Elements of Competitive Programming. Dynamic Programming. 88 Problems with Solutions. A Functional Approach

This book was planned as an aid to students preparing for competitive programming. Written in a problem-solution format, this is exceptionally convenient for analyzing common errors made by the coder in competitive coding sports, for reviewing different methods of solving the same problems and for discussing difficult questions of fundamentals of algorithms with focus on dynamic programming. Attention can be drawn to various aspects of the problem, certain fine points can be made, and a more thorough understanding of the fundamentals can be reached. The art of formulating and solving problems using dynamic programming can be learned only through active participation by the student. Infused with the wisdom of Richard Bellman, the father of Dynamic Programming, this tiny book distills the inherent concepts and techniques in a problem-solution format. A functional approach to a coding problem is beyond the foundational aspect of underlying genetic and computational structures, often termed as π ∞ . A concept becomes not difficult because the complexities built into it are clarified. In a bid to reach the core of the problem, the concept is splitbroken into fragments, complexities are exposed and delicate points are examined. Then the concept is recomposed to make it integral and as a result, this reintegrated concept becomes sufficiently simple and comprehensible. This helps build a coder's insight to reveal the internal structure and internal logic of the concepts, algorithms and mathematical theorems. The student must first discover, by experience, that proper formulation is not quite as trivial as it appears when reading a solution. Then, by considerable practice with solving problems on his own, he will acquire the feel for the subject that ultimately renders proper formulation easy and natural. For this reason, this book contains a large number (88) of instructional problems in a graded way, carefully chosen to allow the student to acquire the art that I seek to convey. The student must do these problems on his own. Solutions are given next to the problem because the reader needs feedback on the correctness of his procedures in order to learn, but any student who reads the solution before seriously attempting the problem does so at this own peril. The primary goal is to convey, by examples, the art of formulating the solution of problems in terms of dynamic-programming recurrence relations. The reader must learn how to identify the appropriate state and stage variables, and how to define and characterize the optimal value function. Corollary to this objective is reader evaluation of the feasibility and computational magnitude of the solution, based on the recurrence relation. The secondary goal is to show how dynamic programming can be used analytically to establish the structure of the optimal solution, or conditions necessarily satisfied by the optimal solution, both for their own interest and as means of reducing computation. Additionally few special techniques have been distilled that have proved useful on certain classes of problems. This book provides a functional approach to solving problems using dynamic programming. Written in an extremely lively form of problems and solutions (including code in modern C++ and pseudo style), this leads to extreme simplification of optimal coding with great emphasis on unconventional and integrated science of dynamic Programming. Though aimed primarily at serious programmers, it imparts the knowledge of deep internals of underlying concepts and beyond to computer scientists alike. Preface 4 List of Algorithms 6 Genesis 12 Optimal Loot Partition 14 Deterministic 14 Stochastic 15 Exam Prep 18 Optimal Coin Tossing 22 Proving Optimality Principle 24 Computation 28 Ascension to Heaven 30 Fibonacci Line Search 34 Coin Change 38 Constrained Subsequence 44 Maximum Sum 44 Minimum Sum 48 Circular Sequence 51 Maximum Product 55 Stock Trading 58 Binary Tree Mall Loot 66 Binary Search Tree Generation 68 Quantify Yogic Effect 72 Path to Heaven 86 Stairway 86 Kriya Grid 90 Kriya Sequence 142 Kriya Catalysis 212 Bibliography 228