Pith. sign in

REVIEW 2 major objections 5 minor 28 references

Tracking Paths in Planar Graphs

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that finding the smallest tracking set in a planar graph is NP-hard, and also gives a polynomial-time 4-approximation algorithm and a linear-time exact algorithm for bounded-clique-width graphs when a decomposition is…

desk verdict Good paper with a repairable proof gap in the 4-approximation lemma; the results stand, but the version needs a fix before citing. read the letter →

arxiv 1908.05445 v2 pith:S4GJUEMD submitted 2019-08-15 cs.DM cs.DS

David Eppstein , Michael T. Goodrich , James A. Liu , Pedro Matias This is my paper · ORCID
classification cs.DMcs.DS MSC 05C8568Q1768Q25
keywords approximationalgorithmtrackingpathsplanargraphsNP-hardnessclique-widthmonadicsecond-orderlogic3-SATgraphalgorithms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the problem of placing the fewest trackers on vertices of a graph so that every path from a source s to a destination t records a distinct sequence of visited trackers, allowing an observer to reconstruct which path was taken. It establishes two main facts for planar graphs: the problem is NP-complete, so no polynomial-time exact algorithm is expected, and yet there is a polynomial-time 4-approximation, meaning the tracker set it outputs is never more than four times larger than an optimal set. The approximation is based on the observation that every cycle in a well-reduced planar graph needs only a bounded number of trackers and that the number of faces gives a lower bound on the optimum. It also shows that if the input has bounded clique-width and a clique decomposition is supplied, the exact optimum can be computed in linear time. These results are relevant to surveillance, network monitoring, and tracing the spread of infections or misinformation, where one wants to reconstruct an unseen path from sparse observations.

What carries the argument

The load-bearing object is the entry-exit pair: for a cycle C, an ordered pair (s',t') of vertices on C such that s connects to s', t' connects to t, the two connecting paths are disjoint, and each meets C only at its endpoint. Such a pair describes two alternative s-t paths that differ only inside C, so any tracking set must put a tracker somewhere on C outside {s',t'}. The approximation algorithm is carried by a cycle-characterization lemma from earlier work, together with a planar face-charging argument: adding faces one at a time shows each optimal tracker can account for at most two faces, giving OPT≥(|F|−1)/2; a sequence of safe reductions turns the graph into one where every face is trivially tracked by at most two trackers per face, giving the upper bound. For NP-hardness, the machinery is a gadget reduction from planar 3-satisfiability in which variable gadgets have exactly two minimum tracking sets, one per truth value, and clause gadgets are satisfied exactly when their face is tracked. For bounded clique-width, the machinery is an MSO1 formula expressing that a vertex set is a tracking set, which invokes a general linear-time algorithm for MSO1-expressible optimization problems on such graphs when a clique decomposition is supplied.

What would settle it

Run Algorithm A on all small planar graphs and compare its output size to the exact minimum tracking set found by brute force; a single instance where Algorithm A returns more than four times the optimum would disprove the 4-approximation claim. More narrowly, the stated lemma $\mathrm{ALG} \le 2(|F|-2)$ is already falsified by a triangle with s and t attached as degree-1 leaves, for which |F|=2 and the algorithm outputs one tracker.

Watch

Extended reading notes

Core claim

The paper's central claim is that Planar-Tracking, the optimization version of finding a minimum tracking set in an undirected planar graph with a source s and destination t, is NP-hard but admits a 4-approximation algorithm. The algorithm first applies safe reductions that remove vertices not on any s-t path, contract chains of degree-2 vertices, and force trackers on degree-2 vertices inside small cycles; it then outputs all remaining vertices of degree at least three except s and t. The paper proves the output is always a tracking set, proves it has size at most 2(|F|−2) on reduced graphs, and proves every tracking set has size at least (|F|−1)/2, yielding the ratio 4. The NP-hardness is shown by a polynomial reduction from planar 3-satisfiability using variable and clause gadgets whose minimum tracking sets encode truth assignments. Finally, because the property of being a tracking set is expressible in monadic second-order logic, the exact problem is solvable in linear time on graphs of bounded clique-width when a clique decomposition is given.

