22 research outputs found
Faster Dynamic Range Mode
In the dynamic range mode problem, we are given a sequence a of length bounded by N and asked to support element insertion, deletion, and queries for the most frequent element of a contiguous subsequence of a. In this work, we devise a deterministic data structure that handles each operation in worst-case O?(N^0.655994) time, thus breaking the O(N^{2/3}) per-operation time barrier for this problem. The data structure is achieved by combining the ideas in Williams and Xu (SODA 2020) for batch range mode with a novel data structure variant of the Min-Plus product
Simpler Reductions from Exact Triangle
In this paper, we provide simpler reductions from Exact Triangle to two
important problems in fine-grained complexity: Exact Triangle with Few
Zero-Weight -Cycles and All-Edges Sparse Triangle.
Exact Triangle instances with few zero-weight -cycles was considered by
Jin and Xu [STOC 2023], who used it as an intermediate problem to show SUM
hardness of All-Edges Sparse Triangle with few -cycles (independently
obtained by Abboud, Bringmann and Fischer [STOC 2023]), which is further used
to show SUM hardness of a variety of problems, including -Cycle
Enumeration, Offline Approximate Distance Oracle, Dynamic Approximate Shortest
Paths and All-Nodes Shortest Cycles. We provide a simple reduction from Exact
Triangle to Exact Triangle with few zero-weight -cycles. Our new reduction
not only simplifies Jin and Xu's previous reduction, but also strengthens the
conditional lower bounds from being under the SUM hypothesis to the even
more believable Exact Triangle hypothesis. As a result, all conditional lower
bounds shown by Jin and Xu [STOC 2023] and by Abboud, Bringmann and Fischer
[STOC 2023] using All-Edges Sparse Triangle with few -cycles as an
intermediate problem now also hold under the Exact Triangle hypothesis.
We also provide two alternative proofs of the conditional lower bound of the
All-Edges Sparse Triangle problem under the Exact Triangle hypothesis, which
was originally proved by Vassilevska Williams and Xu [FOCS 2020]. Both of our
new reductions are simpler, and one of them is also deterministic -- all
previous reductions from Exact Triangle or 3SUM to All-Edges Sparse Triangle
(including P\u{a}tra\c{s}cu's seminal work [STOC 2010]) were randomized.Comment: To appear in SOSA 202
Simpler and Higher Lower Bounds for Shortcut Sets
We provide a variety of lower bounds for the well-known shortcut set problem:
how much can one decrease the diameter of a directed graph on vertices and
edges by adding or of shortcuts from the transitive closure
of the graph. Our results are based on a vast simplification of the recent
construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an
lower bound for the -sized shortcut set
problem. We highlight that our simplification completely removes the use of the
convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous
lower bound constructions. Our simplification also removes the need for
randomness and further removes some log factors. This allows us to generalize
the construction to higher dimensions, which in turn can be used to show the
following results. For -sized shortcut sets, we show an
lower bound, improving on the previous best lower bound. For
all , we show that there exists a such that there
are -vertex -edge graphs where adding any shortcut set of size
keeps the diameter of at . This
improves the sparsity of the constructed graph compared to a known similar
result by Hesse [SODA 2003].
We also consider the sourcewise setting for shortcut sets: given a graph
, a set , how much can we decrease the sourcewise
diameter of , by adding a set of edges from the transitive closure of
? We show that for any integer , there exists a graph on
vertices and with ,
such that when adding or shortcuts, the sourcewise diameter is
.Comment: To appear in SODA 2024. Abstract shortened to fit arXiv requirement
Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths
One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n^{3-o(1)} time, and it is widely open whether it has an O(n^{3-ε}) time algorithm for ε > 0. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product. A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an n-vertex graph with potentially negative edge weights in {-M, …, M}. SSRP with positive integer edge weights bounded by M can be solved in Õ(Mn^ω) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in O(M^{0.7519} n^{2.5286}) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M^{0.8043} n^{2.4957}) time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require n^{2.5-o(1)} time. Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an Õ(M n^{ω}) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges
Fine-Grained Complexity and Algorithms for the Schulze Voting Method
We study computational aspects of a well-known single-winner voting rule
called the Schulze method [Schulze, 2003] which is used broadly in practice. In
this method the voters give (weak) ordinal preference ballots which are used to
define the weighted majority graph (WMG) of direct comparisons between pairs of
candidates. The choice of the winner comes from indirect comparisons in the
graph, and more specifically from considering directed paths instead of direct
comparisons between candidates.
When the input is the WMG, to our knowledge, the fastest algorithm for
computing all winners in the Schulze method uses a folklore reduction to the
All-Pairs Bottleneck Paths problem and runs in time, where is
the number of candidates. It is an interesting open question whether this can
be improved. Our first result is a combinatorial algorithm with a nearly
quadratic running time for computing all winners. This running time is
essentially optimal. If the input to the Schulze winners problem is not the WMG
but the preference profile, then constructing the WMG is a bottleneck that
increases the running time significantly; in the special case when there are
candidates and voters, the running time is , or
if there is a nearly-linear time algorithm for multiplying dense
square matrices. To address this bottleneck, we prove a formal equivalence
between the well-studied Dominance Product problem and the problem of computing
the WMG. We prove a similar connection between the so called Dominating Pairs
problem and the problem of finding a winner in the Schulze method.
Our paper is the first to bring fine-grained complexity into the field of
computational social choice. Using it we can identify voting protocols that are
unlikely to be practical for large numbers of candidates and/or voters, as
their complexity is likely, say at least cubic.Comment: 19 pages, 2 algorithms, 2 tables. A previous version of this work
appears in EC 2021. In this version we strengthen Theorem 6.2 which now holds
also for the problem of finding a Schulze winne
Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths
APSP with small integer weights in undirected graphs [Seidel'95, Galil and
Margalit'97] has an time algorithm, where
is the matrix multiplication exponent. APSP in directed graphs with small
weights however, has a much slower running time that would be
even if [Zwick'02]. To understand this bottleneck, we
build a web of reductions around directed unweighted APSP. We show that it is
fine-grained equivalent to computing a rectangular Min-Plus product for
matrices with integer entries; the dimensions and entry size of the matrices
depend on the value of . As a consequence, we establish an equivalence
between APSP in directed unweighted graphs, APSP in directed graphs with small
integer weights, All-Pairs Longest Paths in DAGs with small
weights, approximate APSP with additive error in directed graphs with small
weights, for and several other graph problems. We also
provide fine-grained reductions from directed unweighted APSP to All-Pairs
Shortest Lightest Paths (APSLP) in undirected graphs with weights and
APSP in directed unweighted graphs (computing counts mod
).
We complement our hardness results with new algorithms. We improve the known
algorithms for APSLP in directed graphs with small integer weights and for
approximate APSP with sublinear additive error in directed unweighted graphs.
Our algorithm for approximate APSP with sublinear additive error is optimal,
when viewed as a reduction to Min-Plus product. We also give new algorithms for
variants of #APSP in unweighted graphs, as well as a near-optimal
-time algorithm for the original #APSP problem in unweighted
graphs. Our techniques also lead to a simpler alternative for the original APSP
problem in undirected graphs with small integer weights.Comment: abstract shortened to fit arXiv requirement