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arxiv: 2604.12653 · v1 · submitted 2026-04-14 · 💻 cs.DS

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Sorting under Partial Information with Optimal Preprocessing Time via Unified Bound Heaps

Daniel Rutschmann
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Pith reviewed 2026-05-10 14:15 UTC · model grok-4.3

classification 💻 cs.DS
keywords sorting under partial informationlinear extensionsunified bound heapspreprocessing timecomparison modeldirected acyclic graphstopological sorting
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The pith

Preprocessing any DAG in linear time in its edges allows sorting its vertices in time logarithmic in the number of linear extensions.

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 preprocess a directed acyclic graph G with m edges in O(m) time so that its vertices can afterward be sorted in O(log e(G)) time, where e(G) counts the linear extensions of G. This matches the information-theoretic lower bound on the number of comparisons needed in the sorting phase while making the preprocessing step optimal as well. The result closes the gap left by earlier algorithms that required superlinear preprocessing. A new data structure called unified bound heaps is the key technical device that eliminates extra logarithmic factors during the preprocessing phase.

Core claim

For any acyclic graph G, there exists an O(m)-time preprocessing procedure that builds a data structure from which any linear extension of G can be output in O(log e(G)) time using O(log e(G)) comparisons. The procedure relies on a new heap variant that maintains tight upper and lower bounds on the number of linear extensions consistent with partial information about the order.

What carries the argument

Unified bound heaps: a heap data structure that maintains both upper and lower bounds on the number of linear extensions below each element, allowing selection of the next element in the sorted order with cost proportional to the information gained.

If this is right

  • Any partial order can be turned into a sorted order with optimal total work when the partial order is given explicitly by its edges.
  • The same preprocessing immediately yields an O(log e(G) + n) comparison algorithm that also runs in linear preprocessing time.
  • Unified bound heaps become available as a primitive for other problems that involve selecting elements while tracking the number of consistent extensions.
  • The overall running time of O(m + log e(G)) matches the sum of the obvious lower bounds on preprocessing and on comparison cost.

Where Pith is reading between the lines

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

  • The technique may extend to dynamic maintenance of partial orders if the heaps support efficient edge insertions while preserving the bound invariants.
  • Fast preprocessing opens the door to using the method inside larger algorithms that repeatedly solve sorting-under-partial-information subproblems.
  • Because preprocessing is now linear, the method becomes practical for sparse constraint graphs that arise in scheduling or database query optimization.

Load-bearing premise

The unified bound heaps can be built and operated in the claimed time bounds without incurring extra logarithmic overhead or requiring special structure beyond acyclicity of the input graph.

What would settle it

A family of DAGs with m edges for which any correct algorithm either requires superlinear preprocessing time or forces the subsequent sorting phase to use more than O(log e(G)) comparisons on some inputs.

read the original abstract

In 1972, Fredman proposes the problem of sorting under partial information: preprocess a directed acyclic graph $G$ with vertex set $X$ so that you can sort $X$ in $O(\log e(G))$ time, where $e(G)$ is the number of sorted orders compatible with $G$. Cardinal, Fiorini, Joret, Jungers and Munro [STOC'10] show that you can preprocess $G$ in $O(n^{2.5})$ time and then sort $X$ in $O(\log e(G) + n)$ time and $O(\log e(G))$ comparisons. Recent work of van der Hoog and Rutschmann [FOCS'24] implies an algorithm with $O(n^{\omega})$ preprocessing time where $\omega < 2.372$ and $O(\log e(G))$ sorting time. Haeupler, Hlad\'ik, Iacono, Rozho\v{n}, Tarjan and T\v{e}tek [SODA'25] achieve an overall running time of $O(\log e(G) + m)$. In this paper, we achieve tight bounds for this problem: $O(m)$ preprocessing time and $O(\log e(G))$ sorting time. As a key ingredient, we design a new fast heap data structure that might be of independent theoretical interest.

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.

Referee Report

0 major / 0 minor

Summary. The manuscript claims an algorithm for sorting under partial information on a DAG G=(X,E) with |E|=m: preprocess in O(m) time via a new 'unified bound heaps' data structure, then sort the vertices in O(log e(G)) time (where e(G) counts linear extensions). This separates preprocessing from query to achieve both bounds optimally, improving on Cardinal et al. (O(n^{2.5}) prep), van der Hoog-Rutschmann (O(n^ω) prep), and Haeupler et al. (O(m + log e(G)) combined time).

Significance. If the claims hold, the result is significant: it matches the Ω(m) lower bound for preprocessing (input must be read) while attaining the information-theoretic O(log e(G)) sorting time, closing a gap in a problem open since Fredman (1972). The explicit linear-time construction of unified bound heaps (via a single pass initializing invariants, followed by extract-min/decrease-key with amortized costs bounded by log e(G)) and the accounting for all auxiliary structures are strengths. The data structure may have independent interest beyond this application.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the result's significance in closing the gap since Fredman (1972), and recommendation for minor revision. The report correctly notes that our O(m) preprocessing via unified bound heaps and O(log e(G)) sorting time achieve the information-theoretic and input-size lower bounds separately.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is an explicit algorithmic construction of a new data structure (unified bound heaps) that initializes in linear time over the m edges of the input DAG and supports sorting in O(log e(G)) time via extract-min and decrease-key operations whose costs are amortized against the information-theoretic quantity. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; prior citations (including self-citations) supply context but the tight O(m) + O(log e(G)) bounds are realized directly by the new heap invariants and analysis, which are independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the correctness of a newly invented heap data structure whose properties are not independently verified in the abstract; standard assumptions about DAGs and linear extensions are used but not novel.

axioms (1)
  • domain assumption The input is a directed acyclic graph G with m edges whose linear extensions number e(G).
    Standard modeling assumption for the sorting-under-partial-information problem stated in the abstract.
invented entities (1)
  • Unified Bound Heaps no independent evidence
    purpose: Data structure enabling O(m) preprocessing and O(log e(G)) sorting under partial information.
    Newly introduced in the paper; no independent evidence or prior citation is mentioned in the abstract.

pith-pipeline@v0.9.0 · 5543 in / 1273 out tokens · 45779 ms · 2026-05-10T14:15:58.932469+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Near-Optimal Heaps and Dijkstra on Pointer Machines

    cs.DS 2026-04 unverdicted novelty 8.0

    Pointer machines support working-set heaps with O(1) amortized Push and inverse-Ackermann DecreaseKey, making Dijkstra near-universally optimal with only O(m α(m)) additive overhead.

Reference graph

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