DPFn

DPFn is an acronym used in some computational literature to denote a Dynamic Programming Function, a class of functions defined and evaluated using dynamic programming techniques to solve optimization problems characterized by optimal substructure and overlapping subproblems. A DPFn is typically specified by a base case and a recurrence relation that expresses the value of a subproblem in terms of smaller subproblems. The computation can be performed either top-down with memoization or bottom-up through tabulation, ensuring each subproblem is solved once and reused.

In practice, DPFn captures the core of many classic algorithms: in the knapsack problem, f(i,w) represents the maximum value using the first i items with weight limit w; in edit distance, f(i,j) is the minimum edits to convert prefixes; in shortest path on a DAG, f(i) is the minimum cost to reach node i.

Applications span computer science, operations research, bioinformatics, and economics, including sequence alignment, resource allocation, scheduling, and risk optimization.

Implementation considerations include memory usage, time complexity, and the choice between rolling arrays for space optimization and full tabulation for simple access. DPFn can be expressed in many languages using arrays or hash maps, with memoization libraries or explicit tabulation.

Historically, dynamic programming was introduced by Richard Bellman in the mid-20th century; the term DPFn in some texts emphasizes the function-centric view of solving problems by recording subproblem results and composing them to form solutions.

See also dynamic programming, memoization, tabulation, optimization.