arxiv: 2605.14112 · v1 · submitted 2026-05-13 · 💻 cs.DS · cs.CC

Recognition: no theorem link

Fast Leaf-to-Ancestor Minimum Query in the Oracle Model

Aleksey Upirvitskiy , Aleksandr Levin

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:51 UTC · model grok-4.3

classification 💻 cs.DS cs.CC

keywords data structurestree queriesoracle modelminimum queriespreprocessingconstant timeladder decompositionbinary lifting

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The pith

A static data structure preprocesses a rooted tree in O(n log h) oracle comparisons to answer any leaf-to-ancestor minimum query in O(1) time with no comparisons at query time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to build a data structure for minimum queries on paths from any leaf up to any ancestor in a rooted weighted tree, where the only way to access weights is through a comparison oracle. After O(n log h) preprocessing time, space, and oracle calls, every such query is answered in constant time using only precomputed indices. The construction first converts edge weights to node weights with a deterministic tie-break to create a total order, then decomposes the tree into ladders, applies binary lifting, and stores a sparse table of minimum indices over the ladder arrays. The preprocessing bound is shown to be tight in the deterministic comparison model.

Core claim

By converting edge weights to node weights via a deterministic tie-break to obtain a total order, performing a ladder decomposition of the tree, applying binary lifting to locate ladder heads, and building a sparse-table RMQ over the arrays of ladder representatives while storing the indices chosen by the oracle during preprocessing, one obtains a static structure that uses O(n log h) preprocessing oracle calls and answers every leaf-to-ancestor minimum query in O(1) worst-case time with zero oracle calls at query time.

What carries the argument

Ladder decomposition combined with sparse-table RMQ on ladder arrays, where indices are selected by the comparison oracle during preprocessing.

If this is right

Every leaf-to-ancestor minimum query is answered in constant time after the stated preprocessing.
The number of oracle comparisons required for preprocessing is O(n log h) and this bound is tight.
The same structure works whether weights are initially on edges or nodes after the conversion step.
Query-time access requires only array lookups and no further comparisons or oracle calls.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same ladder-plus-sparse-table pattern could be tested on related path aggregates such as maximum or range-sum if the oracle model is relaxed.
Because the structure is entirely static, any dynamic tree setting would require additional mechanisms to maintain the ladders and sparse tables under insertions or deletions.
The O(1) query bound makes the structure attractive for repeated minimum lookups along rootward paths in large trees where oracle calls are the dominant cost.

Load-bearing premise

The input is a static rooted tree whose weights admit a total order through a consistent, deterministic comparison oracle with a fixed tie-breaking rule.

What would settle it

A family of trees on which any structure answering leaf-to-ancestor minima in O(1) time must perform omega(n log h) oracle comparisons in the worst case during preprocessing would falsify the claimed tightness.

read the original abstract

We study leaf-to-ancestor path-minimum queries on a rooted, weighted tree in the oracle model, where the only allowed value operation is a comparison oracle on edge (or node) weights. We give a static data structure that, after O(n log h) preprocessing time, space, and oracle calls (where $n$ is the number of nodes and $h$ is the tree height), answers any leaf-to-ancestor query in O(1) worst-case time with zero oracle calls at query time. The method combines (I) an edge-to-node weight conversion with a deterministic tie-break to obtain a total order; (II) ladder (longest-path) decomposition; (III) binary lifting; and (IV) sparse-table RMQ built over ladder arrays, storing indices selected via the oracle during preprocessing. We also show that the preprocessing oracle-comparison bound is tight in the deterministic comparison model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an O(n log h) preprocessing structure for leaf-to-ancestor min queries with O(1) zero-oracle query time via ladders, lifting, and sparse tables, plus a matching lower bound, but the zero-oracle query claim rests on avoiding a final comparison in the RMQ step.

read the letter

The main takeaway is that this paper builds a static structure for leaf-to-ancestor minimum queries on weighted trees in the comparison oracle model. After O(n log h) preprocessing time, space, and oracle calls it answers any such query in O(1) time with no oracle calls during the query itself, and it shows the preprocessing comparison count is tight in the deterministic model. The approach converts weights to a total order with tie-breaking, decomposes the tree into ladders, uses binary lifting to reach the right ladder segment, and runs sparse-table RMQ on precomputed indices from those ladders. The lower bound on preprocessing comparisons is a clean addition that pins down the cost of the setup phase. The combination is new for this exact query type and model, and the high-level reuse of standard primitives is handled cleanly in the abstract. The tie-break rule and consistency assumption on the oracle are standard and do not add hidden cost. The soft spot is the zero-oracle query claim. A plain sparse table on two candidate indices normally requires one extra comparison to pick the smaller weight, yet the paper asserts zero calls at query time while staying inside the O(n log h) budget. The construction must therefore arrange that every query reduces to a single pre-resolved index or uses an auxiliary structure that encodes the winner without extra comparisons or space. The abstract does not spell out how the ladder-plus-lifting step guarantees this, so the full proofs need to show the mechanism explicitly. If that detail holds, the result is solid; if it reduces to one oracle call, the claim would need a small adjustment. This is aimed at people working on comparison-model data structures and tree path aggregates. A reader who already uses ladder decompositions or needs tight oracle bounds would find the technique and the lower bound useful. The paper deserves a serious referee because the stated bounds are clean and the combination is plausible once the query-time detail is checked.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a static data structure for leaf-to-ancestor path-minimum queries on a rooted, weighted tree in the comparison oracle model. After O(n log h) preprocessing time, space, and oracle calls, any query is answered in O(1) worst-case time using zero oracle calls at query time. The construction combines edge-to-node weight conversion with deterministic tie-breaking for a total order, ladder (longest-path) decomposition, binary lifting, and sparse-table RMQ built over ladder arrays that store indices pre-selected via the oracle. The paper also proves that the O(n log h) preprocessing oracle bound is tight in the deterministic comparison model.

