Nanocrossbars (i.e., nanowire crossbars) offer extreme logic densities but come with very high defect rates; stuck-open/closed, broken nanowires. Achieving reasonable yield and utilization requires logic mapping that is defect-aware even at the crosspoint level. Such logic mapping works with a defect map per each manufactured chip. The problem can be expressed as matching of two bipartite graphs; one for the logic to be implemented and other for the nanocrossbar. This article shows that the problem becomes a Bipartite SubGraph Isomorphism (BSGI) problem within sub-nanocrossbars free of stuck-closed faults. Our heuristic KNS-2DS is an iterative rough canonizer with approximately O(N 2 ) complexity followed by an O(N 3 ) matching algorithm. Canonization brings a partial or full order to graph nodes. It is normally used for solving the regular Graph Isomorphism (GI) problem, while we apply it to BSGI. KNS stands for K-Neighbor Sort and is used for initializing our main contribution 2-Dimensional-Sort (2DS). 2DS operates on the adjacency matrix of a bipartite graph. Radix-2 2DS solves the problem in the absence of stuck-closed faults. With the addition of Radix-3 and our novel Radix-2.5 sort, we solve problems that also have stuck-closed faults. We offer very short runtimes (due to canonization) compared to previous work and have success on all benchmarks. KNS-2DS is also novel from the perspective of BSGI problem as it is based on canonization but not on a search tree with backtracking.

This paper addresses the NP-complete problem of mapping a logic function on to a nanocrossbar with a known defect map. We first show that this problem can be transformed into a Bipartite SubGraph Isomorphism (BSGI) problem. Then we present our proposed KNS-2DS algorithm, which canonizes both graphs in N 2 time (N being the number of nodes) and then matches them in N 3 time in the worst case. KNS-2DS uses a KNeighbor Sort (KNS) to initialize our main contribution 2D-Sort (2DS). 2DS is an iterative rough canonizer that lets a straightforward matching algorithm complete the job. Our algorithm offers very short run-times (due to canonization) compared to previous work and has success on all benchmarks. KNS-2DS is also novel from the perspective of the BSGI problem in the sense that it is based on canonization but not on a search tree with backtracking.

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