Load-bearing premise

The upper-bound proof of the 4-approximation assumes that after the initial reduction no vertex has degree 1, but the source s and destination t may themselves have degree 1; because of that, the lemma's stated bound $\mathrm{ALG} \le 2(|F|-2)$ is false as written, though a corrected Euler-argument that accounts for the two endpoints still yields the 4-approximation ratio.

Editorial extensions

If this is right

  • Planar-Tracking is NP-complete, so unless P=NP there is no polynomial-time exact algorithm for planar instances; approximation and parameterized algorithms are the meaningful targets.
  • The 4-approximation is analytically tight up to constants: there are planar instances where an optimal set has |F|/2 trackers and the algorithm outputs 2(|F|−2), so the ratio cannot be improved just by tightening the two lemmas.
  • The reduction shows that tracking-set decisions are at least as hard as planar 3-satisfiability, placing the problem among other planar NP-hard optimization problems.
  • For graphs of bounded clique-width, exact tracking-set optimization is fixed-parameter tractable in the clique width: cubic time without a decomposition, linear time when a bounded-width clique decomposition is given.
  • The entry-exit-pair characterization means that checking whether a proposed vertex set is a tracking set can be reduced to checking every cycle, which is why the planar face structure can be exploited.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The face-charging lower bound likely extends to graphs on surfaces of bounded genus with a constant that grows with the genus, since Euler's formula still relates vertices, edges, and faces; the paper leaves this as an open direction.
  • The degree-1 endpoint oversight in the upper-bound proof is repairable: a corrected Euler count that charges the endpoints separately still preserves the 4-approximation ratio, so the theorem's conclusion is probably safe even though the stated lemma is not.
  • Because the MSO1 formulation is the only graph-structure requirement, exact linear-time algorithms should also hold for any graph class of bounded clique-width if one is willing to pay cubic preprocessing, and on small networks brute force can validate the approximation ratio in practice.
  • The gadget reduction suggests a natural testable extension: one can attempt to strengthen the inapproximability ratio beyond the factor 4 by composing the planar 3-SAT reduction with gap amplification, which the paper does not do.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the tracking-paths problem on planar graphs, in which one seeks a minimum set of vertices whose intersections with every s-t path are pairwise distinct sequences. The authors claim three main results: (1) a 4-approximation algorithm for Planar-Tracking based on reductions and an Euler-formula counting argument, (2) NP-hardness of Planar-Tracking via a reduction from Planar-3-SAT, and (3) solvability of the general problem in linear time on graphs of bounded clique-width when a clique decomposition is provided, by an MSO1 formulation and Courcelle's theorem. The NP-hardness reduction is detailed and plausible, and the clique-width application is a standard, correct use of Courcelle's theorem. The main issue is in the approximation section: Lemma 14's stated upper bound is false as written because it neglects degree-1 terminals, although the argument can be repaired to yield the same 4-approximation ratio.

Significance. If the approximation proof is repaired, the paper makes a solid contribution: it establishes that Planar-Tracking is NP-hard and admits a constant-factor approximation, closing a natural gap left by earlier work on general graphs. The bounded-clique-width result is a clean application of Courcelle's theorem and is interesting as a theoretical algorithmic result. The NP-hardness construction is detailed and appears sound. The 4-approximation is the weakest link: the key counting lemma is false in the form stated, but the failure is local and the corrected bound ALG <= 2|F|-2 still combines with Lemma 6's lower bound OPT >= (|F|-1)/2 to give the claimed approximation ratio. Thus the central claim is defensible but the manuscript as written contains a load-bearing error that must be fixed.

