Optimal In-place Algorithms for Basic Graph Problems
Pith reviewed 2026-05-24 17:50 UTC · model grok-4.3
The pith
Linear time in-place algorithms solve depth-first search, biconnectivity, and other core graph problems by exploiting input sortedness.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By detecting and carefully exploiting the sortedness present in the input representation for any graph without loss of generality, linear time in-place algorithms can be obtained for graph search methods including depth-first search, breadth-first search and maximum cardinality search, for connectivity problems including biconnectivity and 2-edge connectivity, and for decomposition problems including chain decomposition.
What carries the argument
Detection and exploitation of sortedness in the input graph representation without loss of generality.
If this is right
- All listed problems can be solved in O(n + m) time with O(1) extra space.
- The algorithms improve the prior O(n^3 lg n) in-place bounds by a polynomial factor.
- The obtained running times are optimal because they match the time needed to read the input.
- The same sortedness observation yields basic linear time in-place solutions for several problems directly.
Where Pith is reading between the lines
- The sortedness technique may extend to additional graph problems that currently lack linear in-place solutions.
- In memory-constrained settings the approach allows full graph processing without allocating auxiliary arrays.
- Similar assumptions about detectable order in input data could be tested for other combinatorial structures.
Load-bearing premise
The input representation for any graph contains detectable sortedness that can be exploited without loss of generality.
What would settle it
An input graph format in which adjacency lists cannot be detected or made sorted in linear time without extra space or asymptotic slowdown.
read the original abstract
We present linear time {\it in-place} algorithms for several basic and fundamental graph problems including the well-known graph search methods (like depth-first search, breadth-first search, maximum cardinality search), connectivity problems (like biconnectivity, $2$-edge connectivity), decomposition problem (like chain decomposition) among various others, improving the running time (by polynomial multiplicative factor) of the recent results of Chakraborty et al. [ESA, 2018] who designed $O(n^3 \lg n)$ time in-place algorithms for a strict subset of the above mentioned problems. The running times of all our algorithms are essentially optimal as they run in linear time. One of the main ideas behind obtaining these algorithms is the detection and careful exploitation of sortedness present in the input representation for any graph without loss of generality. This observation alone is powerful enough to design some basic linear time in-place algorithms, but more non-trivial graph problems require extra techniques which, we believe, may find other applications while designing in-place algorithms for different graph problems in the future.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims linear-time in-place algorithms for fundamental graph problems including DFS, BFS, maximum cardinality search, biconnectivity, 2-edge connectivity, and chain decomposition. These improve the O(n³ log n) time in-place algorithms of Chakraborty et al. (ESA 2018) for a subset of the problems. The central technique is the detection and exploitation of sortedness present in any graph input representation, which is asserted to hold without loss of generality and to suffice for the linear-time in-place bounds.
Significance. If the central claims hold, the results would be significant: they would establish optimal (linear-time) in-place algorithms for a broad suite of basic graph primitives, closing a substantial gap from the prior polynomial-factor overhead. The sortedness-exploitation idea, if rigorously justified, could serve as a reusable primitive for other space-constrained graph algorithms.
major comments (1)
- [Abstract (final paragraph)] Abstract, final paragraph: The claim that 'detectable sortedness present in the input representation for any graph' can be exploited WLOG to obtain linear-time in-place algorithms is load-bearing for all stated running-time bounds. No argument is supplied showing that the detection step itself runs in O(n+m) time using O(1) (or O(log n)) extra space on a standard adjacency-list encoding; standard inputs are not required to be sorted, and any implicit sorting or certification step would normally require Ω(m log n) comparisons or auxiliary space, violating the claimed bounds.
Simulated Author's Rebuttal
Thank you for the careful review and for identifying the load-bearing claim in the abstract. We address the concern point-by-point below and will revise the manuscript to strengthen the presentation.
read point-by-point responses
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Referee: Abstract, final paragraph: The claim that 'detectable sortedness present in the input representation for any graph' can be exploited WLOG to obtain linear-time in-place algorithms is load-bearing for all stated running-time bounds. No argument is supplied showing that the detection step itself runs in O(n+m) time using O(1) (or O(log n)) extra space on a standard adjacency-list encoding; standard inputs are not required to be sorted, and any implicit sorting or certification step would normally require Ω(m log n) comparisons or auxiliary space, violating the claimed bounds.
Authors: We agree that the abstract states the claim without supplying a self-contained argument for the detection step, and that this justification must be made explicit. In the revision we will add a dedicated paragraph (and a short subsection in the Preliminaries) that shows detection consists of a single linear scan: for each adjacency list we compare consecutive neighbor IDs with a constant number of index variables, using only O(1) extra words and O(n+m) time. We will also clarify the WLOG statement: any undirected graph admits an adjacency-list representation in which neighbor lists appear in sorted vertex-ID order; when the given input happens to be in that form the algorithms exploit it directly. When the input lists are unsorted the paper’s additional techniques (described in Sections 3–5) still achieve linear time without performing an explicit sort. The revised text will make the assumption and the fallback explicit so that the linear-time in-place bounds are rigorously supported. revision: yes
Circularity Check
No significant circularity in derivation chain
full rationale
The paper presents explicit algorithmic constructions for linear-time in-place graph algorithms (DFS, BFS, biconnectivity, etc.) that improve on a cited prior result. The key observation about detectable sortedness in input representations is introduced as an independent WLOG claim rather than a self-definition or fitted parameter renamed as output. Self-citation to Chakraborty et al. [ESA 2018] functions only as a performance baseline, not as the load-bearing justification for the new linear-time bounds. No equation or step reduces the claimed results to the inputs by construction, and the derivation remains self-contained against external algorithmic benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Input graph representations contain detectable sortedness exploitable without loss of generality.
discussion (0)
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