DOCSLIB.ORG

Constrained Unscented Dynamic Programming

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constrained Unscented Dynamic Programming Constrained Unscented Dynamic Programming Brian Plancher, Zachary Manchester, and Scott Kuindersma Abstract— Differential Dynamic Programming (DDP) has be- using this control law to update the input trajectory. This come a popular approach to performing trajectory optimization process is repeated until convergence. for complex, underactuated robots. However, DDP presents two Second derivatives of the dynamics (rank-three tensors) practical challenges. First, the evaluation of dynamics deriva- tives during optimization creates a computational bottleneck, are required to compute the quadratic cost-to-go approxi- particularly in implementations that capture second-order dy- mations used in DDP. This computational bottleneck led to namic effects. Second, constraints on the states (e.g., boundary the development of the iterative Linear Quadratic Regulator conditions, collision constraints, etc.) require additional care (iLQR) algorithm [10], which uses only first derivatives of since the state trajectory is implicitly defined from the inputs the dynamics to reduce computation time at the expense of and dynamics. This paper addresses both of these problems by building on recent work on Unscented Dynamic Programming slower convergence. Recent work has proposed a completely (UDP)—which eliminates dynamics derivative computations in derivative-free variant of DDP that uses a deterministic DDP—to support general nonlinear state and input constraints sampling scheme inspired by the unscented Kalman filter. using an augmented Lagrangian. The resulting algorithm has The resulting algorithm, Unscented Dynamic Programming the same computational cost as first-order penalty-based DDP (UDP) [11], has the same computational complexity per- variants, but can achieve constraint satisfaction to high pre- cision without the numerical ill-conditioning associated with iteration as iLQR with finite-difference derivatives, but pro- penalty methods. We present results demonstrating its favorable vides empirical convergence approaching that of the full performance on several simulated robot systems including a second-order DDP algorithm. The primary contribution of quadrotor and 7-DoF robot arm. this paper is an extension of UDP that supports nonlinear constraints on states and inputs. I. INTRODUCTION Several authors have proposed methods for adding con- Trajectory optimization algorithms [1] are a powerful straints to DDP methods. For the special case of box con- set of tools for synthesizing dynamic motions for complex straints on input variables, quadratic programs (QPs) can robots [2], [3], [4], [5], [6]. Most robot tasks of interest be solved in the backwards pass [12], [13]. In this setting, include state constraints relating to, e.g., obstacle avoidance, bounds on the inputs are enforced by projecting the feedback reaching a desired goal state, and maintaining contact with terms onto the constraint surface. Farshidian et al. [14] the environment. Direct transcription methods for trajectory extends this to equality constraints on both the state and optimization parameterize both the state and input trajecto- input through a similar projection framework while pure state ries, allowing them to easily handle such constraints by for- constraints are still handled via penalty functions. mulating a large (and sparse) nonlinear program that can be Designing effective penalty functions and continuation solved using off-the-shelf Sequential Quadratic Programming schedules can be difficult.One common approach is to in- (SQP) packages. In contrast, Differential Dynamic Program- crease penalty weighting coefficients in the cost function ming (DDP) parameterizes only the input trajectory and uses until convergence to a result that satisfies the constraints [14]. Bellman’s optimality principle to iteratively solve a sequence However, it is well known that these methods often lead to of much smaller optimization problems [7], [8]. Recently, numerical ill-conditioning before reaching the desired con- DDP and its variants have received increased attention due straint tolerance [15], [16]. In practice, this leads to infeasible to growing evidence that online planning is possible for trajectories or collisions when run on hardware. Despite this, high-dimensional robots [4], [9]. However, constraints are penalty methods have seen broad application in robotics. typically addressed approximately by augmenting the native For example, van den Berg [17] uses exponential barrier cost with constraint penalty functions. This paper introduces cost terms in the LQR Smoothing algorithm to prevent a a variant of DDP that captures nonlinear constraints on states quadrotor from colliding with cylindrical obstacles. and inputs with high accuracy while maintaining favorable The augmented Lagrangian approach to solving con- convergence properties. strained nonlinear optimization problems was conceived to The classical DDP algorithm begins by simulating the dy- overcome the conditioning problems