Title: A Generalized Binary Tree Mechanism for Private Approximation of All-Pair Shortest Distances
Venue: NeurIPS 2025
Accepted: yes
Keywords: differential privacy

graph theory
Abstract: We study the problem of approximating all-pair distances in a weighted undirected graph with differential privacy, introduced by Sealfon [Sea16]. Given a publicly known undirected graph, we treat the weights of edges as sensitive information, and two graphs are neighbors if their edge weights differ in one edge by at most one. We obtain efficient algorithms with significantly improved bounds on a broad class of graphs which we refer to as *recursively separable*. In particular, for any $n$-vertex $K_h$-minor-free graph, our algorithm achieve an additive error of $ \widetilde{O}(h(nW)^{1/3} ) $, where $ W $ represents the maximum edge weight; For grid graphs, the same algorithmic scheme achieve additive error of $ \widetilde{O}(n^{1/4}\sqrt{W}) $.

Our approach can be seen as a generalization of the celebrated binary tree mechanism for range queries, as releasing range queries is equivalent to computing all-pair distances on a path graph. In essence, our approach is based on generalizing the binary tree mechanism to graphs that are *recursively separable*.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a fundamental differential privacy problem on graph, namely estimating all pairs shortest distances privately. The main contribution is to design a generic adaptation of the binary tree mechanism to a class of graphs that could be recursively separated, e.g., the grid graph. In particular, they show that if the graph is $(p, q, q')$ recursively separable, then their algorithm achieves an additive error of $p\\log^3(n/\\delta)/\\epsilon$ for $(\\epsilon, \\delta)$-DP. For planar and general $K_h$ minor free graphs, their algorithm achieves an error of $\\tilde O_\\delta(h\\cdot (\\frac{(nW)^{1/3}}{\\epsilon^{2/3}}))$ and the bound is further improved for grid graphs to $\\tilde O_\\delta(\\frac{n^{1/4}W^{1/2}}{\\epsilon^{1/2}})$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The results are quite important, as it improves upon the additive error for a large class of graphs and it introduces the binary tree mechanism into the picture of DP APSP. It is known that binary tree mechanism works for path graph due to the range counting \"1D\" nature of APSP on a path. This paper demonstrates that for graphs that have good separability structures, you can essentially reduce to the case of a path (to some extent). While natural, this gives us better understanding for the class of graphs that could leverage the powerful binary tree mechanism.\n\n2. Conceptually, the idea is quite neat and simple, and the paper is well-written. I find it easy to read and parse.\n\nWeaknesses:\n\n1. As illustrated in strengths, the main contribution of this paper is figuring out the correct class of algorithm that resembles the \"path\" behavior in order to apply binary tree mechanism. As such, once the property of recursively separable is spelled out, both the algorithms are the analysis become relatively straightforward. It does not bring much more insights beyond that. I think it would be interesting to abstract the general binary tree mechanism beyond DP APSP.\n\n2. The paper is quite technically dense, authors could consider adding more discussions and a conclusion section in the appendix.\n\nQUESTIONS:\n1. What do you think is the \"correct\" answer to DP APSP? At least from the results of this paper, it seems for a large class of graphs, the correct answer might be closer to the previous proved lower bound?\n\nLIMITATIONS:\nAuthors didn't discuss limitations. One source of possible improvements is suggested in the weakness section.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

comments: SUMMARY:
The problem of releasing All-Pair Shortest Distances (APSD) on a (general) weighted graph with differential privacy has been well-studied. However, when the given graph has special properties, the error bound can be further improved. In this paper, the authors propose efficient algorithms with significantly improved bounds on graphs that are recursively separable (**Definition 1**). Typical examples of such graphs are trees or planar graphs. The property of recursive separability allows one to use the divide-and-conquer method (or a generalization of the binary tree mechanism) to recursively compute the shortest path, thus improving the bound for APSD on these graphs.

STRENGTHS AND WEAKNESSES:
# Strengths

1. The proposed solutions seem reasonable and correct.
2. The paper is well-written, including intuitive explanations and sufficient theoretical analysis.

# Weaknesses

1. It would be better to open-source the implementations that can reproduce the results shown in Appendix E.

QUESTIONS:
No.