Significance. If the central claims hold, the result is significant for oracle-model data structures: it delivers constant-time queries with no runtime comparisons, which is non-trivial because standard RMQ techniques incur at least one comparison per query. The explicit use of ladder decomposition plus binary lifting to pre-resolve all minima, together with the matching lower bound, strengthens the contribution and provides a clean optimality statement. The work usefully demonstrates how standard primitives can be combined to eliminate query-time oracle calls within a tight budget.

major comments (2)

[Query algorithm (likely §4)] The zero-oracle query claim is load-bearing for the main result, yet the abstract and high-level description leave open how the final step of sparse-table RMQ is performed without a runtime comparison. Standard sparse tables retrieve two candidate indices i1 and i2 for the two overlapping blocks and must compare their weights; the manuscript must show explicitly (in the query algorithm section) that the leaf-to-ancestor restriction plus precomputed ladder indices reduces every query to a single pre-resolved index, or that an auxiliary O(n log h)-space winner table resolves the choice without an oracle call.
[Lower-bound section] The tightness proof in the deterministic comparison model is stated but its argument is not cross-referenced to the concrete upper-bound construction. The lower-bound section should contain a short paragraph showing that any data structure achieving O(1) query time with zero query oracle calls must perform Ω(n log h) preprocessing comparisons in the worst case, matching the ladder height and binary-lifting levels used in the upper bound.

minor comments (2)

[Introduction or §3] A small worked example (e.g., a 10-node tree with explicit ladders and sparse-table entries) would clarify how a leaf-to-ancestor query is reduced to a single index lookup.
[Preliminaries] The notation for tree height h and the precise definition of a 'ladder' should appear in the first paragraph of the preliminaries rather than being introduced piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: The zero-oracle query claim is load-bearing for the main result, yet the abstract and high-level description leave open how the final step of sparse-table RMQ is performed without a runtime comparison. Standard sparse tables retrieve two candidate indices i1 and i2 for the two overlapping blocks and must compare their weights; the manuscript must show explicitly (in the query algorithm section) that the leaf-to-ancestor restriction plus precomputed ladder indices reduces every query to a single pre-resolved index, or that an auxiliary O(n log h)-space winner table resolves the choice without an oracle call.

Authors: We agree that the query procedure needs more explicit detail. In our construction the ladder decomposition combined with binary lifting produces, for every leaf-to-ancestor path, a contiguous segment of a precomputed array whose minimum index is already selected by oracle calls during preprocessing. Because every query reduces to a single such pre-resolved index, the sparse-table lookup returns that index directly with no weight comparison at query time. We will add a dedicated subsection with pseudocode and a step-by-step walk-through of the query algorithm to make this reduction explicit. revision: yes
Referee: The tightness proof in the deterministic comparison model is stated but its argument is not cross-referenced to the concrete upper-bound construction. The lower-bound section should contain a short paragraph showing that any data structure achieving O(1) query time with zero query oracle calls must perform Ω(n log h) preprocessing comparisons in the worst case, matching the ladder height and binary-lifting levels used in the upper bound.

Authors: We will insert a short bridging paragraph in the lower-bound section that directly links the Ω(n log h) bound to the parameters of the upper-bound construction. The argument will note that supporting O(1)-time, zero-oracle queries requires pre-resolving the minimum for every possible ladder segment and every binary-lifting jump; distinguishing these configurations in the worst case necessitates Ω(n log h) comparisons, matching the height-h ladders and log h lifting levels used in the data structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent standard primitives.

full rationale

The claimed data structure is assembled from externally validated components (ladder decomposition, binary lifting, and sparse-table RMQ) whose correctness does not depend on the target O(1) zero-oracle query bound. Preprocessing selects minimum indices via the allowed O(n log h) oracle calls and stores them; the query procedure then directly returns a pre-selected index without further reduction to fitted parameters or self-referential equations. The separate lower-bound argument for preprocessing tightness is a standard information-theoretic comparison-model argument and does not invoke the upper-bound construction. No self-citation is load-bearing for the central claim, and no step equates the output bounds to the input assumptions by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definition of a rooted tree and the existence of a consistent comparison oracle; no free parameters are introduced and no new entities are postulated.

axioms (1)

domain assumption The input is a finite rooted tree with n nodes and height h whose edge or node weights can be compared by an oracle that returns a consistent total order after deterministic tie-breaking.
This is the standard model assumption stated in the abstract for the oracle setting.

pith-pipeline@v0.9.0 · 5455 in / 1218 out tokens · 47877 ms · 2026-05-15T01:51:31.411426+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

arXiv:2105.01864 [cs.DS] (2021).https://arxiv.org/abs/2105.01864

Yang, T.: Tree Path Minimum Query Oracle via Borůvka Trees. arXiv:2105.01864 [cs.DS] (2021).https://arxiv.org/abs/2105.01864

work page arXiv 2021
[2]

Algorithmica 18, 263–270 (1997).https://doi.org/10.1007/BF02526037

King, V.: A Simpler Minimum Spanning Tree Verification Algorithm. Algorithmica 18, 263–270 (1997).https://doi.org/10.1007/BF02526037

work page doi:10.1007/bf02526037 1997
[3]

Theo- retical Computer Science

Bender, M.A., Farach-Colton, M.: The Level Ancestor Problem simplified. Theo- retical Computer Science. 321(1), 5–12 (2004).https://doi.org/10.1016/j.tcs. 2003.05.002

work page doi:10.1016/j.tcs 2004