of penalty methods namics forward from an initial state with an initial guess for by adding a linear Lagrange multiplier term to the objec- the input trajectory. An affine control law is then computed tive [16]. As pointed out by other researchers [18], these in a backwards pass using local quadratic approximations methods may be particularly well-suited to trajectory opti- of the cost-to-go. The forward pass is then performed again mization problems as they allow the solver to temporarily traverse infeasible regions and aggressively move towards School of Engineering and Applied Sciences, Harvard University, Cambridge, MA. brian [email protected], local optima before making incremental adjustments to sat- fzmanchester, [email protected] isfy constraints. A theoretical formulation of this method was proposed for use in the DDP context over two decades Taking the second-order Taylor expansion of Q, we have: ago [19] and was later used to develop a hybrid-DDP T 2 1 3 2 0 QT QT 3 2 1 3 algorithm [20], but this work appears to have received 1 x u little attention in the robotics community. We compare our Q(δx; δu) ≈ 4δx5 4Qx Qxx Qxu5 4δx5 ; (5) 2 T constrained UDP algorithm against this method. δu Qu Qxu Quu δu In the remainder of the paper, we review key concepts where the block matrices are computed as: from DDP, augmented Lagrangian methods, and the un- T 0 0 scented transform (Section II), introduce the constrained Qxx = `xx + fx Vxxfx + Vx · fxx UDP algorithm (Section III), and describe our experimental T 0 0 Quu = `uu + fu Vxxfu + Vx · fuu results on an inverted pendulum, a quadrotor flying through Q = ` + f T V 0 f + V 0 · f (6) a virtual forest, and a manipulator avoiding obstacles (Sec- xu xu x xx u x xu T 0 tion IV). Several practical considerations are also discussed. Qx = `x + fx Vx T 0 Qu = `u + fu Vx: II. BACKGROUND Following the notation used elsewhere [4], we have dropped In the following subsections, we summarize key ideas explicit time indices and used a prime to indicate the next from DDP, augmented Lagrangian methods, and the un- timestep. Derivatives with respect to x and u are are denoted scented transform, all of which form the basis for the with subscripts. The rightmost terms in the equations for constrained UDP algorithm described in Section III. Qxx, Quu, and Qxu involve second derivatives of the dynam- ics, which are rank-three tensors. As mentioned previously, A. Differential Dynamic Programming these tensor calculations are relatively expensive and are We assume a discrete-time nonlinear dynamical system of often omitted, resulting in the iLQR algorithm [10]. the form: Minimizing equation (5) with respect to δu results in the following correction to the control trajectory: xk+1 = f(xk; uk); (1) −1 δu = −Quu (Quxδx + Qu) ≡ Kδx + d; (7) where x 2 Rn is the system state and u 2 Rm is a control input. The goal is to find an input trajectory, U = which consists of an affine term d and a linear feedback term fu0; : : : ; uN−1g, that minimizes an additive cost function, Kδx. These terms can be substituted back into equation (5) to obtain an updated quadratic model of V : N−1 X J(x0;U) = `f (xN ) + `(xk; uk); (2) 1 −1 ∆V = − QuQuu Qu k=0 2 −1 (8) Vx = Qx − QuQuu Qux where x0 is the initial state and x1; : : : ; xN are computed by −1 integrating forward the dynamics (1). Vxx = Qxx − QxuQuu Qux: Using Bellman’s principle of optimality [21], we can de- Therefore, a backward update pass can be performed starting fine the optimal cost-to-go, Vk(x), by the recurrence relation: at the final state, xN , by setting VN = `f (xN ) and iteratively applying the above computations. A forward simulation pass VN (x) = `f (xN ) is then performed to compute a new state trajectory using the (3) updated controls. This forward-backward process is repeated Vk(x) = min `(x; u) + Vk+1(f(x; u)): u until the algorithm converges within a specified tolerance. When interpreted as an update procedure, this relationship DDP, like other variants of Newton’s method, can achieve leads to classical dynamic programming algorithms [21]. quadratic convergence near a local optimum [8], [22]. How- However, the curse of dimensionality prevents direct applica- ever, care must be taken to ensure good convergence behavior tion of dynamic programming to most systems of interest to from arbitrary initialization: A line search parameter, α, must the robotics community. In addition, while VN (x) = `f (xN ), be added to the forward pass to ensure a satisfactory decrease and often has a simple analytical form, Vk(x) will typically in cost, and a regularization term, ρ, must be added to Quu have complex geometry that is difficult to represent due in equation (7) to ensure positive-definitiveness [23]. to the nonlinearity of the dynamics (1). DDP avoids these difficulties by settling for local approximations to the cost- B. Unscented Dynamic Programming to-go along a trajectory. UDP replaces the gradient and Hessian calculations in We define Q(δx; δu) as the local change in the minimiza- equation (6) with approximations computed from a set of tion argument in (3) under perturbations, δx; δu: sample points [11].
