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Dynamic programming is both a mathematical optimization method and an algorithmic paradigm. ... the strategy is called "divide and conquer" instead. [1]
The divide-and-conquer paradigm is often used to find an optimal solution of a problem. Its basic idea is to decompose a given problem into two or more similar, but simpler, subproblems, to solve them in turn, and to compose their solutions to solve the given problem. Problems of sufficient simplicity are solved directly.
The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory; though the basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior and Abraham Wald's ...
The one-shot deviation principle (also known as single-deviation property [1]) is the principle of optimality of dynamic programming applied to game theory. [2] It says that a strategy profile of a finite multi-stage extensive-form game with observed actions is a subgame perfect equilibrium (SPE) if and only if there exist no profitable single deviation for each subgame and every player.
Dynamic problem For an initial set of N numbers, dynamically maintain the maximal one when insertion and deletions are allowed. A well-known solution for this problem is using a self-balancing binary search tree. It takes space O(N), may be initially constructed in time O(N log N) and provides insertion, deletion and query times in O(log N).
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
The coin of the highest value, less than the remaining change owed, is the local optimum. (In general, the change-making problem requires dynamic programming to find an optimal solution; however, most currency systems are special cases where the greedy strategy does find an optimal solution.)
The probabilistic convolution tree-based dynamic programming method also efficiently solves the probabilistic generalization of the change-making problem, where uncertainty or fuzziness in the goal amount W makes it a discrete distribution rather than a fixed quantity, where the value of each coin is likewise permitted to be fuzzy (for instance ...
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