major comments (2)
  1. [Section 3.2, Lemma 14] Lemma 14 states ALG <= 2(|F|-2), but the proof explicitly assumes that after Reduction 1 there are no vertices of degree 1. This is false because the source s and destination t may themselves have degree 1. A concrete counterexample is a triangle with pendant edges from s to one triangle vertex and from t to another: after Reduction 1 the graph has |F|=2, and Algorithm A places one tracker (on the remaining degree-2 triangle vertex via Reduction 3, or on a degree-3 vertex after further suppression), so ALG=1, violating the claimed bound 2(|F|-2)=0. The Euler-formula argument omits the contribution of degree-1 vertices from the degree sum. Including them yields ALG <= 2|F|-2 rather than 2(|F|-2). Since Lemma 6 gives OPT >= (|F|-1)/2, the corrected bound still implies the 4-approximation in Theorem 3, but the lemma and its proof must be revised before Theorem 3 is valid as written.
  2. [Section 3.2, Reduction 2 and Appendix A, Lemma 24] Reduction 2 as stated removes an arbitrary one of two adjacent degree-2 vertices and does not exclude s and t. If a terminal has degree 2 and is adjacent to another degree-2 vertex, the algorithm can delete the source or destination, after which the output of Algorithm A is not a tracking set for the original s-t instance. The proof of Lemma 24 implicitly assumes the removed vertex is not a terminal when it argues that moving a tracker from u to u' preserves the tracking property; this argument fails when u=s or v=t. The reduction rule and its safety proof must explicitly forbid removing s and t (or otherwise handle terminal vertices), and the effect of this restriction on the degree-counting argument in Lemma 14 should be rechecked.
minor comments (5)
  1. [Abstract and Introduction] The attribution of the origin of the tracking-paths problem is inconsistent: the abstract says it was first introduced in [3], while the introduction cites both [4] and [3]. These references should be reconciled.
  2. [Section 1, problem definition] The problem is stated as an optimization problem ('What is the smallest tracking set?'), but Corollary 18 claims NP-completeness. The corresponding decision version (does there exist a tracking set of size at most k?) should be stated explicitly.
  3. [Figure 3(b)] Figure 3(b) is labeled as a tight example for ALG = 2(|F|-2). If Lemma 14 is corrected to ALG <= 2|F|-2, the tight-example claim and caption should be updated to match the corrected bound.
  4. [Section 3.2, Claim 7] Claim 7 states that a vertex on an entry-exit pair has degree greater than 2, with proof 'Trivial by Definition 1.' This is not true for s and t: a terminal of degree 2 can lie on a cycle and serve as an entry or exit vertex. The claim should be restricted to non-terminal vertices, which is all that the later reduction proofs actually need.
  5. [Section 5] The introduction says Tracking can be solved in cubic time for bounded clique-width, but Theorem 22 and the surrounding text state polynomial time (via Oum's cubic-time decomposition construction) and linear time only when the decomposition is given. The wording should be made consistent and precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is anchored to independent external results (Banik et al., Lichtenstein, Courcelle) and introduces no self-referential definitions or fitted predictions.

full rationale

The paper's claims are built on independent external results rather than on its own conclusions. The tracking-set characterization (Lemma 2), the safety of Reduction 1 (Lemma 4), and the tracking-set/feedback-vertex-set and kernel facts are cited from Banik et al. [3], with Reductions 9-11 explicitly extending an independent lemma from that same source. The NP-hardness reduction uses Lichtenstein's NP-completeness of Planar-3-SAT and Knuth-Raghunathan's rectilinear drawing theorem. The bounded-clique-width result applies Courcelle's theorem to an MSO1 formula whose correctness is argued directly (Remarks 20-21) and whose underlying equivalence (Lemma 19) is derived from Lemma 2, not from the theorem being proved. No quantity is fit to data, no parameter is renamed as a prediction, and no uniqueness theorem from the present authors is invoked to force a choice. The paper does not cite any prior work by its own authors in a load-bearing way. The identified flaw in Lemma 14 — that after Reduction 1 the endpoints s and t may have degree 1, so the stated bound ALG ≤ 2(|F|−2) can fail on a triangle with pendant s and t — is a correctness gap in the proof, not a circularity: the lemma's inequality is derived from Euler's formula and degree counting, not from the approximation statement or from an input-dependent fit. The derivation chain is therefore self-contained with respect to the circularity criteria, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new mathematical objects or free parameters. It relies on standard planar graph theory, previously established lemmas about tracking sets, and the meta-theorems of Courcelle and Oum. The false degree-1 assertion in Lemma 14 is not an invented entity but an incorrect proof step, captured in the red flags.

