Does Your Result Checker Really Check?

A result checker is a program that checks the output of the computation of the observed program for correctness. Introduced originally by Blum, the result checking paradigm has provided a powerful platform assuring the reliability of software. However, constructing result checkers for most problems requires not only significant domain knowledge but also ingenuity and can be error prone. In this paper we present our experience in validating result checkers using formal methods. We have conducted several case studies in validating result checkers from the commercial LEDA system for combinatorial and geometric computing. In one of our case studies, we detected a logical error in a result checker for a program computing max flow of a graph. 1

On the Design, Analysis, and Implementation of Algorithms for Selected Problems in Graphs and Networks

This thesis studies three problems in network optimization, viz., the minimum spanning tree verification (MSTV) problem, the undirected negative cost cycle detection (UNCCD) problem, and the negative cost girth (NCG) problem. These problems find applications in several domains including program verification, proof theory, real-time scheduling, social networking, and operations research.;The MSTV problem is defined as follows: Given an undirected graph G = (V,E) and a spanning tree T, is T a minimum spanning tree of G? We focus on the case where the number of distinct edge weights is bounded. Using a bucketed data structure to organize the edge weights, we present an efficient algorithm for the MSTV problem, which runs in O (| E| + |V| · K) time, where K is the number of distinct edge weights. When K is a fixed constant, this algorithm runs in linear time. We also profile our MSTV algorithm with the current fastest known MSTV implementation. Our results demonstrate the superiority of our algorithm when K ≤ 24.;The UNCCD problem is defined as follows: Given an undirected graph G = (V,E) with arbitrarily weighted edges, does G contain a negative cost cycle? We discuss two polynomial time algorithms for solving the UNCCD problem: the b-matching approach and the T-join approach. We obtain new results for the case where the edge costs are integers in the range {lcub}--K ·· K{rcub}, where K is a positive constant. We also provide the first extensive empirical study that profiles the discussed UNCCD algorithms for various graph types, sizes, and experiments.;The NCG problem is defined as follows: Given a directed graph G = (V,E) with arbitrarily weighted edges, find the length, or number of edges, of the negative cost cycle having the least number of edges. We discuss three strongly polynomial NCG algorithms. The first NCG algorithm is known as the matrix multiplication approach in the literature. We present two new NCG algorithms that are asymptotically and empirically superior to the matrix multiplication approach for sparse graphs. We also provide a parallel implementation of the matrix multiplication approach that runs in polylogarithmic parallel time using a polynomial number of processors. We include an implementation profile to demonstrate the efficiency of the parallel implementation as we increase the graph size and number of processors. We also present an NCG algorithm for planar graphs that is asymptotically faster than the fastest topology-oblivious algorithm when restricted to planar graphs