# The Extended Partitioning Problem

Asawaree Kalavade and Edward A. Lee,Prepublished version |

Published version |

## Abstract

In system-level design, applications are represented as task graphs where tasks (called nodes) have moderate to large granularity and each node has several implementation options differing in area and execution time. We define the extended partitioning problem as the joint determination of the mapping (hardware or software), the implementation option (called implementation bin), as well as the schedule, for each node, so that the overall area allocated to nodes in hardware is minimum and a deadline constraint is met. This problem is considerably harder (and richer) than the traditional binary partitioning problem that determines just the best mapping and schedule. Both binary and extended partitioning problems are constrained optimization problems and are NP-hard.We first present an efficient (O(N2)) heuristic, called GCLP, to solve the binary partitioning problem. The heuristic reduces the greediness associated with traditional list-scheduling algorithms by formulating a global measure, called global criticality (GC). The GC measure also permits an adaptive selection of the optimization objective at each step of the algorithm; since the optimization problem is constrained by a deadline, either area or time is optimized at a given step based on the value of GC. The selected objective is used to determine the mapping of nodes that are "normal", i.e. nodes that do not exhibit affinity for a particular mapping. To account for nodes that are not "normal", we define "extremities" and "repellers". Extremities consume disproportionate amounts of resources in hardware and software. Repellers are inherently unsuitable to either hardware or software based on certain structural properties. The mapping of extremities and repellers is determined jointly by GC and their local preference.

We then present an efficient (O(N3 + N2B), for N nodes and B bins per node) heuristic for extended partitioning, called MIBS, that alternately uses GCLP and an implementation-bin selection procedure. The implementation-bin selection procedure chooses, for a node with already determined mapping, an implementation bin that maximizes the area-reduction gradient of as-yet unmapped nodes. Solutions generated by both heuristics are shown to be reasonably close to optimal. Extended partitioning generates considerably smaller overall hardware as compared to binary partitioning.