assumptions (6)
  • domain assumption Banik et al. Lemma 2: a vertex set T is a tracking set iff every simple cycle has no untracked entry-exit pair
    Used as the working characterization throughout Sections 2 and 5, cited from [3] without proof.
  • standard math Planar-3-SAT is NP-complete, and planar rectilinear drawings with variables on a line exist (Lichtenstein; Knuth and Raghunathan)
    Foundation of the NP-hardness reduction in Section 4.
  • standard math Courcelle's theorem: MSO1-expressible optimization problems are linear-time on bounded clique-width graphs given a clique decomposition
    Used to prove Theorem 22 in Section 5.
  • standard math Euler's formula and the face-edge inequality 2m >= 3f for simple planar graphs
    Used in Lemmas 6 and 14 to relate the number of faces to the size of tracking sets.
  • domain assumption Reduction 1 (removing edges and vertices not on any s-t path) is safe and polynomial-time (Banik et al.)
    Used as the first step of Algorithm A and in Lemma 5.
  • standard math Oum's cubic-time algorithm approximates rank-width, giving a bounded-width clique decomposition in polynomial time
    Used in Theorem 22 to claim polynomial time even when no clique decomposition is supplied.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tracking Paths in Planar Graphs." pith.science (2026). https://pith.science/paper/S4GJUEMD

@misc{pith2026190805445,
  author       = {Pith},
  title        = {Pith review of: Tracking Paths in Planar Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4GJUEMD}},
  note         = {Machine review of arXiv:1908.05445}
}
abstract

We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle's theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 12 canonical work pages

  1. [1]

    The Mathematical Theory of Infectious Diseases and its Applications

    Norman TJ Bailey. The Mathematical Theory of Infectious Diseases and its Applications . Charles Griffin & Company Ltd, High Wycombe, United Kingdom, 2nd edition, 1975

    work page 1975

  2. [2]

    Fixed-parameter tractable algorithms for tracking set problems

    Aritra Banik and Pratibha Choudhary. Fixed-parameter tractable algorithms for tracking set problems. In B. S. Panda and Partha P. Goswami, editors, Algorithms and Discrete Applied Mathematics - 4th International Conference, CALDAM 2018, Guwahati, India, February 15-17, 2018, Proceedings , volume 10743 of Lecture Notes in Computer Science , pages 93--104. ...

    doi:10.1007/978-3-319-74180-2 2018

  3. [3]

    A polynomial sized kernel for tracking paths problem

    Aritra Banik, Pratibha Choudhary, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. A polynomial sized kernel for tracking paths problem. In Latin American Symposium on Theoretical Informatics , volume 10807 of Lecture Notes in Computer Science , pages 94--107. Springer, 2018. https://doi.org/10.1007/978-3-319-77404-6_8 doi:10.1007/978-3-319-77404-6_8

    work page doi:10.1007/978-3-319-77404-6_8 2018

  4. [4]

    Tracking paths

    Aritra Banik, Matthew J Katz, Eli Packer, and Marina Simakov. Tracking paths. In International Conference on Algorithms and Complexity , volume 10236 of Lecture Notes in Computer Science , pages 67--79. Springer, 2017. https://doi.org/10.1007/978-3-319-57586-5_7 doi:10.1007/978-3-319-57586-5_7

    work page doi:10.1007/978-3-319-57586-5_7 2017

  5. [5]

