Row-hammering flips bits in a victim DRAM row by frequently activating its adjacent rows, compromising DRAM integrity. Several studies propose to prevent row-hammering by counting the number of activates to a DRAM row and refreshing the corresponding victim rows before the count surpasses a rowhammer threshold. However, these approaches either incur a significant area overhead or a large number of additional activations (ACT) that could degrade the system performance. In this paper, we propose CAT-TWO, a time-window-optimized version of the existing Counter-based Adaptive Tree (CAT) scheme for row-hammer prevention. We first ensure that the victim rows are always refreshed at the last level of the tree without counter overflow by configuring the threshold and the number of CAT-TWO counters based on the fact that the maximum number of ACTs is limited within the refresh window. We further reduce the size and latency of CAT-TWO by applying high-radix rank-level CAT-TWO with multiple tree roots. CAT-TWO incurs less than 0.7% energy overhead on a baseline DDR4 DRAM device, and generates less than 0.03% additional ACTs to refresh victim rows in the worst case, which hardly affects system performance. INDEX TERMS DRAM chips, DRAM reliability, row-hammering.
SUMMARYThis article presents a class of spline algorithms for generating orientation trajectories that approximately minimize angular acceleration. Each algorithm constructs a twice-di erentiable curve on the rotation group SO(3) that interpolates a given ordered set of rotation matrices at speciÿed knot times. Rotation matrices are parametrized, respectively, by the unit quaternion, canonical co-ordinate, and Cayley-Rodrigues representations. All the algorithms share the common feature of (i) being invariant with respect to choice of ÿxed and moving frames (bi-invariant), and (ii) being cubic in the parametrized co-ordinates. We assess the performance of these algorithms by comparing the resulting trajectories with the minimum angular acceleration curve.
This article presents a cubic spline algorithm for interpolation on the rotation group SO(3). Given an ordered set of rotation matrices and knot times, the algorithm generates a twice-differentiable curve on SO(3) that interpolates the given rotation matrices at their specified times. In our approach SO(3) is locally parametrized by the Cayley parameters, and the generated curve is cubic in the sense that the Cayley parameter representation is a cubic polynomial. The resulting algorithm is a computationally efficient way of generating bi-invariant (i.e., invariant with respect to choice of both inertial and body-fixed frames) trajectories on the rotation group that does not require the evaluation of transcendental functions, and can also be viewed as an approximation to a minimum angular acceleration trajectory. Because the Cayley parameters provide a one-to-one correspondence between R3 and a dense set of SO(3), the resulting trajectories do not have the “multiple winding” effect that occurs in several existing methods.
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