Load more

Recommended publications
  • Metaheuristics1
    METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming.
    [Show full text]
  • Lecture 4 Dynamic Programming
    1/17 Lecture 4 Dynamic Programming Last update: Jan 19, 2021 References: Algorithms, Jeff Erickson, Chapter 3. Algorithms, Gopal Pandurangan, Chapter 6. Dynamic Programming 2/17 Backtracking is incredible powerful in solving all kinds of hard prob- lems, but it can often be very slow; usually exponential. Example: Fibonacci numbers is defined as recurrence: 0 if n = 0 Fn =8 1 if n = 1 > Fn 1 + Fn 2 otherwise < ¡ ¡ > A direct translation in:to recursive program to compute Fibonacci number is RecFib(n): if n=0 return 0 if n=1 return 1 return RecFib(n-1) + RecFib(n-2) Fibonacci Number 3/17 The recursive program has horrible time complexity. How bad? Let's try to compute. Denote T(n) as the time complexity of computing RecFib(n). Based on the recursion, we have the recurrence: T(n) = T(n 1) + T(n 2) + 1; T(0) = T(1) = 1 ¡ ¡ Solving this recurrence, we get p5 + 1 T(n) = O(n); = 1.618 2 So the RecFib(n) program runs at exponential time complexity. RecFib Recursion Tree 4/17 Intuitively, why RecFib() runs exponentially slow. Problem: redun- dant computation! How about memorize the intermediate computa- tion result to avoid recomputation? Fib: Memoization 5/17 To optimize the performance of RecFib, we can memorize the inter- mediate Fn into some kind of cache, and look it up when we need it again. MemFib(n): if n = 0 n = 1 retujrjn n if F[n] is undefined F[n] MemFib(n-1)+MemFib(n-2) retur n F[n] How much does it improve upon RecFib()? Assuming accessing F[n] takes constant time, then at most n additions will be performed (we never recompute).
    [Show full text]
  • Dynamic Programming Via Static Incrementalization 1 Introduction
    Dynamic Programming via Static Incrementalization Yanhong A. Liu and Scott D. Stoller Abstract Dynamic programming is an imp ortant algorithm design technique. It is used for solving problems whose solutions involve recursively solving subproblems that share subsubproblems. While a straightforward recursive program solves common subsubproblems rep eatedly and of- ten takes exp onential time, a dynamic programming algorithm solves every subsubproblem just once, saves the result, reuses it when the subsubproblem is encountered again, and takes p oly- nomial time. This pap er describ es a systematic metho d for transforming programs written as straightforward recursions into programs that use dynamic programming. The metho d extends the original program to cache all p ossibly computed values, incrementalizes the extended pro- gram with resp ect to an input increment to use and maintain all cached results, prunes out cached results that are not used in the incremental computation, and uses the resulting in- cremental program to form an optimized new program. Incrementalization statically exploits semantics of b oth control structures and data structures and maintains as invariants equalities characterizing cached results. The principle underlying incrementalization is general for achiev- ing drastic program sp eedups. Compared with previous metho ds that p erform memoization or tabulation, the metho d based on incrementalization is more powerful and systematic. It has b een implemented and applied to numerous problems and succeeded on all of them. 1 Intro duction Dynamic programming is an imp ortant technique for designing ecient algorithms [2, 44 , 13 ]. It is used for problems whose solutions involve recursively solving subproblems that overlap.
