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We construct n-node graphs on which any O(n)-size spanner has additive error at least +Ω(n^{3/17}), improving on the previous best lower bound of Ω(n^{1/7}) [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for O(n)-size additive emulators to +Ω(n^{1/14}). 

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 extend recent results regarding finding shortest unique substrings (SUSs) to obtain new time-space tradeoffs for this problem and the generalization of finding k-mismatch SUSs. Our new results include the first algorithm for finding a k-mismatch SUS in sublinear space, which we obtain by extending an algorithm by Senanayaka (2019) and combining it with a result on sketching by Gawrychowski and Starikovskaya (2019). We first describe how, given a text T of length n and m words of workspace, with high probability we can find an SUS of length L in O(n(L/m)logL) time using random access to T, or in O(n(L/m)log2(L)loglogσ) time using O((L/m)log2L) sequential passes over T. We then describe how, for constant k, with high probability, we can find a k-mismatch SUS in O(n1+ϵL/m) time using O(nϵL/m) sequential passes over T, again using only m words of workspace. Finally, we also describe a deterministic algorithm that takes O(nτlogσlogn) time to find an SUS using O(n/τ) words of workspace, where τ is a parameter.