46 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Analysing trajectory similarity and improving graph dilation
In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph
Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics
The "short cycle removal" technique was recently introduced by Abboud,
Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of
approximation. Its main technical result is that listing all triangles in an
-regular graph is -hard under the 3-SUM conjecture even
when the number of short cycles is small; namely, when the number of -cycles
is for .
Abboud et al. achieve by applying structure vs. randomness
arguments on graphs. In this paper, we take a step back and apply conceptually
similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve
the best possible and the following lower bounds under the 3-SUM
conjecture:
* Approximate distance oracles: The seminal Thorup-Zwick distance oracles
achieve stretch after preprocessing a graph in
time. For the same stretch, and assuming the query time is Abboud et
al. proved an lower bound on the
preprocessing time; we improve it to which is only a
factor 2 away from the upper bound. We also obtain tight bounds for stretch
and and higher lower bounds for dynamic shortest paths.
* Listing 4-cycles: Abboud et al. proved the first super-linear lower bound
for listing all 4-cycles in a graph, ruling out time
algorithms where is the number of 4-cycles. We settle the complexity of
this basic problem by showing that the
upper bound is tight up to factors.
Our results exploit a rich tool set from additive combinatorics, most notably
the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key
ingredient that may be of independent interest is a subquadratic algorithm for
3-SUM if one of the sets has small doubling.Comment: Abstract shortened to fit arXiv requirement
Algorithms for sparse convolution and sublinear edit distance
In this PhD thesis on fine-grained algorithm design and complexity, we investigate output-sensitive and sublinear-time algorithms for two important problems. (1) Sparse Convolution: Computing the convolution of two vectors is a basic algorithmic primitive with applications across all of Computer Science and Engineering. In the sparse convolution problem we assume that the input and output vectors have at most t nonzero entries, and the goal is to design algorithms with running times dependent on t. For the special case where all entries are nonnegative, which is particularly important for algorithm design, it is known since twenty years that sparse convolutions can be computed in near-linear randomized time O(t log^2 n). In this thesis we develop a randomized algorithm with running time O(t \log t) which is optimal (under some mild assumptions), and the first near-linear deterministic algorithm for sparse nonnegative convolution. We also present an application of these results, leading to seemingly unrelated fine-grained lower bounds against distance oracles in graphs. (2) Sublinear Edit Distance: The edit distance of two strings is a well-studied similarity measure with numerous applications in computational biology. While computing the edit distance exactly provably requires quadratic time, a long line of research has lead to a constant-factor approximation algorithm in almost-linear time. Perhaps surprisingly, it is also possible to approximate the edit distance k within a large factor O(k) in sublinear time O~(n/k + poly(k)). We drastically improve the approximation factor of the known sublinear algorithms from O(k) to k^{o(1)} while preserving the O(n/k + poly(k)) running time.In dieser Doktorarbeit über feinkörnige Algorithmen und Komplexität untersuchen wir ausgabesensitive Algorithmen und Algorithmen mit sublinearer Lauf-zeit für zwei wichtige Probleme. (1) Dünne Faltungen: Die Berechnung der Faltung zweier Vektoren ist ein grundlegendes algorithmisches Primitiv, das in allen Bereichen der Informatik und des Ingenieurwesens Anwendung findet. Für das dünne Faltungsproblem nehmen wir an, dass die Eingabe- und Ausgabevektoren höchstens t Einträge ungleich Null haben, und das Ziel ist, Algorithmen mit Laufzeiten in Abhängigkeit von t zu entwickeln. Für den speziellen Fall, dass alle Einträge nicht-negativ sind, was insbesondere für den Entwurf von Algorithmen relevant ist, ist seit zwanzig Jahren bekannt, dass dünn besetzte Faltungen in nahezu linearer randomisierter Zeit O(t \log^2 n) berechnet werden können. In dieser Arbeit entwickeln wir einen randomisierten Algorithmus mit Laufzeit O(t \log t), der (unter milden Annahmen) optimal ist, und den ersten nahezu linearen deterministischen Algorithmus für dünne nichtnegative Faltungen. Wir stellen auch eine Anwendung dieser Ergebnisse vor, die zu scheinbar unverwandten feinkörnigen unteren Schranken gegen Distanzorakel in Graphen führt. (2) Sublineare Editierdistanz: Die Editierdistanz zweier Zeichenketten ist ein gut untersuchtes Ähnlichkeitsmaß mit zahlreichen Anwendungen in der Computerbiologie. Während die exakte Berechnung der Editierdistanz nachweislich quadratische Zeit erfordert, hat eine lange Reihe von Forschungsarbeiten zu einem Approximationsalgorithmus mit konstantem Faktor in fast-linearer Zeit geführt. Überraschenderweise ist es auch möglich, die Editierdistanz k innerhalb eines großen Faktors O(k) in sublinearer Zeit O~(n/k + poly(k)) zu approximieren. Wir verbessern drastisch den Approximationsfaktor der bekannten sublinearen Algorithmen von O(k) auf k^{o(1)} unter Beibehaltung der O(n/k + poly(k))-Laufzeit
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs
We consider the question of approximating Max 2-CSP where each variable
appears in at most constraints (but with possibly arbitrarily large
alphabet). There is a simple -approximation algorithm for the
problem. We prove the following results for any sufficiently large :
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish
the following hardness results for approximating Maximum Independent Set on
-claw-free graphs:
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
In comparison, known approximation algorithms achieve -approximation in polynomial time [Neuwohner, STACS 2021; Thiery
and Ward, SODA 2023] and -approximation in
quasi-polynomial time [Cygan et al., SODA 2013]
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum