Figure 4

Partitioning the pairwise means in linear time. A partition is indicated by a dotted line such that all elements above and to the left of the line are strictly less than six. The pairwise averages that are required to be computed for determining the partition are indicated by the path. Specifically, imagine that the current best guess at the pseudomedian is six – that is, we now want to divide S₀ into those averages less than six and those greater than or equal to six. We start at the top and right of the matrix and encounter a value of 5. This is less than six so we move down one row. Again, we encounter a value, 5.5, that is less than six so we move down one more row. Note that we now know every element in these previous two rows are less than our partitioning element and that we determined this to be so by computing just two pairwise averages. Returning to the partitioning, we next encounter the value 6.5 which is greater than six, so we scan the row to the left until we find the first element, 5.5, which is less than our partitioning element. When this occurs, we move down a row from six, encounter 6.5, move left, find the value six and reach the diagonal, which completes the partitioning. Importantly, we reached this diagonal by computing only seven pairwise averages. The facilitating requirement of the input data is that it is sorted. This is the partitioning technique implemented in Monahan's algorithm.