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### Paper:

TR23-186 | 28th November 2023 04:42

#### A VLSI Circuit Model Accounting For Wire Delay

TR23-186
Authors: Ce Jin, Ryan Williams, Nathaniel Young
Publication: 28th November 2023 04:42
Keywords:

Abstract:

Given the need for ever higher performance, and the failure of CPUs to keep providing single-threaded performance gains, engineers are increasingly turning to highly-parallel custom VLSI chips to implement expensive computations. In VLSI design, the gates and wires of a logical circuit are placed on a 2-dimensional chip with a small number of layers.
Traditional VLSI models use gate delay to measure the time complexity of the chip, ignoring the lengths of wires. However, as technology has advanced, wire delay is no longer negligible; it has become an important measure in the design of VLSI chips [Markov, Nature (2014)].

Motivated by this situation, we define and study a model for VLSI chips, called wire-delay VLSI, which takes wire delay into account, going beyond an earlier model of Chazelle and Monier [JACM 1985].

$\bullet$ We prove nearly tight upper bounds and lower bounds (up to logarithmic factors) on the time delay of this chip model for several basic problems. For example, AND, OR, and PARITY require $\Theta(n^{1/3})$ delay, while ADDITION and MULTIPLICATION require $\tilde \Theta(n^{1/2})$ delay, and TRIANGLE DETECTION on (dense) $n$-node graphs requires $\tilde \Theta(n)$ delay. Interestingly, when we allow input bits to be read twice, the delay for ADDITION can be improved to $\Theta(n^{1/3})$.

$\bullet$ We also show that proving significantly higher lower bounds in our wire-delay VLSI model would imply breakthrough results in circuit lower bounds. Motivated by this barrier, we also study conditional lower bounds on the delay of chips based on the Orthogonal Vectors Hypothesis from fine-grained complexity.

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