    Nonserial Dynamic Programming

    Umberto Bertele and Francesco Brioschi. Nonserial Dynamic Programming . Academic Press, 1972. URL: https://www.elsevier.com/books/nonserial-dynamic-programming/bertele/978-0-12-093450-8

    work page 1972

  6. [6]

    Survey of target tracking protocols using wireless sensor network

    Sania Bhatti and Jie Xu. Survey of target tracking protocols using wireless sensor network. In 2009 Fifth International Conference on Wireless and Mobile Communications , pages 110--115. IEEE, 2009. https://doi.org/10.1109/ICWMC.2009.25 doi:10.1109/ICWMC.2009.25

    work page doi:10.1109/icwmc.2009.25 2009

  7. [7]

    Tracking routes in communication networks

    Davide Bil \` o , Luciano Gual \` a , Stefano Leucci, and Guido Proietti. Tracking routes in communication networks. In Keren Censor - Hillel and Michele Flammini, editors, Structural Information and Communication Complexity - 26th International Colloquium, SIROCCO 2019, L'Aquila, Italy, July 1-4, 2019, Proceedings , volume 11639 of Lecture Notes in Compu...

    doi:10.1007/978-3-030-24922-9 2019

  8. [8]

    A linear-time algorithm for finding tree-decompositions of small treewidth

    Hans L Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing , 25(6):1305--1317, 1996. https://doi.org/10.1137/S0097539793251219 doi:10.1137/S0097539793251219

    doi:10.1137/s0097539793251219 1996

Show all 28 references
  1. [9]

    The monadic second-order logic of graphs

    Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs . Information and Computation , 85(1):12--75, 1990. https://doi.org/10.1016/0890-5401(90)90043-H doi:10.1016/0890-5401(90)90043-H

    1990 doi

  2. [10]

    Automata for the verification of monadic second-order graph properties

    Bruno Courcelle and Ir \` e ne Durand. Automata for the verification of monadic second-order graph properties. J. Applied Logic , 10(4):368--409, 2012. https://doi.org/10.1016/j.jal.2011.07.001 doi:10.1016/j.jal.2011.07.001

    2012 doi

  3. [11]

    Computations by fly-automata beyond monadic second-order logic

    Bruno Courcelle and Ir \` e ne Durand. Computations by fly-automata beyond monadic second-order logic. Theor. Comput. Sci. , 619:32--67, 2016. https://doi.org/10.1016/j.tcs.2015.12.026 doi:10.1016/j.tcs.2015.12.026

    2016 doi

  4. [12]

    Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach , volume 138 of Encyclopedia of mathematics and its applications

    Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach , volume 138 of Encyclopedia of mathematics and its applications . Cambridge University Press, 2012. URL: http://www.cambridge.org/fr/knowledge/isbn/item5758776...

    2012

  5. [13]

    Handle-rewriting hypergraph grammars

    Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. Journal of Computer and System Sciences , 46(2):218--270, 1993. https://doi.org/10.1016/0022-0000(93)90004-G doi:10.1016/0022-0000(93)90004-G

    1993 doi

  6. [14]

    Linear time solvable optimization problems on graphs of bounded clique-width

    Bruno Courcelle, Johann A Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems , 33(2):125--150, 2000. https://doi.org/10.1007/s002249910009 doi:10.1007/s002249910009

    doi

  7. [15]

    Monadic second-order evaluations on tree-decomposable graphs

    Bruno Courcelle and Mohamed Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science , 109(1-2):49--82, 1993. https://doi.org/10.1016/0304-3975(93)90064-Z doi:10.1016/0304-3975(93)90064-Z

    1993 doi

  8. [16]

    Upper bounds to the clique width of graphs

    Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics , 101(1-3):77--114, 2000. https://doi.org/10.1016/S0166-218X(99)00184-5 doi:10.1016/S0166-218X(99)00184-5

    doi

  9. [17]

    Epidemiology and control of SARS in Singapore

    Kee-Tai Goh, Jeffery Cutter, Bee-Hoon Heng, Stefan Ma, Benjamin KW Koh, Cynthia Kwok, Cheong-Mui Toh, and Suok-Kai Chew. Epidemiology and control of SARS in Singapore . Annals of the Academy of Medicine, Singapore , 35(5):301, 2006. http://www.ncbi.nlm.nih.gov/pubmed/16829997 ...

