Range-to-Prefix Conversion Algorithm

• Range-to-prefix conversion algorithms transform range queries into a bounded set of prefix queries, enabling efficient data structure mapping and query execution.
• They underpin key methodologies in succinct string indexing and integer range filtering by leveraging ordered binary representations and dyadic intervals.
• These reductions offer bidirectional query support, preserving order-theoretic properties and reducing query times through translatable prefix operations.

Searching arXiv for the cited papers to ground the article in current records. Range-to-prefix conversion denotes a family of reductions in which a range query is recast as a bounded number of prefix queries on a derived representation. In current arXiv usage, the term has two principal technical realizations. In succinct string indexing, lexicographic-range queries on the suffix array of a text $T$ are reduced to prefix-select queries on a sequence $W$ of short binary strings, yielding an equivalence between suffix-array functionality and prefix-query structures up to an additive $O(\log\log n)$ query-time term (Kempa et al., 22 Oct 2025). In approximate range filtering, an integer interval $[\ell,r]$ is converted into a minimum-cardinality family of dyadic intervals, each represented by a binary prefix, so that range lookups become a small number of prefix probes (Mößner et al., 2022). A closely related dual construction converts colored-prefix queries into colored-range queries in the external-memory model, showing that prefix/range conversion is a bidirectional design pattern rather than a single domain-specific routine (Larsen et al., 2011).

1. Formal query models

In the suffix-array setting, the source problem is the lexicographic-range query. Given a text $T \in [0..\sigma)^n$, one builds a data structure on $T$ of size $S(n)$, and on input two patterns $P_1,P_2 \in [0..\sigma)^*$ returns

$$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$

where

$$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$

Equivalently, one may ask for reporting all positions whose suffixes lie in the half-open interval $W$0 in lexicographic order, or just its minimum. The target problem is prefix-select on a sequence $W$1 of $W$2 binary strings of length exactly $W$3; given a query bit-pattern $W$4 and a rank $W$5, the query returns

$W$6

the $W$7-th smallest $W$8 such that $W$9 is a prefix of $O(\log\log n)$0 (Kempa et al., 22 Oct 2025).

A second formulation arises in integer range filtering. There the universe is

$O(\log\log n)$1

and the task is to cover a query interval $O(\log\log n)$2 exactly by a small collection of dyadic intervals. Each dyadic interval corresponds to a binary prefix of length $O(\log\log n)$3 for some $O(\log\log n)$4, so the range query is transformed into a prefix set. The required cover is exact, and the number of prefixes is minimized by the standard greedy decomposition (Mößner et al., 2022).

These formulations differ in object type—suffix-array intervals versus integer intervals—but they share a common abstraction: a range is replaced by prefix descriptors whose order-theoretic structure is easier to index or hash.

2. Embedded sequence construction and occurrence mapping

The core construction underlying the string-theoretic reduction starts from a sequence $O(\log\log n)$5 of bit-strings of length $O(\log\log n)$6. Let $O(\log\log n)$7. For $O(\log\log n)$8, define

$O(\log\log n)$9

where `%%%%3$T$3%%%%1\mathrm{bin}(x,k)$[\ell,r]$2k$[\ell,r]$3x$[\ell,r]$4$[\ell,r]$5$[\ell,r]$6$[\ell,r]$7$[\ell,r]$8W$[\ell,r]$9T$$(<a href="/papers/2510.19815" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Kempa et al., 22 Oct 2025</a>).</p> <p>For a query prefix$$T \in [0..\sigma)^n$0 with $T \in [0..\sigma)^n$1, let $T \in [0..\sigma)^n$2, and let $T \in [0..\sigma)^n$3 denote the $T \in [0..\sigma)^n$4-th index of $T \in [0..\sigma)^n$5 carrying $T \in [0..\sigma)^n$6 as a prefix. The key observation is

$T \in [0..\sigma)^n$7

These occurrences appear in the same order in $T \in [0..\sigma)^n$8 as the increasing indices $T \in [0..\sigma)^n$9. If

$T$0

then for any $T$1,

$T$2

This occurrence mapping is the structural bridge between suffix-array intervals and prefix-select answers.

The same construction also supports the converse viewpoint. The summary states that any lex-range query on $T$3 can be answered by one $T$4 call plus repeated prefix-select queries on $T$5, while any prefix-select can be implemented by two $T$6 calls and one small-range access on $T$7. The reduction is therefore bidirectional rather than merely one-way.

3. Answering lexicographic ranges through prefix-select

The operational form of the range-to-prefix reduction assumes that one can answer $T$8 in $T$9 time. A lex-range query proceeds by computing

$S(n)$0

If $S(n)$1, the output is empty. In the simple case emphasized in the description, the patterns share a short $S(n)$2-prefix, handled by taking a common prefix $S(n)$3 up to $S(n)$4 or truncated patterns, with the remaining cases handled by a nonperiodic/periodic split. Let

$S(n)$5

Instead of enumerating suffix-array entries directly, one performs $S(n)$6 prefix-select queries of the form $S(n)$7,i)$ and returns the corresponding outputs (Kempa et al., 22 Oct 2025).

