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The ranked (or top-k) document retrieval problem is defined as follows: preprocess a collection{T₁,T₂,… ,T_d}ofdstrings (called documents) of total lengthninto a data structure, such that for any given query(P,k), wherePis a string (called pattern) of lengthp ≥ 1andk ∈ [1,d]is an integer, the identifiers of thosekdocuments that are most relevant toPcan be reported, ideally in the sorted order of their relevance. The seminal work by Hon et al. [FOCS 2009 and Journal of the ACM 2014] presented anO(n)-space (in words) data structure withO(p+klogk)query time. The query time was later improved toO(p+k)[SODA 2012] and further toO(p/log_σn+k)[SIAM Journal on Computing 2017] by Navarro and Nekrich, whereσis the alphabet size. We revisit this problem in the external memory model and present three data structures. The first one takesO(n)-space and answer queries inO(p/B+ log_Bn + k/B+log^*(n/B)) I/Os, whereBis the block size. The second one takesO(nlog^*(n/B)) space and answer queries in optimalO(p/B+ log_Bn + k/B)I/Os. In both cases, the answers are reported in the unsorted order of relevance. To handle sorted top-kdocument retrieval, we present anO(nlog(d/B))space data structure with optimal query cost.

 

Two equal length strings are a parameterized match (p-match) iff there exists a one-to-one function that renames the symbols in one string to those in the other. The Parameterized Suffix Tree (PST) [Baker, STOC' 93] is a fundamental data structure that handles various string matching problems under this setting. The PST of a text T[1,n] over an alphabet Σ of size σ takes O(nlog n) bits of space. It can report any entry in (parameterized) (i) suffix array, (ii) inverse suffix array, and (iii) longest common prefix (LCP) array in O(1) time. Given any pattern P as a query, a position i in T is an occurrence iff T[i,i+|P|-1] and P are a p-match. The PST can count the number of occurrences of P in T in time O(|P|log σ) and then report each occurrence in time proportional to that of accessing a suffix array entry. An important question is, can we obtain a compressed version of PST that takes space close to the text’s size of nlogσ bits and still support all three functionalities mentioned earlier? In SODA' 17, Ganguly et al. answered this question partially by presenting an O(nlogσ) bit index that can support (parameterized) suffix array and inverse suffix array operations in O(log n) time. However, the compression of the (parameterized) LCP array and the possibility of faster suffix array and inverse suffix array queries in compact space were left open. In this work, we obtain a compact representation of the (parameterized) LCP array. With this result, in conjunction with three new (parameterized) suffix array representations, we obtain the first set of PST representations in o(nlog n) bits (when logσ = o(log n)) as follows. Here ε > 0 is an arbitrarily small constant. - Space O(n logσ) bits and query time O(log_σ^ε n); - Space O(n logσlog log_σ n) bits and query time O(log log_σ n); and - Space O(n logσ log^ε_σ n) bits and query time O(1). The first trade-off is an improvement over Ganguly et al.’s result, whereas our third trade-off matches the optimal time performance of Baker’s PST while squeezing the space by a factor roughly log_σ n. We highlight that our trade-offs match the space-and-time bounds of the best-known compressed text indexes for exact pattern matching and further improvement is highly unlikely. 

Hydrogel microparticles (microgels) are an attractive approach for therapeutic delivery because of their modularity, injectability, and enhanced integration with the host tissue. Multiple microgel fabrication strategies and chemistries have been implemented, yet manipulation of microgel degradability and its effect on in vivo tissue responses remains underexplored. Here, the authors report a facile method to synthesize microgels crosslinked with ester‐containing junctions to afford tunable degradation kinetics. Monodisperse microgels of maleimide‐functionalized poly(ethylene‐glycol) are generated using droplet microfluidics crosslinked with thiol‐terminated, ester‐containing molecules. Tunable mechanics are achievable based on the ratio of degradable to nondegradable crosslinkers in the continuous phase. Degradation in an aqueous medium leads to microgel deformation based on swelling and a decrease in elastic modulus. Furthermore, degradation byproducts are cytocompatible and do not cause monocytic cell activation under noninflammatory conditions. These injectable microgels possess time‐dependent degradation on the order of weeks in vivo. Lastly, the evaluation of tissue responses in a subcutaneous dorsal pocket shows a dynamic type‐1 like immune response to the synthetic microgels, driven by interferon gamma (IFN‐γ  ) expression, which can be moderated by tuning the degradation properties. Collectively, this study demonstrates the development of a hydrolytic microgel platform that can be adapted to desired host tissue immune responses.