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The hereditary discrepancy of a set system is a quantitative measure of the pseudorandom properties of the system. Roughly speaking, hereditary discrepancy measures how well one can 2-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has numerous applications in computational geometry, communication complexity and derandomization. More recently, the hereditary discrepancy of the set system of shortest paths has found applications in differential privacy [Chen et al. SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy O(n^{1/4}), and we construct lower bound examples demonstrating that this bound is tight up to polylog n factors. Our lower bounds hold even for planar graphs and bipartite graphs, and improve a previous lower bound of Ω(n^{1/6}) obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from Ω(n^{1/6}) [Chen et al. SODA 23] to Ω̃(n^{1/4}), and we improve the lower bound on additive error for the differentially-private all sets range queries problem to Ω̃(n^{1/4}), which is tight up to polylog n factors [Deng et al. WADS 23]. 

We perform a rigorous study of private matrix analysis when only the last 𝑊 updates to matrices are considered useful for analysis. We show the existing framework in the non-private setting is not robust to noise required for privacy. We then propose a framework robust to noise and use it to give first efficient 𝑜(𝑊) space differentially private algorithms for spectral approximation, principal component analysis (PCA), multi-response linear regression, sparse PCA, and non-negative PCA. Prior to our work, no such result was known for sparse and non-negative differentially private PCA even in the static data setting. We also give a lower bound to demonstrate the cost of privacy in the sliding window model. 

In this paper, we focus on answering queries, in a differentially private manner, on graph streams. We adopt the sliding window model of privacy, where we wish to perform analysis on the last W updates and ensure that privacy is preserved for the entire stream. We show that in this model, the price of ensuring differential privacy is minimal. Furthermore, since differential privacy is preserved under post-processing, our results can be used as a subroutine in many tasks, most notably solving cut functions and spectral clustering.