[Paper Review] Comments on "Resource placement in Cartesian product of networks" [Imani, Sarbazi-Azad and Zomaya, J. Parallel Distrib. Comput., 70 (2010) 481-495]

Pranava K. Jha|arXiv (Cornell University)|Feb 16, 2013

Interconnection Networks and Systems5 references3 citations

TL;DR

This paper critically evaluates the 2010 paper on resource placement in Cartesian product networks, identifying multiple technical flaws including incorrect or unsupported claims, redundant algorithms, misuse of terminology, and failure to cite foundational literature. The critique highlights that core results were already established in prior work, particularly in Imrich and Klavźžr's 2000 book on product graphs, and that key definitions—such as volume and homogeneity—are inconsistently or incorrectly applied.

ABSTRACT

The present note points out a number of errors, omissions, redundancies and arbitrary deviations from the standard terminology in the paper "Resource placement in Cartesian product of networks," by N. Imani, H. Sarbazi-Azad and A.Y. Zomaya [J. Parallel Distrib. Comput. 70 (2010) 481-495].

Motivation & Objective

• To identify and correct technical inaccuracies in the 2010 paper on resource placement in Cartesian product networks.
• To highlight the omission of foundational citations, particularly to Imrich and Klavźžr's 2000 book on product graphs.
• To correct the misuse of key graph-theoretic terminology such as 'homogeneous' and 'volume'.
• To expose redundant algorithms and proofs that undermine the paper's originality and clarity.
• To correct mathematical errors in theorems, definitions, and algorithmic descriptions, including incorrect set membership and matrix notation.

Proposed method

• Conduct a line-by-line, section-by-section critique of the 2010 paper using formal mathematical reasoning.
• Identify and correct incorrect or misleading definitions, such as the volume function Vol_G(d,c), which is not invariant across vertices.
• Point out the misuse of the term 'homogeneous' to mean 'isomorphic', contradicting standard graph theory terminology.
• Demonstrate that key results—such as vertex partitioning into isomorphic subgraphs—were already established in prior literature.
• Reveal algorithmic redundancies, such as steps in HMP, HTP, and IHTP that are proven to be unnecessary.
• Correct mathematical errors in proofs, including incorrect set intersections (e.g., union vs. intersection of vertex sets) and misused matrix notation.

Experimental results

Research questions

• RQ1 Are the results on resource placement in Cartesian product networks truly novel, or are they based on previously published work?
• RQ2 Is the definition of volume (Vol_G(d,c)) valid for arbitrary graphs, or does it require symmetry assumptions?
• RQ3 Why does the paper use non-standard terminology like 'homogeneous' to mean 'isomorphic', and what are the consequences?
• RQ4 Are the algorithms presented (e.g., HMP, HTP, IHTP) truly necessary, or are they redundant as proven in the critique?
• RQ5 What are the specific mathematical errors in the proofs of Theorems 4, 5, and 9, and how do they affect the validity of the results?

Key findings

• The concept of volume (Vol_G(d,c)) is incorrectly used without acknowledging that it depends on the choice of vertex c, contradicting the assumption of invariance in Theorem 1.
• The term 'homogeneous' is incorrectly used to mean 'isomorphic', conflicting with standard graph theory where 'homogeneous' refers to a different, stronger property.
• The paper fails to cite Imrich and Klavźžr's 2000 book, which already contains the vertex partitioning result later claimed as original.
• The proof of Theorem 2 shows that Step (b) of Algorithm 2-HMP is redundant, yet the algorithm is still presented in full, indicating poor editorial quality.
• The matrix M in Theorem 4 is ambiguously defined—referred to as a 'cubic matrix' but later as M_{|Q_H|×|Q_H|} and M_{a×b}, with no clarification on a and b.
• In Theorem 5, the assumption that v belongs to both R_{1,1} and R_{H,1} leads to a logical contradiction, invalidating the proof.