Extracting exact and complete information using range queries is not only time-inefficient, on average, but also susceptible to privacy breaches. As a remedy, Demertzis et al.~\cite{demertzis2016practical} (SIGMOD'16) introduced, in the context of searchable symmetric encryption, the principle of covering a query with a single canonical range using a DAG structure (TDAG). It was later formalized by Falzon et al.~\cite{falzon2022range} (VLDB'22) as the single range cover (SRC) search algorithm -- a faster and potentially privacy-preserving search principle. SRC-search can, however, produce false positives. In this work we study how a family of specific Directed Acyclic Graphs (DAGs) could substantially decrease the false-positive ratio, without asymptotic increase of SRC-search time and storage, compared to the classical 1D-Tree. We perform a competitive analysis of a $c$-DAG with respect to the 1D-Tree and derive only an {\bf\em additive constant Returned Level Overhead} while reaching a substantial {\bf\em multiplicative logarithmic improvement in the Accuracy Ratio}, on average. We also provide a generic framework extending our results to non-uniform datasets whose level overhead distribution is close to that of a uniform dataset. We define and analyze several single-path local selection rules that can substantially speed up the original SRC-search algorithm, which was based on exhaustive search, and their privacy implications. Experiments on the Gowalla and NYC Yellow Taxi datasets show that the level overhead stays at or below the proven constant, up to sampling error, and that the level overhead distribution of real data is close to a uniform baseline in most configurations.

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