We prove an elementary yet powerful combinatorial lemma: in any rooted tree with L leaves, the number of nodes whose depth is smaller than the number of their leaf descendants is at most L. For any string T of length n, a direct application of this lemma to the suffix trie of T yields that the number of substrings of T whose length is smaller than their number of occurrences in T is at most n. This combinatorial insight leads to space-efficient data structures with optimal query times for string counting problems via the following algorithmic framework: store the counts for the at most n "frequent" substrings of T in a preprocessing step, and use a reporting query to count for the "infrequent" substrings. Our framework acts as a convenient black box, lifting indexes with reporting time 𝒪(|P|+|Occ_T(P)|) to support counting queries in time 𝒪(|P|), where P is the queried pattern and Occ_T(P) is the set of occurrences of P in T. As applications, we show efficient indexes for consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping occurrences.

We introduce a weighted and unconstrained variant of the well-known minimum$k$union problem: Given a bipartite graph$\mathcal {G}(U,V,E)$with weights for all nodes in$V$, find a set$S\subseteq V$such that the ratio between the total weight of the nodes in$S$and the number of theirdistinctadjacent nodes in$U$is maximized. Our problem, which we termHeavy Nodes in a Small Neighborhood(HNSN), finds applications in marketing, team formation, and money laundering detection. For example, in the latter application,$S$represents bank account holders who obtain illicit money from some peers of a criminal and route it through their accounts to a target account belonging to the criminal. We prove thatHNSNcan be solved exactly in polynomial time via linear programming. We also develop several algorithms offering different effectiveness/efficiency trade-offs: an exact algorithm, based on node contraction, graph decomposition, and linear programming, as well as three peeling algorithms. The first peeling algorithm is a near-linear time approximation algorithm with a tight approximation ratio, the second is an iterative algorithm that converges to an optimal solution in a very small number of iterations in practice, and the third is a near-linear time greedy heuristic. In addition, we formalize a money laundering scenario involving multiple target accounts and show how our algorithms can be extended to deal with it. Our experiments on real and synthetic datasets show that our algorithms find (near-)optimal solutions, outperforming a natural baseline, and that they can detect money laundering more effectively and efficiently than two state-of-the-art methods.

Minimizers sampling is one of the most widely-used mechanisms for sampling strings [Roberts et al., Bioinformatics 2004]. Let S = S[1] . . . S[n] be a string over a totally ordered alphabet Σ. Further let w ≥ 2 and k ≥ 1 be two integers. The minimizer of S[i . . i + w + k − 2] is the smallest position in [i, i + w − 1] where the lexicographically smallest length-k substring of S[i . . i + w + k − 2] starts. The set of minimizers over all i ∈ [1, n − w − k + 2] is the set Mw,k(S) of the minimizers of S. We consider the following basic problem: Given S, w, and k, can we efficiently compute a total order on Σ that minimizes |Mw,k(S)|? We show that this is unlikely by proving that the problem is NP-hard for any w ≥ 3 and k ≥ 1. Our result provides theoretical justification as to why there exist no exact algorithms for minimizing the minimizers samples, while there exists a plethora of heuristics for the same purpose.