Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

Forest Stand Structure and Primary Production in relation to Ecosystem Development, Disturbance, and Canopy Composition

Temperate forests are complex ecosystems that sequester carbon (C) in biomass. C storage is related to ecosystem-scale forest structure, changing over succession, disturbance, and with community composition. We quantified ecosystem biological and physical structure in two forest chronosequences varying in disturbance intensity, and three late successional functional types to examine how multiple structural expressions relate to ecosystem C cycling. We quantified C cycling as wood net primary production (NPP), ecosystem structure as Simpson’s Index, and physical structure as leaf quantity (LAI) and arrangement (rugosity), examining how wood NPP-structure relates to light distribution and use-efficiency. Relationships between structural attributes of biodiversity, LAI, and rugosity differed. Development of rugosity was conserved regardless of disturbance and composition, suggesting optimization of vegetation arrangement over succession. LAI and rugosity showed significant positive productivity trends over succession, particularly within deciduous broadleaf forests, suggesting these measures of structure contain complementary, not redundant, information related to C cycling

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$

Semi-algebraic Ramsey numbers

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real variables. The description complexity of such a relation is at most $t$ if the number of polynomials and their degrees are all bounded by $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

The visual preferences for forest regeneration and field afforestation : four case studies in Finland

The overall aim of this dissertation was to study the public's preferences for forest regeneration fellings and field afforestations, as well as to find out the relations of these preferences to landscape management instructions, to ecological healthiness, and to the contemporary theories for predicting landscape preferences. This dissertation includes four case studies in Finland, each based on the visualization of management options and surveys. Guidelines for improving the visual quality of forest regeneration and field afforestation are given based on the case studies. The results show that forest regeneration can be connected to positive images and memories when the regeneration area is small and some time has passed since the felling. Preferences may not depend only on the management alternative itself but also on the viewing distance, viewing point, and the scene in which the management options are implemented. The current Finnish forest landscape management guidelines as well as the ecological healthiness of the studied options are to a large extent compatible with the public's preferences. However, there are some discrepancies. For example, the landscape management instructions as well as ecological hypotheses suggest that the retention trees need to be left in groups, whereas people usually prefer individually located retention trees to those trees in groups. Information and psycho-evolutionary theories provide some possible explanations for people's preferences for forest regeneration and field afforestation, but the results cannot be consistently explained by these theories. The preferences of the different stakeholder groups were very similar. However, the preference ratings of the groups that make their living from forest - forest owners and forest professionals - slightly differed from those of the others. These results provide support for the assumptions that preferences are largely consistent at least within one nation, but that knowledge and a reference group may also influence preferences.Väitöskirjassa tutkittiin ihmisten maisemapreferenssejä (maisemallisia arvostuksia) metsänuudistamishakkuiden ja pellonmetsitysten suhteen sekä analysoitiin näiden preferenssien yhteyksiä maisemanhoito-ohjeisiin, vaihtoehtojen ekologiseen terveyteen ja preferenssejä ennustaviin teorioihin. Väitöskirja sisältää neljä tapaustutkimusta, jotka perustuvat hoitovaihtoehtojen visualisointiin ja kyselytutkimuksiin. Tapaustutkimusten pohjalta annetaan ohjeita siitä, kuinka uudistushakkuiden ja pellonmetsitysten visuaalista laatua voidaan parantaa. Väitöskirjan tulokset osoittavat, että uudistamishakkuut voivat herättää myös myönteisiä mielikuvia ja muistoja, jos uudistusala on pieni ja hakkuun välittömät jäljet ovat jo peittyneet. Preferensseihin vaikuttaa hoitovaihtoehdon lisäksi mm. katseluetäisyys, katselupiste ja ympäristö, jossa vaihtoehto on toteutettu. Eri viiteryhmien (metsäammattilaiset, pääkaupunkiseudun asukkaat, ympäristönsuojelijat, tutkimusalueiden matkailijat, paikalliset asukkaat sekä metsänomistajat) maisemapreferenssit olivat hyvin samankaltaisia. Kuitenkin ne ryhmät, jotka saavat ainakin osan elannostaan metsästä - metsänomistajat ja metsäammattilaiset - pitivät metsänhakkuita esittävistä kuvista hieman enemmän kuin muut ryhmät. Nämä tulokset tukevat oletusta, että maisemapreferenssit ovat laajalti yhteneväisiä ainakin yhden kansan tai kulttuurin keskuudessa, vaikka myös viiteryhmä saattaa vaikuttaa preferensseihin jonkin verran. Nykyiset metsämaisemanhoito-ohjeet ovat pitkälti samankaltaisia tässä väitöskirjassa havaittujen maisemapreferenssien kanssa. Myöskään tutkittujen vaihtoehtoisten hoitotapojen ekologisen paremmuuden ja niihin kohdistuvien maisemallisten arvostusten välillä ei ollut suurta ristiriitaa. Kuitenkin joitakin eroavaisuuksia oli; esimerkiksi sekä maisemanhoito-ohjeiden että ekologisten hypoteesien mukaan säästöpuut tulisi jättää ryhmiin, kun taas ihmiset pitivät eniten yksittäin jätetyistä puista. Informaatiomalli ja psyko-evolutionaarinen teoria tarjoavat mahdollisia selityksiä uudistushakkuisiin ja pellonmetsitykseen kohdistuville preferensseille, vaikkakaan tutkimuksen tuloksia ei voida täysin selittää näillä teorioilla