TY - GEN

T1 - Lower bound for succinct range minimum query

AU - Liu, Mingmou

AU - Yu, Huacheng

N1 - Funding Information:
∗This research was supported by National Key R&D Program of China 2018YFB1003202 and the NSFC Nos. 61722207 and 61672275. †Part of the research was done when Mingmou Liu was visiting the Harvard University.
Publisher Copyright:
© 2020 ACM.

PY - 2020/6/8

Y1 - 2020/6/8

N2 - Given an integer array A[1..n], the Range Minimum Query problem (RMQ) asks to preprocess A into a data structure, supporting RMQ queries: given a,bg [1,n], return the index ig[a,b] that minimizes A[i], i.e., argminig[a,b] A[i]. This problem has a classic solution using O(n) space and O(1) query time by Gabow, Bentley, Tarjan (STOC, 1984) and Harel, Tarjan (SICOMP, 1984). The best known data structure by Fischer, Heun (SICOMP, 2011) and Navarro, Sadakane (TALG, 2014) uses 2n+n/(logn/t)t+Õ(n3/4) bits and answers queries in O(t) time, assuming the word-size is w=(logn). In particular, it uses 2n+n/polylogn bits of space as long as the query time is a constant. In this paper, we prove the first lower bound for this problem, showing that 2n+n/polylogn space is necessary for constant query time. In general, we show that if the data structure has query time O(t), then it must use at least 2n+n/(logn)Õ(t2) space, in the cell-probe model with word-size w=(logn).

AB - Given an integer array A[1..n], the Range Minimum Query problem (RMQ) asks to preprocess A into a data structure, supporting RMQ queries: given a,bg [1,n], return the index ig[a,b] that minimizes A[i], i.e., argminig[a,b] A[i]. This problem has a classic solution using O(n) space and O(1) query time by Gabow, Bentley, Tarjan (STOC, 1984) and Harel, Tarjan (SICOMP, 1984). The best known data structure by Fischer, Heun (SICOMP, 2011) and Navarro, Sadakane (TALG, 2014) uses 2n+n/(logn/t)t+Õ(n3/4) bits and answers queries in O(t) time, assuming the word-size is w=(logn). In particular, it uses 2n+n/polylogn bits of space as long as the query time is a constant. In this paper, we prove the first lower bound for this problem, showing that 2n+n/polylogn space is necessary for constant query time. In general, we show that if the data structure has query time O(t), then it must use at least 2n+n/(logn)Õ(t2) space, in the cell-probe model with word-size w=(logn).

KW - Cell-probe complexity

KW - Data structure

KW - Lower bound

KW - Range minimum query

UR - http://www.scopus.com/inward/record.url?scp=85086761821&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85086761821&partnerID=8YFLogxK

U2 - 10.1145/3357713.3384260

DO - 10.1145/3357713.3384260

M3 - Conference contribution

AN - SCOPUS:85086761821

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

SP - 1402

EP - 1415

BT - STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Makarychev, Konstantin

A2 - Makarychev, Yury

A2 - Tulsiani, Madhur

A2 - Kamath, Gautam

A2 - Chuzhoy, Julia

PB - Association for Computing Machinery

T2 - 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020

Y2 - 22 June 2020 through 26 June 2020

ER -