A ternary-arithmetic topological based algebraic method for networks traffic observability

In this paper an algebraic method, which shares all the advantages of the topological methods and allows us to obtain the same results as the standard algebraic method with a substantial reduction in memory and cpu requirements, is presented. The main idea consists of writing the link, OD and scanned flows in terms of route instead of OD flows. This alternative permits starting the algebraic and topological processes with identical matrices of zeros and ones. In addition, in most iterations the pivots can be selected in such a way that the resulting matrices after each iteration contain only zeroes, ones and minus ones. This allows us to design a ternary arithmetic which reproduces the algebraic results exactly, requires only two bits to store each matrix entry and have no precision or non-zero pivot identification problems. Only when this process cannot be continued, the pure algebraic method is used, but only with a very reduced size matrix when compared with the size of the initial matrix. The method is illustrated by its application to a very simple network and to a real network example (the city of Cuenca, Spain).