Dispelling the N^3 myth for the Kt jet-finder

TL;DR

Problem: the kt jet-finder's O($N^3$) scaling hindered high-multiplicity analyses. Approach: recast clustering as a two-dimensional nearest-neighbor problem using Voronoi/Delaunay structures to obtain an efficient O($N \log N$) FastJet algorithm, implemented with CGAL and a binary-tree minimum structure. Findings: FastJet significantly outperforms existing jet finders across a range of $N$, enabling fast kt clustering for LHC/heavy-ion environments and facilitating jet-area-based pileup subtraction. Significance: broadens practical use of kt clustering and suggests further geometry-driven optimizations and related fast, infrared-safe jet-finders.

Abstract

At high-energy colliders, jets of hadrons are the observable counterparts of the perturbative concepts of quarks and gluons. Good procedures for identifying jets are central to experimental analyses and comparisons with theory. The Kt family of successive recombination jet finders has been widely advocated because of its conceptual simplicity and flexibility and its unique ability to approximately reconstruct the partonic branching sequence in an event. Until now however, it had been believed that for an ensemble of N particles the algorithmic complexity of the Kt jet finder scaled as N^3, a severe issue in the high multiplicity environments of LHC and heavy-ion colliders. We here show that the computationally complex part of Kt jet-clustering can be reduced to two-dimensional nearest neighbour location for a dynamic set of points. Borrowing techniques developed for this extensively studied problem in computational geometry, Kt jet-finding can then be performed in N ln N time. Code based on these ideas is found to run faster than all other jet finders in current use.

Figures (3)

• Figure 1: The Voronoi diagram for ten random points. The Delaunay triangulation (red) connecting the ten points is also shown. In this example the points 1, 4, 2, 8 and 3 are the 'Voronoi' neighbours of 7, and 3 is its nearest neighbour.
• Figure 2: The running times (on a 3 GHz Pentium 4 processor with $1$ GB of memory, 512 kB of cache, and version 3.4 of the GNU g++ compiler) of the KtJet KtImplementation and FastJet implementations of the $k_t$-clustering jet-finder versus the number of initial particles. Different values of $N$ have been obtained by taking a LHC dijet event with $p_t \simeq 60$ GeV and adding on variable numbers of minimum bias events. Both kinds of events have been simulated with Pythia 6.3 Pythia.
• Figure 3: The running times of various jet-finders versus the number of initial particles. JetClu is a widely-used cone-type jet-finder, however it is 'almost infrared unsafe', i.e. perturbative predictions have large logarithmic dependence on small parameters (e.g. seed threshold) MidpointIdeaSeymour:1997kj. MidPoint MidpointIdea is an infrared safe cone-type jet finder. For both we use code and parameters from CDF CDFCones. The Optimal Jet Finder Tkachov (OJF) has been run with $\Omega_{cut} = 0.15$ and a maximum of 8 jets, so as to produce a final state similar to that returned by the $k_t$ and cone jet-finders and to limit its run time.