Amortized Complexity Verified

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

“…The proof of inequality (3) usually involves arithmetic, and sometimes the correctness of the functional algorithm. For skew heaps and splay trees, the analogous results are already proved in [18], and only slight modifications are necessary to bring them into the right form for our use.…”

“…bst t =⇒ splay time x t + splay tree P (splay x t) ≤ splay tree P t + splay atime (size1 t), which follows from the corresponding lemma in [18]. Note the requirement that t is a binary search tree.…”

“…Skew heap and splay tree For these two examples, the bulk of the analysis (functional correctness and justification of amortized runtime) is done in [18]. Our work is primarily to define the imperative version of the algorithm and verifying its agreement with the functional version.…”

“…We also make use of existing work by Nipkow [18] on analysis of amortized complexity for functional programs. Based on this work, we verify the amortized complexity of imperative implementations of two data structures: skew heaps and splay trees.…”

Exponential Automatic Amortized Resource Analysis

“…Techniques and tools for automatic and semi-automatic resource analysis have been extensively studied. The applied methods range from deriving and analyzing recurrence relations [55,1,16,2,12,36,10,37], to abstract interpretation and static analysis [18,7,49,39], to type systems [11,56,53], to proof assistants and program logics [4,9,8,48,19,45,42], to term rewriting [6,5,47]. Many techniques focus on upper bounds on the worst-case bounds, but average-case bounds [15,35,43,54] and lower-bounds have also been studied [3,17,44].…”