    2006

  10. [18]

    Tracking moving targets in a smart sensor network

    Rahul Gupta and Samir R Das. Tracking moving targets in a smart sensor network. In 2003 IEEE 58th Vehicular Technology Conference. VTC 2003-Fall , volume 5, pages 3035--3039. IEEE, 2003. https://doi.org/10.1109/VETECF.2003.1286181 doi:10.1109/VETECF.2003.1286181

    2003 arXiv

  11. [19]

    S-functions for graphs

    Rudolf Halin. S-functions for graphs. Journal of Geometry , 8(1-2):171--186, 1976. https://doi.org/10.1007/BF01917434 doi:10.1007/BF01917434

    1976 doi

  12. [20]

    The problem of compatible representatives

    Donald E Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM Journal on Discrete Mathematics , 5(3):422--427, 1992. https://doi.org/10.1137/0405033 doi:10.1137/0405033

    1992 doi

  13. [21]

    Planar formulae and their uses

    David Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing , 11(2):329--343, 1982. https://doi.org/10.1137/0211025 doi:10.1137/0211025

    1982 doi

  14. [22]

    Epidemics and percolation in small-world networks

    Cristopher Moore and Mark EJ Newman. Epidemics and percolation in small-world networks. Physical Review E , 61(5):5678, 2000. https://doi.org/10.1103/PhysRevE.61.5678 doi:10.1103/PhysRevE.61.5678

    doi

  15. [23]

    Spread of epidemic disease on networks

    Mark EJ Newman. Spread of epidemic disease on networks. Physical Review E , 66(1):016128, 2002. https://doi.org/10.1103/PhysRevE.66.016128 doi:10.1103/PhysRevE.66.016128

    2002 doi

  16. [24]

    Approximating rank-width and clique-width quickly

    Sang-Il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms , 5(1):10, 2008. https://doi.org/10.1145/1435375.1435385 doi:10.1145/1435375.1435385

    2008

  17. [25]

    Survey of network-based defense mechanisms countering the DoS and DDoS problems

    Tao Peng, Christopher Leckie, and Kotagiri Ramamohanarao. Survey of network-based defense mechanisms countering the DoS and DDoS problems . ACM Computing Surveys , 39(1):3, 2007. https://doi.org/10.1145/1216370.1216373 doi:10.1145/1216370.1216373

    2007

  18. [26]

    Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width . Journal of Algorithms , 7(3):309--322, 1986. https://doi.org/10.1016/0196-6774(86)90023-4 doi:10.1016/0196-6774(86)90023-4

    1986 doi

  19. [27]

    Rumors in a network: Who's the culprit? IEEE Transactions on Information Theory , 57(8):5163--5181, 2011

    Devavrat Shah and Tauhid Zaman. Rumors in a network: Who's the culprit? IEEE Transactions on Information Theory , 57(8):5163--5181, 2011. https://doi.org/10.1109/TIT.2011.2158885 doi:10.1109/TIT.2011.2158885

    2011

  20. [28]

    Hash-based IP traceback

    Alex C Snoeren, Craig Partridge, Luis A Sanchez, Christine E Jones, Fabrice Tchakountio, Stephen T Kent, and W Timothy Strayer. Hash-based IP traceback . ACM SIGCOMM Computer Communication Review , 31(4):3--14, 2001. https://doi.org/10.1145/383059.383060 doi:10.1145/383059.383060

    2001

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.