1. A Mapping Algorithm for Defect- Tolerance of Reconfigurable Nano- Architectures By: Ibis Benito M.B. Tahoori, “A Mapping Algorithm for Defect-Tolerance of Reconfigurable Nano-Architectures”

2. Why more defects? Nanowires are only a few atoms long. Crosspoints are more fragile, resulting in more defects. Solution: customized chip configuration However, this requires large space to store defect map and testing, diagnosis and design efforts.

3. Crossbar Faults Switch stuck-open faults: missing switch at crosspoint. Switch stuck-closed faults: a horizontal and a vertical wire are shorted together and become unusable. Nanowire open, bridging faults: open fault on a nanowire; two or more nanowires shorted together, all become unusable.

4. Defect-unaware design flow Almost all design steps are unaware of the existence and location of defects within the nano-chip. Steps: Identify universal defect-free subsets within the original nxn partially defective fabric. Store information on defect-free subsets in a compact defect map. Map used resources into kxk defect-free crossbar within original nxn fabric. Size of maximum defect-free crossbar will be used for all chips manufactured in the same process environment (approximately same defect density level).

5. Identifying the defect-free subsets Greedy Mapping Algorithm Sample of manufactured chips Max kxk defect-free crossbar Bipartite graph representation to illustrate a nxn crossbar. Finding the maximum kxk defect- free crossbar corresponds to the maximum biclique of a bipartite graph.

6. Greedy Mapping Algorithm U: input nanowires V: output nanowires E: crosspoints Nodes in each partition are arranged in decreasing order according to their degree. Iterate alternatively between set U and V, adding zero-degree nodes to the corresponding solution list and removing the highest-degree nodes from the original U and V sets. Output of this algorithm is the maximum square biclique UxV. Worst case complexity: O(nlogn)

7. Greedy Algorithm vs. Exact Method 1000 crossbars randomly generated for each data point Greedy Algorithm: O(nlogn) Exact Method: Exponential

8. Conclusions Defect map size reduced from O(n 2 ) to O(n). No per-chip customized design. Algorithm to determine the maximum defect- free crossbar with O(nlogn) complexity, as opposed to the exponential complexity of an exact method.