Adaptive negative cycle detection in dynamic graphs

Abstract

We examine the problem of detecting negative cycles in a dynamic graph, which is a fundamental problem that arises in electronic design automation and systems theory.We introduce the concept of adaptive negative cycle detection, in which a graph changes over time, and negative cycle detection needs to be done periodically, but not necessarily after every individual change. Such scenarios arise, for example, during iterative design space exploration for hardware and software synthesis. We present an algorithm for this problem, called the Adaptive Bellman-Ford (ABF) algorithm, based on a novel extension of the well known Bellman-Ford algorithm. The ABF algorithm allows us to systematically adapt information for a given graph to a modified version of the graph. We show that the ABF algorithm significantly outperforms previously available approaches for dynamic graphs, which either recompute negative cycle information from scratch whenever a graph is modified, or process the modifications one at a time ("incrementally").As an application of the ABF technique, we show that it can be used to obtain a very fast implementation of Lawler's technique for the computation of the maximum-cycle mean (MCM) of a graph, especially for a certain important kind of sparse graph. We further illustrate the application of the ABF technique to design-space exploration by developing automated search techniques for scheduling iterative data-flow graphs.

The hierarchical timing pair model for multirate DSP applications

“…It performs a sequence of successive approximations to find a close approximation to the iteration period . Since we have shown [18] the efficiency of Lawler's algorithm on graphs of bounded degree, this algorithm provides an effective way of computing the iteration period for graphs described using the hierarchical model. It may be possible to find other algorithms that can operate directly on the timing pair lists and compute a closed-form analytical expression for the maximum cycle mean of the system.…”

The hierarchical timing pair model

“…when its outputs become stable once the inputs are applied). By using these constraints, additional metrics can be obtained relating to the throughput and latency of the system, such as the iteration period bound, which is the same as the maximum cycle mean [8] for single rate graphs. The constraints are used for determining the feasibility of different schedules of the system, where a schedule consists of an ordering of the vertices on resources that can provide the required functionality.…”

“…Lawler's method [9] combined with the adaptive negative cycle detection techniques from [8] provides an efficient method of computing the maximum cycle mean of the system, since it operates by fixing and testing the system for consistency, using a binary search to iteratively improve the estimated value of . Because the constraint time of each path depends on the iteration period which is as yet unknown, it is not obvious how other algorithms for the MCM can be extended to this model.…”

Physical design algorithms for asynchronous circuits

“…Then, combining Equation (6) with the reformulated Equations (13) and 14, we can easily transform AT DP to GT DP.…”

“…For the incremental MCR problem, only very few researches have been done. In [13], the authors developed an adaptive negative cycle detection algorithm and incorporated it into the Lawler's MCR algorithm [43]. However, the experiments in [13] are performed only on very small graphs, and thus the efficiency of the algorithm cannot be confirmed.…”