TY - GEN

T1 - Static data structure lower bounds imply rigidity

AU - Dvir, Zeev

AU - Golovnev, Alexander

AU - Weinstein, Omri

N1 - Publisher Copyright:
© 2019 Association for Computing Machinery.

PY - 2019/6/23

Y1 - 2019/6/23

N2 - We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of t ≥ ω(log2 n) on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space (s = (1 + ε)n), would already imply a semi-explicit (PNP) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial (t ≥ nδ) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime (s = n + o(n)), we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the “inner" and “outer" dimensions of a matrix (Paturi and Pudlák, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest.

AB - We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of t ≥ ω(log2 n) on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space (s = (1 + ε)n), would already imply a semi-explicit (PNP) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial (t ≥ nδ) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime (s = n + o(n)), we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the “inner" and “outer" dimensions of a matrix (Paturi and Pudlák, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest.

KW - Circuit lower bound

KW - Codes

KW - Data structures

KW - Rigidity

UR - http://www.scopus.com/inward/record.url?scp=85068766763&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068766763&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316348

DO - 10.1145/3313276.3316348

M3 - Conference contribution

AN - SCOPUS:85068766763

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 967

EP - 978

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

T2 - 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019

Y2 - 23 June 2019 through 26 June 2019

ER -