1,908 research outputs found
A Simple and Scalable Static Analysis for Bound Analysis and Amortized Complexity Analysis
We present the first scalable bound analysis that achieves amortized
complexity analysis. In contrast to earlier work, our bound analysis is not
based on general purpose reasoners such as abstract interpreters, software
model checkers or computer algebra tools. Rather, we derive bounds directly
from abstract program models, which we obtain from programs by comparatively
simple invariant generation and symbolic execution techniques. As a result, we
obtain an analysis that is more predictable and more scalable than earlier
approaches. Our experiments demonstrate that our analysis is fast and at the
same time able to compute bounds for challenging loops in a large real-world
benchmark. Technically, our approach is based on lossy vector addition systems
(VASS). Our bound analysis first computes a lexicographic ranking function that
proves the termination of a VASS, and then derives a bound from this ranking
function. Our methodology achieves amortized analysis based on a new insight
how lexicographic ranking functions can be used for bound analysis
Labeling Schemes with Queries
We study the question of ``how robust are the known lower bounds of labeling
schemes when one increases the number of consulted labels''. Let be a
function on pairs of vertices. An -labeling scheme for a family of graphs
\cF labels the vertices of all graphs in \cF such that for every graph
G\in\cF and every two vertices , the value can be inferred
by merely inspecting the labels of and .
This paper introduces a natural generalization: the notion of -labeling
schemes with queries, in which the value can be inferred by inspecting
not only the labels of and but possibly the labels of some additional
vertices. We show that inspecting the label of a single additional vertex (one
{\em query}) enables us to reduce the label size of many labeling schemes
significantly
Non-polynomial Worst-Case Analysis of Recursive Programs
We study the problem of developing efficient approaches for proving
worst-case bounds of non-deterministic recursive programs. Ranking functions
are sound and complete for proving termination and worst-case bounds of
nonrecursive programs. First, we apply ranking functions to recursion,
resulting in measure functions. We show that measure functions provide a sound
and complete approach to prove worst-case bounds of non-deterministic recursive
programs. Our second contribution is the synthesis of measure functions in
nonpolynomial forms. We show that non-polynomial measure functions with
logarithm and exponentiation can be synthesized through abstraction of
logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem
using linear programming. While previous methods obtain worst-case polynomial
bounds, our approach can synthesize bounds of the form
as well as where is not an integer. We present
experimental results to demonstrate that our approach can obtain efficiently
worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the
divide-and-conquer algorithm for the Closest-Pair problem, where we obtain
worst-case bound, and (ii) Karatsuba's algorithm for
polynomial multiplication and Strassen's algorithm for matrix multiplication,
where we obtain bound such that is not an integer and
close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201
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