Large-scale search and recommendation systems typically adopt a cascaded architecture of retrieval, pre-ranking, ranking, and re-ranking to balance efficiency and accuracy. However, pre-ranking still faces challenges of behavioral sparsity, limited interest diversity, and computational latency. We propose the Query-Attention Dual-Stream (QADS) framework to address these issues. QADS partitions user behaviors into strongly and weakly correlated streams and further decomposes them into fine-grained subsequences guided by domain knowledge. A query-centric attention mechanism reduces complexity from O(N) to O(1), enabling efficient inter- and intra-sequence modeling. A contrastive cross-category transfer module propagates dense patterns from weakly correlated to sparse domains, while a latency-aware parallel inference architecture further reduces delay by 36%. Experiments on public and industrial datasets show that QADS delivers significant performance improvements and has been successfully deployed in large-scale e-commerce search systems.

The Secretary problem is a classical sequential decision-making question that can be succinctly described as follows: a set of rank-ordered applicants are interviewed sequentially for a single position. Once an applicant is interviewed, an immediate and irrevocable decision is made if the person is to be offered the job or not and only applicants observed so far can be used in the decision process. The problem of interest is to identify the stopping rule that maximizes the probability of hiring the highest-ranked applicant. A multiple-choice version of the Secretary problem, known as the Dowry problem, assumes that one is given a fixed integer budget for the total number of selections allowed to choose the best applicant. It has been solved using tools from dynamic programming and optimal stopping theory. We provide the first combinatorial proof for a related new query-based model for which we are allowed to solicit the response of an expert to determine if an applicant is optimal. Since the selection criteria differ from those of the Dowry problem, we obtain nonidentical expected stopping times.

Trades, introduced by Hedayat [9], are two sets of blocks of elements which may be exchanged (traded) without altering the counts of certain subcollections of elements within their constituent blocks. They are of importance in applications where certain combinations of elements dynamically become prohibited from being placed in the same group of elements, since in this case one can trade the offending blocks with allowed ones. This is particularly the case in distributed storage systems, where due to privacy and other constraints, data of some groups of users cannot be stored together on the same server. We introduce a new class of balanced trades, important for access balancing of servers, and perturbation resilient balanced trades, important for studying the stability of server access frequencies with respect to changes in data popularity. The constructions and bounds on our new trade schemes rely on specialized selections of defining sets in minimal trades and number-theoretic analyses.

The well-known secretary problem in sequential analysis and optimal stopping theory asks one to maximize the probability of finding the optimal candidate in a sequentially examined list under the constraint that accept/reject decisions are made in real-time. The problem is related to practical questions arising in online search, data streaming, daily purchase modeling and multi-arm bandit mechanisms. An extension is the postdoc problem, for which one aims to identify the second-best candidate with highest possible probability of success. We solve the postdoc problem for the nontraditional setting where the candidates are not presented uniformly at random but rather according to permutations drawn from the Mallows distribution. The optimal stopping criteria depend on the choice of the Mallows model parameter$\theta$: For$\theta > 1$, we reject the first$k^{\prime}(\theta)$candidates and then accept the next left-to-right second-best candidate (second-best ranked when comparing with all appeared candidates). This coincides with the optimal strategy for the classical postdoc problem, where the rankings being drawn uniformly at random$(\boldsymbol{i}.\boldsymbol{e}. \theta=1)$. For$0 < \theta\leqslant 1/2$, we reject the first$k^{\prime \prime}(\theta)$candidates and then accept the next left-to-right best candidate; if no selection is made before the last candidate, then the last candidate is accepted. For$1/2 < \theta < 1$, we reject the first$k_{1}(\theta)$candidates and then accept the next left-to-right maximum, or reject the first$k_{2}(\theta)\geqslant k_{1}(\theta)$candidates and then accept the next left-to-right second-maximum, whichever comes first.

Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to determine the smallest possible size of the union of the color sets that allows for such a digraph representation. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to introduce the directed intersection number (DIN), the smallest number of colors needed to represent a DAG. Our main results are upper bounds on the DIN of DAGs based on what we call the longest terminal path decomposition of the vertex set, and constructive lower bounds.

Consider a directed graph (digraph) in which two user vertices are connected if and only if they share at least one unit of common information content and the head vertex has a strictly smaller content than the tail. We seek to estimate the smallest possible global information content that can explain the observed digraph topology. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to describe the directed intersection number (DIN), the smallest number of information units needed to represent the DAG. Our main result is a nontrivial upper bound on the DIN number of DAGs based on the longest terminal path decomposition of the vertex set. In addition, we compute the exact values of the DIN number for several simple yet relevant families of connected DAGs and construct digraphs that have near-optimal DIN values.