    [Show full text]
  • Automatic Code Generation Using Dynamic Programming Techniques
    ! Automatic Code Generation using Dynamic Programming Techniques MASTERARBEIT zur Erlangung des akademischen Grades Diplom-Ingenieur im Masterstudium INFORMATIK Eingereicht von: Igor Böhm, 0155477 Angefertigt am: Institut für System Software Betreuung: o.Univ.-Prof.Dipl.-Ing. Dr. Dr.h.c. Hanspeter Mössenböck Linz, Oktober 2007 Abstract Building compiler back ends from declarative specifications that map tree structured intermediate representations onto target machine code is the topic of this thesis. Although many tools and approaches have been devised to tackle the problem of automated code generation, there is still room for improvement. In this context we present Hburg, an implementation of a code generator generator that emits compiler back ends from concise tree pattern specifications written in our code generator description language. The language features attribute grammar style specifications and allows for great flexibility with respect to the placement of semantic actions. Our main contribution is to show that these language features can be integrated into automatically generated code generators that perform optimal instruction selection based on tree pattern matching combined with dynamic program- ming. In order to substantiate claims about the usefulness of our language we provide two complete examples that demonstrate how to specify code generators for Risc and Cisc architectures. Kurzfassung Diese Diplomarbeit beschreibt Hburg, ein Werkzeug das aus einer Spezi- fikation des abstrakten Syntaxbaums eines Programms und der Spezifika- tion der gewuns¨ chten Zielmaschine automatisch einen Codegenerator fur¨ diese Maschine erzeugt. Abbildungen zwischen abstrakten Syntaxb¨aumen und einer Zielmaschine werden durch Baummuster definiert. Fur¨ diesen Zweck haben wir eine deklarative Beschreibungssprache entwickelt, die es erm¨oglicht den Baummustern Attribute beizugeben, wodurch diese gleich- sam parametrisiert werden k¨onnen.
    [Show full text]
  • 1 Introduction 2 Dijkstra's Algorithm
    15-451/651: Design & Analysis of Algorithms September 3, 2019 Lecture #3: Dynamic Programming II last changed: August 30, 2019 In this lecture we continue our discussion of dynamic programming, focusing on using it for a variety of path-finding problems in graphs. Topics in this lecture include: • The Bellman-Ford algorithm for single-source (or single-sink) shortest paths. • Matrix-product algorithms for all-pairs shortest paths. • Algorithms for all-pairs shortest paths, including Floyd-Warshall and Johnson. • Dynamic programming for the Travelling Salesperson Problem (TSP). 1 Introduction As a reminder of basic terminology: a graph is a set of nodes or vertices, with edges between some of the nodes. We will use V to denote the set of vertices and E to denote the set of edges. If there is an edge between two vertices, we call them neighbors. The degree of a vertex is the number of neighbors it has. Unless otherwise specified, we will not allow self-loops or multi-edges (multiple edges between the same pair of nodes). As is standard with discussing graphs, we will use n = jV j, and m = jEj, and we will let V = f1; : : : ; ng. The above describes an undirected graph. In a directed graph, each edge now has a direction (and as we said earlier, we will sometimes call the edges in a directed graph arcs). For each node, we can now talk about out-neighbors (and out-degree) and in-neighbors (and in-degree). In a directed graph you may have both an edge from u to v and an edge from v to u.
    [Show full text]
  • Language and Compiler Support for Dynamic Code Generation by Massimiliano A
    Language and Compiler Support for Dynamic Code Generation by Massimiliano A. Poletto S.B., Massachusetts Institute of Technology (1995) M.Eng., Massachusetts Institute of Technology (1995) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1999 © Massachusetts Institute of Technology 1999. All rights reserved. A u th or ............................................................................ Department of Electrical Engineering and Computer Science June 23, 1999 Certified by...............,. ...*V .,., . .* N . .. .*. *.* . -. *.... M. Frans Kaashoek Associate Pro essor of Electrical Engineering and Computer Science Thesis Supervisor A ccepted by ................ ..... ............ ............................. Arthur C. Smith Chairman, Departmental CommitteA on Graduate Students me 2 Language and Compiler Support for Dynamic Code Generation by Massimiliano A. Poletto Submitted to the Department of Electrical Engineering and Computer Science on June 23, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Dynamic code generation, also called run-time code generation or dynamic compilation, is the cre- ation of executable code for an application while that application is running. Dynamic compilation can significantly improve the performance of software by giving the compiler access to run-time infor- mation that is not available to a traditional static compiler. A well-designed programming interface to dynamic compilation can also simplify the creation of important classes of computer programs. Until recently, however, no system combined efficient dynamic generation of high-performance code with a powerful and portable language interface. This thesis describes a system that meets these requirements, and discusses several applications of dynamic compilation.
