How can you design a custom algorithm pattern for a novel problem?

To identify the problem, you must understand the context and the expected benefits that will be asked from the solution. The identification consists in a clear requirement specification of the input data, outputs and a metric to gauge the effectiveness of the solution. Problem identification, in its early stages, will typically require feasibility studies and research to identify edge cases, special scenarios and most common failure modes and trade-offs. Feasibility studies should give a sense of the resources needed to solve the problem and an approximate expected complexity of the solution.

Analysis and development go hand in hand. Conceptual work must be accompanied by proofs-of-concept, even if they are not efficient. The first priority is to get an algorithm that can solve the problem/subproblem, subject to testing constraints. You should aim to produce something you can test and that solves the problem, albeit partially or approximately. However, with proofs-of-concept, the objective is a working solution, not an optimal one. So, as long as you can test your solution effectively, keep in the "analysis - hypotesis - testing - improving/pivoting" iterations. Don't attempt to decouple analysis from development. This ends in disaster. Algorithm design is iterative in nature.

Analyzing the problem in-depth is essential for designing a custom algorithm pattern. This process unveils the intricacies, relationships, and patterns within the problem, providing insights into potential algorithmic solutions. Techniques like abstraction and decomposition help identify key elements, making it possible to tailor a unique algorithm that precisely addresses the novel problem and its subproblems. This analytical phase greatly enhances the likelihood of creating an effective and innovative solution.

Algorithm development typically happens in iterations consisting of: Analysis - Algorithm Coding - Testing - improving/pivoting. Implementation and refinement are not final steps, are intermediate steps towards an ever improving metric. Once you have a proof of concept, the objective is to improve the metric defined in the requirements until reaching satisfactory performance, while meeting trade-offs and restrictions. Refinements can be in multiple dimensions: speed, memory, edge case handling, documentation, etc. Not all will impact the business-defined metric, some of them will be necessary to increase algorithm robustness, and even to collaborate effectively in the organization through good coding practices and detailed documentation.

Generally, you can use the known techniques as divide-and conquer's, greedy's or dynamic programming. Recognize also if it is a NP-hard problem. Then you can make an analysis taking into account 'worst-case' and 'average-case' analysis. You might not pay attention to constant factors operations within the algorithms and of course, asymptotic analysis.