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 -