    [Show full text]
  • Bellman-Ford Algorithm
    The many cases offinding shortest paths Dynamic programming Bellman-Ford algorithm We’ve already seen how to calculate the shortest path in an Tyler Moore unweighted graph (BFS traversal) We’ll now study how to compute the shortest path in different CSE 3353, SMU, Dallas, TX circumstances for weighted graphs Lecture 18 1 Single-source shortest path on a weighted DAG 2 Single-source shortest path on a weighted graph with nonnegative weights (Dijkstra’s algorithm) 3 Single-source shortest path on a weighted graph including negative weights (Bellman-Ford algorithm) Some slides created by or adapted from Dr. Kevin Wayne. For more information see http://www.cs.princeton.edu/~wayne/kleinberg-tardos. Some code reused from Python Algorithms by Magnus Lie Hetland. 2 / 13 �������������� Shortest path problem. Given a digraph ����������, with arbitrary edge 6. DYNAMIC PROGRAMMING II weights or costs ���, find cheapest path from node � to node �. ‣ sequence alignment ‣ Hirschberg's algorithm ‣ Bellman-Ford 1 -3 3 5 ‣ distance vector protocols 4 12 �������� 0 -5 ‣ negative cycles in a digraph 8 7 7 2 9 9 -1 1 11 ����������� 5 5 -3 13 4 10 6 ������������� ���������������������������������� 22 3 / 13 4 / 13 �������������������������������� ��������������� Dijkstra. Can fail if negative edge weights. Def. A negative cycle is a directed cycle such that the sum of its edge weights is negative. s 2 u 1 3 v -8 w 5 -3 -3 Reweighting. Adding a constant to every edge weight can fail. 4 -4 u 5 5 2 2 ���������������������� c(W) = ce < 0 t s e W 6 6 �∈ 3 3 0 v -3 w 23 24 5 / 13 6 / 13 ���������������������������������� ���������������������������������� Lemma 1.
    [Show full text]
  • Notes on Dynamic Programming Algorithms & Data Structures 1
    Notes on Dynamic Programming Algorithms & Data Structures Dr Mary Cryan These notes are to accompany lectures 10 and 11 of ADS. 1 Introduction The technique of Dynamic Programming (DP) could be described “recursion turned upside-down”. However, it is not usually used as an alternative to recursion. Rather, dynamic programming is used (if possible) for cases when a recurrence for an algorithmic problem will not run in polynomial-time if it is implemented recursively. So in fact Dynamic Programming is a more- powerful technique than basic Divide-and-Conquer. Designing, Analysing and Implementing a dynamic programming algorithm is (like Divide- and-Conquer) highly problem specific. However, there are particular features shared by most dynamic programming algorithms, and we describe them below on page 2 (dp1(a), dp1(b), dp2, dp3). It will be helpful to carry along an introductory example-problem to illustrate these fea- tures. The introductory problem will be the problem of computing the nth Fibonacci number, where F(n) is defined as follows: F0 = 0; F1 = 1; Fn = Fn-1 + Fn-2 (for n ≥ 2). Since Fibonacci numbers are defined recursively, the definition suggests a very natural recursive algorithm to compute F(n): Algorithm REC-FIB(n) 1. if n = 0 then 2. return 0 3. else if n = 1 then 4. return 1 5. else 6. return REC-FIB(n - 1) + REC-FIB(n - 2) First note that were we to implement REC-FIB, we would not be able to use the Master Theo- rem to analyse its running-time. The recurrence for the running-time TREC-FIB(n) that we would get would be TREC-FIB(n) = TREC-FIB(n - 1) + TREC-FIB(n - 2) + Θ(1); (1) where the Θ(1) comes from the fact that, at most, we have to do enough work to add two values together (on line 6).
    [Show full text]
  • Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space
    applied sciences Article Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space Dawid Połap 1,* ID , Karolina K˛esik 1, Marcin Wo´zniak 1 ID and Robertas Damaševiˇcius 2 ID 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; [email protected] (K.K.); [email protected] (M.W.) 2 Department of Software Engineering, Kaunas University of Technology, Studentu 50, LT-51368, Kaunas, Lithuania; [email protected] * Correspondence: [email protected] Received: 16 December 2017; Accepted: 13 February 2018 ; Published: 16 February 2018 Abstract: The increasing exploration of alternative methods for solving optimization problems causes that parallelization and modification of the existing algorithms are necessary. Obtaining the right solution using the meta-heuristic algorithm may require long operating time or a large number of iterations or individuals in a population. The higher the number, the longer the operation time. In order to minimize not only the time, but also the value of the parameters we suggest three proposition to increase the efficiency of classical methods. The first one is to use the method of searching through the neighborhood in order to minimize the solution space exploration. Moreover, task distribution between threads and CPU cores can affect the speed of the algorithm and therefore make it work more efficiently. The second proposition involves manipulating the solutions space to minimize the number of calculations. In addition, the third proposition is the combination of the previous two. All propositions has been described, tested and analyzed due to the use of various test functions.
