# Semi-Group Range Sum Revisited: Query-Space Lower Bound Tightened

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## Abstract

Let \(\mathcal {D}\) be a set of

*n*elements \(e_1,\ldots , e_n\) drawn from a commutative semigroup. Given two integers*x*,*y*satisfying \(1 \le x \le y \le n\), a*range sum query*returns the sum of the \(y-x+1\) elements \(e_x\), \(e_{x+1}\),..., \(e_y\). The goal of indexing is to store \(\mathcal {D}\) in a data structure so that all such queries can be answered efficiently in the worst case. This paper proves a new lower bound in the semigroup model on the tradeoff between space and query time for the above problem. We show that, if the query time needs to be at most an integer*t*, a structure must use space. The bound is asymptotically tight for*every*\(t \ge 2\), and is matched by an existing structure. Previously, the best lower bounds either had a substantially smaller non-linear factor (Yao in Space-time tradeoff for answering range queries (extended abstract). In: STOC, pp. 128–136, 1982), or were tight only for constant*t*(Alon and Schieber in Optimal preprocessing for answering on-line product queries. Technical Report TR 71/87, Tel-Aviv University, 1987). Our lower bound is asymptotically tight*bidirectionally*, namely, it also answers the following question: if the space needs to be bounded by an integer*m*, what is the best query time achievable? The techniques behind our lower bound are drastically different from those of Yao (Space-time tradeoff for answering range queries (extended abstract). In: STOC, pp. 128–136, 1982) and Alon and Schieber (Optimal preprocessing for answering on-line product queries. Technical Report TR 71/87, Tel-Aviv University, 1987), and reveal new insight on the characteristics of the problem.## Keywords

Range sum queries Semi-group Lower bound Length decomposition property## Notes

### Acknowledgements

Shuigeng Zhou was supported by the Key Projects of Fundamental Research Program of Shanghai Municipal Commission of Science and Technology under grant No. 14JC1400300.

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