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Data about individuals may contain private and sensitive information. Thedifferential privacy (DP) was proposed to address the problem of protecting theprivacy of each individual while keeping useful information about a population.Sealfon (2016) introduced a private graph model in which the graph topology isassumed to be public while the weight information is assumed to be private.That model can express hidden congestion patterns in a known transportationsystem. In this paper, we revisit the problem of privately releasingapproximate distances between all pairs of vertices in (Sealfon 2016). Our goalis to minimize the additive error, namely the difference between the releaseddistance and actual distance under private setting. We propose improvedsolutions to that problem for several cases.  For the problem of privately releasing all-pairs distances, we show that fortree with depth $h$, we can release all-pairs distances with additive error$O(\log^{1.5} h \cdot \log^{1.5} V)$ for fixed privacy parameter where $V$ thenumber of vertices in the tree, which improves the previous error bound$O(\log^{2.5} V)$, since the size of $h$ can be as small as $O(\log V)$. Ourresult implies that a $\log V$ factor is saved, and the additive error in treecan be smaller than the error on array/path. Additionally, for the grid graphwith arbitrary edge weights, we also propose a method to release all-pairsdistances with additive error $\tilde O(V^{3/4}) $ for fixed privacyparameters. On the application side, many cities like Manhattan are composed ofhorizontal streets and vertical avenues, which can be modeled as a grid graph.

Distances Release with Differential Privacy in Tree and Grid Graph