    [Show full text]
  • Branch and Bound, Divide and Conquer, and More TA: Chinmay Nirkhe ([email protected]) & Catherine Ma ([email protected])
    CS 38 Introduction to Algorithms Week 7 Recitation Notes Branch and Bound, Divide and Conquer, and More TA: Chinmay Nirkhe ([email protected]) & Catherine Ma ([email protected]) 1 Branch and Bound 1.1 Preliminaries So far we have covered Dynamic Programming, Greedy Algorithms, and (some) Graph Algorithms. Other than a few examples with Knapsack and Travelling Salesman, however, we have mostly covered P algorithms. Now we turn to look at some NP algorithms. Recognize, that, we are not going to come up with any P algorithms for these problems but we will be able to radically improve runtime over a na¨ıve algorithm. Specifically we are going to discuss a technique called Branch and Bound. Let's first understand the intu- ition behind the technique. Assume we are trying to maximize a function f over a exponentially large set X. However, not only is the set exponentially large but f might be computationally intensive on this set. Fortunately, we know of a function h such that f ≤ h everywhere. Our na¨ıve strategy is to keep a maximum value m (initially at −∞) and for each x 2 X, update it by m maxfm; f(x)g. However, an alternative strategy would be to calculate h(x) first. If h(x) ≤ m then we know f(x) ≤ h(x) ≤ m so the maximum will not change. So we can effectively avoid computing f(x) saving us a lot on computation! However, if h(x) > m then we cannot tell anything about f(x) so we would have to compute f(x) to check the maximum.
    [Show full text]
  • Toward a Model for Backtracking and Dynamic Programming
    TOWARD A MODEL FOR BACKTRACKING AND DYNAMIC PROGRAMMING Michael Alekhnovich, Allan Borodin, Joshua Buresh-Oppenheim, Russell Impagliazzo, Avner Magen, and Toniann Pitassi Abstract. We propose a model called priority branching trees (pBT ) for backtrack- ing and dynamic programming algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as Interval Scheduling, Knapsack and Satisfiability. Keywords. Greedy Algorithms, Dynamic Programming, Models of Computation, Lower Bounds. Subject classification. 68Q10 1. Introduction The “Design and Analysis of Algorithms” is a basic component of the Computer Science Curriculum. Courses and texts for this topic are often organized around a toolkit of al- gorithmic paradigms or meta-algorithms such as greedy algorithms, divide and conquer, dynamic programming, local search, etc. Surprisingly (as this is often the main “theory course”), these algorithmic paradigms are rarely, if ever, precisely defined. Instead, we provide informal definitional statements followed by (hopefully) well chosen illustrative examples. Our informality in algorithm design should be compared to computability the- ory where we have a well accepted formalization for the concept of an algorithm, namely that provided by Turing machines and its many equivalent computational models (i.e. consider the almost universal acceptance of the Church-Turing thesis). While quantum computation may challenge the concept of “efficient algorithm”, the benefit of having a well defined concept of an algorithm and a computational step is well appreciated.
    [Show full text]
  • Greedy Algorithms CLRS 16.1-16.2
    Greedy Algorithms CLRS 16.1-16.2 Today we discuss a technique called “greedy”. To understand the greedy technique, it’s best to understand the differences between greedy and dynamic programming. Dynamic programming: • The problem must have the optimal substructure property: the optimal solution to the problem contains within it optimal solutions to subproblems. This allows for a recursive solution. • The idea of dynamic programming is the following: At every step , evaluate all choices recursively and pick the best. A choice is evaluated recursively, meaning all its choices are evaluated and all their choices are evaluated, and so on; basically to evaluate a choice, you have to go through the whole recursion tree for that choice. • The table: The subproblems are “tabulated”, that is, they are stored in a table; using the table to store solutions to subproblems means that subproblem are computed only once. Typically the number of different subproblems is polynomial, but the recursive algorithm implementing the recursive formulation of the problem is exponential. This is because of overlapping calls to same subproblem. So if you don’t use the table to cache partial solutions, you incur a significant penalty (Often the difference is polynomial vs. exponential). • Sometimes we might want to take dynamic programming a step further, and eliminate the recursion —- this is purely for eliminating the overhead of recursion, and does not change the Θ() of the running time. We can think of dynamic programming as filling the sub-problems solutions table bottom-up, without recursion. The greedy technique: • As with dynamic programming, in order to be solved with the greedy technique, the problem must have the optimal substructure property • The problems that can be solved with the greedy method are a subset of those that can be solved with dynamic programming.
    [Show full text]