LIMITATIONS:
Thanks for the submission. In its current form, I think the paper is well-written, and I cannot find limitations.
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents an algorithm for solving the differentially private all-pairs shortest distances (APSD) problem on a special class of graphs called recursively separable graphs. The authors further develop an improved algorithm for a subset of these graphs, namely $K_h$​-minor-free graphs, and demonstrate enhanced approximation guarantees on specific graph classes. In particular, they achieve approximation error bounds of ${\\tilde O}((nW)^{1/3}/\\epsilon^{2/3})$ on planar graphs and ${\\tilde O}(n^{1/4}W^{1/2}/\\epsilon^{1/2})$on grid graphs.\n\nA key technical innovation lies in a new recursive decomposition approach: the graph is partitioned into subgraphs using a divide-and-conquer strategy. For $K_h$​-minor-free graphs, the authors further construct a k-covering structure, which plays a crucial role in optimizing the algorithmic performance.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1.The differentially private APSD problem is a classical and well-studied problem in the context of graph algorithms and privacy. Since its introduction by Sealfon[1], it has seen significant attention, and further progress in this area is of broad interest.\n\n2.The proposed use of recursive structures over graphs introduces a novel and elegant algorithmic technique, showcasing theoretical creativity.\n\n3.The algorithm achieves significantly improved approximation error bounds on widely studied graph families such as planar and grid graphs.\n\nWeaknesses\n\n1.The proposed method applies only to graphs with specific structural properties, such as recursively separable graphs and $K_h$-minor-free graphs (including planar and grid graphs). Its applicability to general graphs remains limited.\n\n[1] Adam Sealfon. Shortest paths and distances with differential privacy. In Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 29–41, 2016.\n\nQUESTIONS:\nThe experimental evaluation is limited to randomly generated grid graphs. Could the authors test the algorithm on more diverse datasets or real-world graph structures to better demonstrate its practical performance?\n\nLIMITATIONS:\nThis is a theoretical paper, and I do not see any negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes algorithms to compute all-pair distances in a\nweighted graph under the constraints of differential privacy. Similar\nto previous work, the algorithm proposed here are based on the binary\ntree mechanism, a classical method to achieve good utility in the\ncontinual observation model of differential privacy. In particular,\nthe algorithms proposed in this paper expand the class of graphs,\ncalled recursively separable in the paper, to which this method is\napplicable.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- the paper identifies a large class of graphs, recursively separable,\n  that can be analyzed using the binary tree mechanism\n\n- the paper presents algorithms for the compute all-pair distances\n  problem that provide good utility on a large class of graphs\n\nWeaknesses\n\n- the improvement in utility for planar graphs and grid graphs do not\n  follow directly from the main result but they require some further\n  steps,\n\n- the presentation of the algorithms can be improved with some more\n  explanations and without showing the concrete algorithms.\n\nThe main contribution of the paper is to extend the range of\napplicability of the binary tree mechanism for computing all-pair\ndistances in a weighted graph in a differentially private way.  In\nparticular, the paper interestingly connect it with a specific class\nof graph which the paper calls recursively separable. Intuitively,\nthese are graphs that can be separated recursively in a balanced way.\nThis connection is intuitive and works well in practice providing a\ngeneral framework for the computing all-pair distances problem.  To\nachieve good accuracy, the algorithm uses the topology of the graph to\nidentify private shortcuts that can be used in separators and across\ndiffernet separators. These topology-based shortcuts permits to reduce\nthe error. Using this framework the paper can provide useful\nalgorithms for a large class of graphs, and achieve utility similar to\nprevious works on some classes of graphs (e.g. bounded tree-width\ngraphs).\nHowever, in order to achieve improved utility with respect to previous\nworks on specific families of graphs, such as planar graphs and grid\ngraphs, the algorithmic framework needs some further optimization\n(e.g. finding shortcuts in k-covering sets). This shows that there is\nlimiting trade-off between generality and accuracy of the proposed\nframework. Nevertheless, the general framework and the optimizations\nare both meaningful and provide good utility.\nOne minor negative aspect of the paper is in its presentation of the\nalgorithms. The paper tries to give some intuition about the algorithms\nbefore presenting them formally. However, this intuition is limited\nand a reader is left alone in decoding the algorithms. The presentation\ncould be improved by moving some of the formal details to the appendix,\nleaving more space for the intuition behind them. \nAlso, as another minor negative point, some of the references are\nmissing important info. E.g. [1] has only title and year, [29] is just\nan unpublished reference, [45] the journal title seems part of the\npaper title.\n\nQUESTIONS:\nCan you elaborate on which other interesting graph classes can be captured by recursively separable graphs? \n\nDo you expect that one could design other optimized version of your framework for them?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-15"}