Voting Algorithms

Sum& Conclusions Voting is important in the realization of ultrareliable systems based on the multi-channel computation paradigm. In an earlier paper (1991 Aug) I dealt with voting networks, viz, hardware implementation of certain voting schemes. A voting algorithm specifies how the voting result is obtained from the input data and can be the basis for implementing a hardware voting network or a software voting routine. This paper presents efficient n-way plurality and threshold voting algorithms based on the type of voting (exact, inexact, or approval), rule for output selection (plurality or threshold) and properties of the input object space (size & structure). Exact voting is the most common voting method and is the easiest to implement. Inexact voting algorithms are more complicated due to intransitivity of approximate equality. As an example, when approximate equality a = b for numerical inputs a & b is defined as la bl 5 E , then a n b and b n c do not imply a P c. In approval voting, each input to the voting process consists of a finite or infinite set of values that have been “approved” by the corresponding computation channel and the value, or set of values, with the highest approval voting must emerge as output. Multiple approved values can result from non-unique answer to a given problem or from uncertainties in the solution process. For exact voting, the complexity of an n-way voting algorithm depends on the structure of the input object space. Threshold voting often requires less time & space, except when the threshold is very small. I extend the techniques for designing efficient exact voting algorithms to inexact and approval voting schemes. My results show that optimal linear-time (in n) voting algorithms are available when the input object space is small. Next in the time-complexity hierarchy is the case of a totally-ordered object space that supports worstcase order+. log(n)]-time algorithms for both exact and inexact voting as well as for certain approval-voting schemes. These algorithms are intimately related to sorting and have the same time complexity. An unordered input object space leads to worst-case quadratic-time exact and inexact voting algorithms, even when a distance metric can be defined on pairs of input objects.