Threshold Voting Is Fundamentally Simpler Than Plurality Voting

We show that n-way plurality voting on a large unordered object space has time and space complexities of θ(n2) and θ(n), respectively. If the object space is ordered, then sorting can be used to reduce the time complexity to the optimal θ(n log n). We then prove that weighted t- out-of -∑vi threshold voting on such an object space has time complexity O(np) and needs working storage space for only p objects, where p=[(∑vi)/t]. Thus, unless t is very small, threshold voting is considerably simpler than plurality voting. As a corollary, weighted majority voting can be performed in linear time with working storage for a single input object.