6 research outputs found
Maximum Scatter TSP in Doubling Metrics
We study the problem of finding a tour of points in which every edge is
long. More precisely, we wish to find a tour that visits every point exactly
once, maximizing the length of the shortest edge in the tour. The problem is
known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997),
motivated by applications in manufacturing and medical imaging. Arkin et al.
gave a -approximation for the metric version of the problem and showed
that this is the best possible ratio achievable in polynomial time (assuming ). Arkin et al. raised the question of whether a better approximation
ratio can be obtained in the Euclidean plane.
We answer this question in the affirmative in a more general setting, by
giving a -approximation algorithm for -dimensional doubling
metrics, with running time , where . As a corollary we obtain (i) an
efficient polynomial-time approximation scheme (EPTAS) for all constant
dimensions , (ii) a polynomial-time approximation scheme (PTAS) for
dimension , for a sufficiently large constant , and (iii)
a PTAS for constant and . Furthermore, we
show the dependence on in our approximation scheme to be essentially
optimal, unless Satisfiability can be solved in subexponential time
Optimal Don’t Care Filling for Minimizing Peak Toggles During At-Speed Stuck-At Testing
Due to the increase in manufacturing/environmental uncertainties in the nanometer regime, testing digital chips under different operating conditions becomes mandatory. Traditionally, stuck-at tests were applied at slow speed to detect structural defects and transition fault tests were applied at-speed to detect delay defects. Recently, it was shown that certain cell-internal defects can only be detected using at-speed stuck-at testing. Stuck-at test patterns are power hungry, thereby causing excessive voltage droop on the power grid, delaying the test response, and finally leading to false delay failures on the tester. This motivates the need for peak power minimization during at-speed stuck-at testing. In this article, we use input toggle minimization as a means to minimize a circuit’s power dissipation during at-speed stuck-at testing under the Combinational State Preservation scan (CSP-scan) Design-For-Testability (DFT) scheme. For circuits whose test sets are dominated by don’t cares, this article maps the problem of optimal X-filling for peak input toggle minimization to a variant of the interval coloring problem and proposes a Dynamic Programming (DP) algorithm (DP-fill) for the same along with a theoretical proof for its optimality. For circuits whose test sets are not dominated by don’t cares, we propose a max scatter Hamiltonian path algorithm, which ensures that the ordering is done such that the don’t cares are evenly distributed in the final ordering of test cubes, thereby leading to better input toggle savings than DP-fill. The proposed algorithms, when experimented on ITC99 benchmarks, produced peak power savings of up to 48% over the best-known algorithms in literature. We have also pruned the solutions thus obtained using Greedy and Simulated Annealing strategies with iterative 1-bit neighborhood to validate our idea of optimal input toggle minimization as an effective technique for minimizing peak power dissipation during at-speed stuck-at testing
Algorithms for Power Aware Testing of Nanometer Digital ICs
At-speed testing of deep-submicron digital very large scale integrated (VLSI) circuits
has become mandatory to catch small delay defects. Now, due to continuous shrinking
of complementary metal oxide semiconductor (CMOS) transistor feature size, power
density grows geometrically with technology scaling. Additionally, power dissipation
inside a digital circuit during the testing phase (for test vectors under all fault models
(Potluri, 2015)) is several times higher than its power dissipation during the normal
functional phase of operation. Due to this, the currents that flow in the power grid during
the testing phase, are much higher than what the power grid is designed for (the
functional phase of operation). As a result, during at-speed testing, the supply grid
experiences unacceptable supply IR-drop, ultimately leading to delay failures during
at-speed testing. Since these failures are specific to testing and do not occur during
functional phase of operation of the chip, these failures are usually referred to false
failures, and they reduce the yield of the chip, which is undesirable.
In nanometer regime, process parameter variations has become a major problem.
Due to the variation in signalling delays caused by these variations, it is important to
perform at-speed testing even for stuck faults, to reduce the test escapes (McCluskey
and Tseng, 2000; Vorisek et al., 2004). In this context, the problem of excessive peak
power dissipation causing false failures, that was addressed previously in the context of
at-speed transition fault testing (Saxena et al., 2003; Devanathan et al., 2007a,b,c), also
becomes prominent in the context of at-speed testing of stuck faults (Maxwell et al.,
1996; McCluskey and Tseng, 2000; Vorisek et al., 2004; Prabhu and Abraham, 2012;
Potluri, 2015; Potluri et al., 2015). It is well known that excessive supply IR-drop during
at-speed testing can be kept under control by minimizing switching activity during
testing (Saxena et al., 2003). There is a rich collection of techniques proposed in the past
for reduction of peak switching activity during at-speed testing of transition/delay faults
ii
in both combinational and sequential circuits. As far as at-speed testing of stuck faults
are concerned, while there were some techniques proposed in the past for combinational
circuits (Girard et al., 1998; Dabholkar et al., 1998), there are no techniques concerning
the same for sequential circuits. This thesis addresses this open problem. We
propose algorithms for minimization of peak switching activity during at-speed testing
of stuck faults in sequential digital circuits under the combinational state preservation
scan (CSP-scan) architecture (Potluri, 2015; Potluri et al., 2015). First, we show that,
under this CSP-scan architecture, when the test set is completely specified, the peak
switching activity during testing can be minimized by solving the Bottleneck Traveling
Salesman Problem (BTSP). This mapping of peak test switching activity minimization
problem to BTSP is novel, and proposed for the first time in the literature.
Usually, as circuit size increases, the percentage of don’t cares in the test set increases.
As a result, test vector ordering for any arbitrary filling of don’t care bits
is insufficient for producing effective reduction in switching activity during testing of
large circuits. Since don’t cares dominate the test sets for larger circuits, don’t care
filling plays a crucial role in reducing switching activity during testing. Taking this
into consideration, we propose an algorithm, XStat, which is capable of performing test
vector ordering while preserving don’t care bits in the test vectors, following which, the
don’t cares are filled in an intelligent fashion for minimizing input switching activity,
which effectively minimizes switching activity inside the circuit (Girard et al., 1998).
Through empirical validation on benchmark circuits, we show that XStat minimizes
peak switching activity significantly, during testing.
Although XStat is a very powerful heuristic for minimizing peak input-switchingactivity,
it will not guarantee optimality. To address this issue, we propose an algorithm
that uses Dynamic Programming to calculate the lower bound for a given sequence
of test vectors, and subsequently uses a greedy strategy for filling don’t cares in this
sequence to achieve this lower bound, thereby guaranteeing optimality. This algorithm,
which we refer to as DP-fill in this thesis, provides the globally optimal solution for
minimizing peak input-switching-activity and also is the best known in the literature
for minimizing peak input-switching-activity during testing. The proof of optimality of
DP-fill in minimizing peak input-switching-activity is also provided in this thesis