Justin Dallant

Boris Aronov, Jean Cardinal, Justin Dallant et al.

In recent years, the Fiat-Naor function inversion scheme has been used to disprove conjectures in fine-grained complexity theory and design state of the art data structures for a number of combinatorial problems. We pursue this line of research by considering its application to data structures for searching in implicit sets, defined as the image of a function. We show that, if $f$ is of the form $[N]\to [2^{w}]^d$ for some $w=polylog(N)$ and is computable in constant time, then, for any $0<α<1$, we can obtain a data structure using $Õ(N^{1-α/3})$ space such that, for a given $d$-dimensional axis-aligned box $B$, we can search for some $x\in [N]$ such that $f(x) \in B$ in time $Õ(N^α)$. (Here the $Õ(.)$ notation omits polylogarithmic factors.) Using similar techniques, we further obtain - data structures for range counting and reporting, predecessor, selection, ranking queries, and combinations thereof, on the set $f([N])$, - data structures for preimage size and preimage selection queries for a given value of $f$, and - data structures for selection and ranking queries on geometric quantities computed from tuples of points in $d$-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the $k$th largest area triangle, or the induced hyperplane that is the $k$th furthest from the origin.

Sofia Brenner, Justin Dallant, Linda Kleist et al.

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

Justin Dallant, László Kozma

The traveling salesman problem (TSP) is a cornerstone of combinatorial optimization and has deeply influenced the development of algorithmic techniques in both exact and approximate settings. Yet, improving on the decades-old bounds for solving TSP exactly remains elusive: the dynamic program of Bellman, Held, and Karp from 1962 uses $2^{n+O(\log{n})}$ time and space, and the divide-and-conquer approach of Gurevich and Shelah from 1987 uses $4^{n + O(\log^2{n})}$ time and polynomial space. A straightforward combination of the two algorithms trades off $T^{n+o(n)}$ time and $S^{n+o(n)}$ space at various points of the curve $ST = 4$. An improvement to this tradeoff when $2 < T < 2\sqrt{2}$ was found by Koivisto and Parviainen (SODA 2010), yielding a minimum of $ST \approx 3.93$. Koivisto and Parviainen show their method to be optimal among a broad class of partial-order-based approaches, and to date, no improvement or alternative method has been found. In this paper we give a tradeoff that strictly improves all previous ones for all $2 < T < 4$, achieving a minimum of $ST < 3.572$. A key ingredient is the construction of sparse set systems (hypergraphs) that admit a large number of maximal chains. The existence of such objects is of independent interest in extremal combinatorics, likely to see further applications. Along the way we disprove a combinatorial conjecture of Johnson, Leader, and Russell from 2013, relating it with the optimality of the previous tradeoff schemes for TSP. Our techniques extend to a broad class of permutation problems over arbitrary semirings, yielding improved space-time tradeoffs in these settings as well.