Algebras for weighted search

Weighted search is an essential component of many fundamental and useful algorithms. Despite this, it relatively under explored as a computational effect, receiving not nearly much attention either depth- or breadth-first search. This paper explores the algebraic underpinning weighted search, demonstrates how to implement monad transformer. The development first which can be expressed polynomial over semirings. These polynomials are generalised free semimodule capture wide range applications, including probability monads, monads for Finally, transformer based on introduced. Applying optimisations this type yields implementation pairing heaps, then used Dijkstra's algorithm efficient probabilistic sampling. construction formalised in Cubical Agda implemented Haskell.