TY - GEN

T1 - Optimal succinct rank data structure via approximate nonnegative tensor decomposition

AU - Yu, Huacheng

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

PY - 2019/6/23

Y1 - 2019/6/23

N2 - Given an n-bit array A, the succinct rank data structure problem asks to construct a data structure using space n + r bits for r ≪ n, supporting rank queries of form rank(u) = Íui=−01 A[i]. In this paper, we design a new succinct rank data structure with r = n/(log n)Ω(t) + n1−c and query time O(t) for some constant c > 0, improving the previous best-known by Pǎtraşcu, which has r = n/(logtn )Ω(t) + Õ(n3/4) bits of redundancy. For r > n1−c, our space-time tradeoff matches the cell-probe lower bound by Pǎtraşcu and Viola, which asserts that r must be at least n/(log n)O(t). Moreover, one can avoid an n1−c-bit lookup table when the data structure is implemented in the cell-probe model, achieving r = ⌈n/(log n)Ω(t)⌉. It matches the lower bound for the full range of parameters. En route to our new data structure design, we establish an interesting connection between succinct data structures and approximate nonnegative tensor decomposition. Our connection shows that for specific problems, to construct a space-efficient data structure, it suffices to approximate a particular tensor by a sum of (few) nonnegative rank-1 tensors. For the rank problem, we explicitly construct such an approximation, which yields an explicit construction of the data structure.

AB - Given an n-bit array A, the succinct rank data structure problem asks to construct a data structure using space n + r bits for r ≪ n, supporting rank queries of form rank(u) = Íui=−01 A[i]. In this paper, we design a new succinct rank data structure with r = n/(log n)Ω(t) + n1−c and query time O(t) for some constant c > 0, improving the previous best-known by Pǎtraşcu, which has r = n/(logtn )Ω(t) + Õ(n3/4) bits of redundancy. For r > n1−c, our space-time tradeoff matches the cell-probe lower bound by Pǎtraşcu and Viola, which asserts that r must be at least n/(log n)O(t). Moreover, one can avoid an n1−c-bit lookup table when the data structure is implemented in the cell-probe model, achieving r = ⌈n/(log n)Ω(t)⌉. It matches the lower bound for the full range of parameters. En route to our new data structure design, we establish an interesting connection between succinct data structures and approximate nonnegative tensor decomposition. Our connection shows that for specific problems, to construct a space-efficient data structure, it suffices to approximate a particular tensor by a sum of (few) nonnegative rank-1 tensors. For the rank problem, we explicitly construct such an approximation, which yields an explicit construction of the data structure.

KW - Partial sum

KW - Spillover representation

KW - Succinct data structure

KW - Tensor decomposition

UR - http://www.scopus.com/inward/record.url?scp=85068751877&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068751877&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316352

DO - 10.1145/3313276.3316352

M3 - Conference contribution

AN - SCOPUS:85068751877

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

SP - 955

EP - 966

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 -