Instance: c = (x2 or x5 or x7) and x3 x3 is forced to x3 = 0 x3 0 1 x2 x1 0 1 0 1 x1 x1 x2 x2 0 1 0 1 0 1 0 1 363 Find a Satisfying Assignment Instance: c = (x2 or x5 or x7) and x3 and x3 This is trivially unsatisfiable because x3 can’t be both 0 and 1. x3 0 1 x2 x1 0 1 0 1 x1 x1 x2 x2 0 1 0 1 0 1 0 1 364 365 366 Review Designing Recursive Back Tracking Algorithm •What are instances, solutions, and costs? •Given an instance I, •What question do you ask the little bird? •Given a bird answer k [K], •What sub-Instance do your give your friend? •Assume he gives you optSubSol for sub-Instance. •How do you produce an optSol for I from •the bird’s k and •the friend’s optSubSol? •How do you determine the cost of optSol from •the bird’s k and •the cost of the friend’s optSubSol? •Try all bird’s answers and take best of best. 367 Review Recursive Back Tracking Algorithm Dynamic Programming Algorithm •Given an instance I, •Imagine running the recursive alg on it. • Determine the complete set of sub-Instances ever given to you, your friends, their friends, … • Build a table indexed by these sub-Instances •Fill in the table in order so that nobody waits. •the cost of its optimal solution •advice given by the bird •Run the recursive algorithm with bird’s advice to find the solution to your instance. 368 Optimization Problems • Don’t mix up the following – What is an instance – What are the objects in an instance – What is a solution – What are the objects in a solution – What is the cost of a solution • Greedy algorithm – What does the algorithm do & know – What does the Prover do & know – What does the Fairy God Mother do & know • Recursive Backtracking / Dynamic Programming – What does the algorithm do & know – What does the little bird do & know – What does the friend do & know 369 Dynamic Programming Don’ts •Yes, the code has a basic structure that you should learn. •But don’t copy other code verbatim •Don’t say if(ai = cj) (i.e. Longest Common Subsequence) when our problem does not have cj 370 Dynamic Programming Don’ts •When looping over the sub-instances •be clear what the set of sub-instances are •which is currently being solved, i.e. which instance is cost(i,j)? •If you know that the set of sub-instances are the prefixes of the input, i.e. <a1,a2, …, ai>, then don’t have a two dimensional table. Table[1..n,1..n]. •Don’t loop over i and loop over j if j never gets mentioned again. 371 Dynamic Programming Don’ts •When trying all bird answers •be clear what the set of bird answers are, •which is currently being tried, •& what it says about the solution being looked for. •When getting help from your friend, •be clear what the sub-instance is that you are giving him •How do you use the current instance and the bird' s answer to form his sub-instance? •Don’t simply say cost(i-1,j-1) 372 Dynamic Programming Don’ts •Think about what the base cases should be. •Don’t make an instance a base cases if they can be solved using the general method. •% is used to start a comment. Don’t put it in front of code. 373 The Question For the Little Bird •Eg. The Best Binary Search Tree problem, –“Which key is at the root of the tree?” 38 25 51 17 31 42 63 4 21 28 35 40 49 55 71 • If a solution is a binary tree of objects, –“What object is at the root of the tree?” 374 Matrix Multiplication 375 376 377 378 379 All Pairs Shortest Paths 380 381 382 383 384 385 end 386 Dynamic Programming Construct a table •for storing an optimal solution & cost •for each sub-instance. Map Sub-Instances i “Best path from vi to t?” Index i ϵ [n], i.e. for each node vi Cell of table “Which is the best path from vi to t?” t, v , v , v , v , …., i s 387

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