Full Theoretical Runtime Analysis of Alternating Variable Method on the Triangle Classification Problem

Runtime Analysis is a type of theoretical investigation that aims to determine, via rigorous mathematical proofs,the time a search algorithm needs to find an optimal solution.This type of investigation is useful to understand why a search algorithm could be successful, and it gives insight of how search algorithms work. In previous work,we proved the runtimes of different search algorithms on the test data generation for the Triangle Classification (TC)problem. We theoretically proved that Alternating Variable Method (AVM) has the best performance on the coverage of the most difficult branch in our empirical study. In this paper,we prove that the runtime of AVM on all the branches of TC is O((log n)²). That is necessary and sufficient to prove that AVM has a better runtime on TC compared to the other search algorithms we previously analysed. The theorems in this paper are useful for future analyses. In fact, to state theta search algorithm has worse runtime compared to AVM, it will be just sufficient to prove that its lower bound is higher than Omega((log n)²) on the coverage of at least one branch of TC.