Bounds for min-max heaps

Very recently a new data structure, called a min-max heap, was presented for implementing the double-ended priority queue. A min-max heap onn keys is constructed inO(n) time; the minimum and maximum keys are found in constant time, and the operations of deleting the minimum, deleting the maximum and inserting a new key into the heap are performed inO(logn) time. In addition, the data structure can be stored implicitly, i.e. in an array ofn elements without using any additional pointers.In this paper, we present lower bound results on the number of comparisons required, in the worst case, for the operations i) to construct a min-max heap on a given set of keys; ii) to convert a min-max heap into a max-min heap; and iii) to merge two min-max heaps into one min-max heap. New upper bounds for the convert and merge operations are also derived. It is found that the main difference between traditional heaps and min-max heaps lies in the time needed to perform the merge operation. While traditional heaps can be merged efficiently, it is shown that min-max heaps are not sublinearly mergeable. Even the seemingly simple task of converting a min-max heap into a max-min heap cannot be performed in less than linear time.This research was supported by the Natural Science and Engineering Council of Canada under Grant No. A0392. (A preliminary version of this paper was presented at the 24th Annual Allerton Conference on Communication, Control and Computing.)     

Ultra-small PbS nanocrystals as sensitizers for red-to-blue triplet-fusion upconversion

Photon upconversion is a strategy to generate high-energy excitations from low-energy photon input, enabling advanced architectures for imaging and photochemistry. Here, we show that ultra-small PbS nanocrystals can sensitize red-to-blue triplet-fusion upconversion with a large anti-Stokes shift (ΔE = 1.04 eV), and achieve max-efficiency upconversion at near-solar fluences (I_th = 220 mW cm⁻²) despite endothermic triplet sensitization. This system facilitates the photo-initiated polymerization of methyl methacrylate using only long-wavelength light (λ_exc: 637 nm); a demonstration of nanocrystal-sensitized upconversion photochemistry. Time-resolved spectroscopy and kinetic modelling clarify key loss channels, highlighting the benefit of long-lifetime nanocrystal sensitizers, but revealing that many (48%) excitons that reach triplet-extracting carboxyphenylanthracene ligands decay before they can transfer to free-floating acceptors—emphasizing the need to address the reduced lifetimes that we determine for molecular triplets near the nanocrystal surface. Finally, we find that the inferred thermodynamics of triplet sensitization from these ultra-small PbS quantum dots are surprisingly favourable—completing an advantageous suite of properties for upconversion photochemistry—and do not vary significantly across the ensemble, which indicates minimal effects from nanocrystal heterogeneity. Together, our demonstration and study of red-to-blue upconversion using ultra-small PbS nanocrystals in a quasi-equilibrium, mildly endothermic sensitization scheme offer design rules to advance implementations of triplet fusion, especially where large anti-Stokes wavelength shifts are sought.

We demonstrate the use of ultra-small PbS quantum dots as endothermic sensitizers for red-to-blue triplet-fusion upconversion, achieving nanocrystal-sensitized upconversion photochemistry.     

ZnO nanoparticles photocatalytic activity toward atmospheric toluene under simulated sunlight

Research on Chemical Intermediates - In this study, the photocatalytic degradation of toluene through zinc oxide (ZnO) nanoparticles coated on glass plates was examined under simulated sunlight....