Low-Cost Comparisons of File Copies

The problem of maintaining consistency of replicas of large files has been addressed b y Barbara, Feijoo and Garcia-Molina [l], Barbara and Lipton [2], Fuchs, W u and Abraham [5] and Metzner [6]. Our model is essentially identical to that previously assumed. We present a scheme that provides all capabilities previously obtained as well as being able to detect and identify missing as well as extraneous pages. Our solution is an adaptation of Metzner’s approach. We assume the existence of a signature function for individual pages. A supersignature is obtained as a power series in a primitive root within the Galois field with 2” elements. The coefficients are the signatures of the individual pages. This scheme enables us to detect errors such as missing, altered or incorrectly p l a c e d pages with very high probability and t o find an error diagnosis i f discrepancies have been detected; The problem solved in this paper is typified by the following situation. Consider a large database, replicas of which are situated a t several sites. Each site keeps its own log file of updates and in order to insure consistency, these log files are compared periodically. The physical organization of the log file consists of pages. *This work was supported by the University of California MICRO program as well as the NCR Corporation, Dayton, OH Our model is the one usually adopted for this variety of problem. A file is organized as a sequence of pages and complete copies of the file are maintained a t distinct sites throughout the network. We assume that few discrepancies occur and note that our scheme will not accommodate catastrophic failures. The cost of sending information through the network is high compared t o the cost of calculations a t an individual site. The order of pages is important and there might be missing pages besides faulty ones. As usual, our solution avoids the costly and tedious task of bitwise comparison between the pages stored a t different sites. Rather, we compare “signatures” of the individual pages by means of a supersignature calculated from the individual page signature. This allows conclusions that two copies are identical only with a previously chosen arbitrarily high probability. The cost of higher accuracy will be additional bits within the signature. Our scheme is based on the calculation of the supersignature. We interpret page signatures as elements of the Galois field 2; of 2” elements, but we disallow 0 as a value for a page signature. (We later propose variants that use more common data structures.) Let g be a fixed primitive root of 27, i.e. a generator of the multiplicative group of 2;. The supersignature is then calculated as the weighted sum of the page signatures with the powers of g. CH2878-7/90/0000/0196$01 .OO