The reported complexity is that this answers all $S(n)$8 elements of the lex-range in

$S(n)$9

More broadly, any data structure using $P_1,P_2 \in [0..\sigma)^*$0 bits, query time $P_1,P_2 \in [0..\sigma)^*$1, build time $P_1,P_2 \in [0..\sigma)^*$2, and working space $P_1,P_2 \in [0..\sigma)^*$3 for suffix-array or lex-range queries on a text of length $P_1,P_2 \in [0..\sigma)^*$4 can be transformed in $P_1,P_2 \in [0..\sigma)^*$5 time into a structure that answers prefix-select queries on $P_1,P_2 \in [0..\sigma)^*$6 short strings in

$P_1,P_2 \in [0..\sigma)^*$7

time and

$P_1,P_2 \in [0..\sigma)^*$8

space, and vice versa.

This equivalence is stronger than a single reduction. It states that suffix-array queries are, up to an additive $P_1,P_2 \in [0..\sigma)^*$9 term in query time, equivalent to prefix-select queries in all parameters. The framework also proves analogous equivalences for inverse suffix-array queries, pattern ranking, lexicographic range, and $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$0-interval queries, thereby identifying six core problem pairs connecting string and prefix-query models.

4. Dyadic decomposition as range-to-prefix conversion

In bloomRF, the range-to-prefix conversion is the standard greedy-from-the-left algorithm for covering an interval $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$1 by the minimum number of dyadic intervals. Starting with $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$2, each iteration computes

$$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$3

and sets

$$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$4

The chosen dyadic block has size $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$5, its corresponding prefix length is $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$6 bits, and the prefix value is the top $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$7 bits of $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$8, namely $$\mathrm{LexRange}_T(P_1,P_2) \coloneqq \{\, SA_T[i] : i \in (\mathrm{RangeBeg}_T(P_1)\ldots \mathrm{RangeBeg}_T(P_2)] \,\},$$9. The algorithm appends this prefix and advances $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$0 by the interval size (Mößner et al., 2022).

The decomposition size satisfies two bounds. The loop executes at most

$$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$1

steps in the worst case. If the interval length is at most $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$2, then

$$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$3

The cover is exact, and the algorithm produces the minimum number of dyadic intervals covering $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$4.

In bloomRF these prefixes are consumed by piecewise-monotone hash-functions. A dyadic interval of size $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$5 corresponds exactly to one word in the bit-array. If a prefix has length $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$6, then it lives at level $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$7, which is mapped to layer $$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$8. Because of the final term

$$\mathrm{RangeBeg}_T(P)=|\{\, j : T[j\ldots n]\prec P \,\}|.$$9

the least significant $W$00 bits of the prefix are mapped in order, so that any contiguous set of up to $W$01 items inside a dyadic block sits in adjacent bits in a single word. This is the piecewise-monotone property.

The resulting range-to-prefix conversion runs in $W$02, and a range query performs one word lookup per prefix. The formal analysis further shows that, because the structure probes at most two words per layer, a range lookup takes $W$03 time, independent of $W$04.

5. External-memory dual: prefix-to-range conversion

A closely related dual reduction appears in external-memory colored reporting. The colored range reporting problem is defined on sets

$W$05

with total size $W$06, and asks for

$W$07

on input $W$08. The colored prefix reporting problem is defined on a static collection

$W$09

of strings, each of length $W$10, with a function

$W$11

and on input pattern $W$12 asks for

$W$13

The reduction sorts $W$14 lexicographically, assigns each string its rank, and sets

$W$15

All strings having prefix $W$16 occupy a contiguous lexicographic interval $W$17, so colored-prefix reporting becomes colored-range reporting on that interval (Larsen et al., 2011).

The interval computation is performed by a compacted trie, or Patricia trie, storing rank-range pointers at internal nodes. The theorem cited from Belazzougui et al. gives a linear-space structure that, for any pattern $W$18 of length $W$19, returns $W$20 in $W$21 I/Os. The colored-range side is answered by a linear-space structure with query cost

$W$22

I/Os, where $W$23 is the output size. Combining the two stages yields colored prefix reporting in

$W$24

I/Os and $W$25 words of space when $W$26.

The model assumptions are explicit: the standard Aggarwal–Vitter I/O model with block size $W$27 words and unlimited disk, strings of length $W$28 so that each occupies $W$29 blocks, no indivisibility assumption, and static data with no insertions or deletions.

6. Relation to suffix-array functionality and prior reductions

The range-to-prefix viewpoint reorganizes several previously separate reductions into a single prefix-query perspective. The FM-index reduction from suffix-array access to rank on the Burrows–Wheeler Transform is identified as the special case $W$30 of prefix-select, namely $W$31 equal to the single BWT symbol and $W$32$W$33\Psi$-function also reduces to prefix operations on the BWT array. Recent sublinear-build structures, described as using a variant of prefix-rank/select on synchronizing sets, fit the same template once the range-to-prefix conversion is made explicit (Kempa et al., 22 Oct 2025).

The principal misconception is to treat range-to-prefix conversion as a single algorithmic trick tied to one data structure. The literature instead exhibits a family of reductions with different ambient models: succinct suffix-array functionality on arbitrary-alphabet texts, approximate range filters on integer universes, and external-memory colored reporting. What remains invariant is the reduction principle: encode a range by prefix descriptors that preserve the order needed by the target structure.

A plausible implication is that prefix queries serve as a canonical intermediate representation for several string and range-reporting tasks. The 2025 equivalence result makes that interpretation explicit by showing that essentially all efficient suffix-array representations can be expressed via prefix-select structures, while the 2011 external-memory result and the 2022 bloomRF construction show that analogous prefix encodings also govern optimal I/O-efficient reporting and compact approximate range filters.