1. A method of performing an allreduce operation on a plurality of compute nodes of a parallel computer, the compute nodes connected through a data communication network, each compute node comprising at least two processing cores, each processing core having contribution data for the allreduce operation, the method comprising: establishing, for each compute node, a plurality of logical rings, each logical ring including a different set of at least one processing core on that compute node, each logical ring including the processing cores on at least two of the compute nodes; iteratively, for each compute node, until each processing core of that compute node is assigned to each logical ring established for that compute node: assigning each processing core of that compute node to one of the logical rings established for that compute node to which the processing core has not previously been assigned during the allreduce operation, and performing, for each logical ring for that compute node, a global allreduce operation using the contribution data for the processing cores assigned to that logical ring or any global allreduce results from previous global allreduce operations in which the processing cores assigned to that logical ring may have participated, yielding current global allreduce results for each processing core on that compute node; and performing, for each compute node, a local allreduce operation using the global allreduce results for each processing core on that compute node.