We show how programming pearls can be systematically derived via fusion, followed by tupling transformations. By focusing on the elimination of intermediate data structures (fusion) followed by the elimination of redundant calls (tupling), we systematically realise both space and time efficient algorithms from naive specifications. We illustrate our approach using a well-known maximum segment sum (MSS) problem, and a less-known maximum segment product (MSP) problem. While the two problems share similar specifications, their optimised codes are significantly different. This divergence in the transformed codes do not pose any difficulty. By relying on modular techniques, we are able to systematically reuse both code and transformation in our derivation.

In existing work on graph algorithms, it is known that a linear time algorithm can be derived mechanically from a logical formula for a class of optimization problems. But this has a serious problem that the derived algorithm has huge constant factor. In this work, we redene this problem on recursive data structures as a maximum marking problem and propose method for deriving a linear time algorithm for that. In this method, speci cation is given using recursive functions instead of logical formula, which results in a practical linear time algorithm. This method is mechanical and in fact, based on this deriving method, we make a system which automatically generates a practical linear time algorithm from specication for a maximum marking problem.