Modelling dynamic programming problems by generalized d-graphs

TL;DR

This paper introduces generalized d-graphs with cycles as a new way to model dynamic programming problems, enabling classification and analysis of DP strategies through graph theory.

Contribution

It presents a novel generalized d-graph framework for modeling DP problems, extending existing algorithms and enabling problem classification via graph-theoretic methods.

Findings

Generalized d-graphs can model DP problems with cycles.

Reformulation of shortest path algorithms as DP strategies.

Framework allows classification of DP problems based on graph properties.

Abstract

In this paper we introduce the concept of generalized d-graph (admitting cycles) as special dependency-graphs for modelling dynamic programming (DP) problems. We describe the d-graph versions of three famous single-source shortest algorithms (The algorithm based on the topological order of the vertices, Dijkstra algorithm and Bellman-Ford algorithm), which can be viewed as general DP strategies in the case of three different class of optimization problems. The new modelling method also makes possible to classify DP problems and the corresponding DP strategies in term of graph theory.