Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within $A$. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall expected time and space requirement in the average case are proven to be $O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.Comment: 11 pages, 1 figur

Effect of Coriaria arborea on seed banks during primary succession on Mt Tarawera, New Zealand

An experiment was conducted over two years to investigate the effect of Coriaria arborea, a native nitrogen-fixing shrub, on soil seed banks at sites representing a post-volcanic successional sequence on Mt Tarawera, New Zealand. The sites ranged from bare volcanic ash and lapilli substrate, through low-growing pre-Coriaria vegetation, to dense stands of Coriaria scrub. Soils (to a depth of 50 mm) under recently established Coriaria and older stands had more seedlings (1096 and 1585 seedlings 0.4 m-2, respectively) and species (37 and 45 species 0.4 m-2, respectively) emerge than where there was no Coriaria (243-320 seedlings 0.4 m-2, 14-25 species 0.4 m-2) and were the only soils with Coriaria seedlings. In total, 3488 seedlings representing 63 taxa were recorded. Seeds were still germinating after 24 months but rates declined markedly in the second year. For example, Coriaria reached a germination peak at 8 weeks but continued to germinate sporadically over the 2-year period. Tree species present in young forest within 0.5 km of the sites were absent. Establishment of Coriaria greatly accelerated an underlying trend of gradually increasing abundance and diversity of seeds in the soil with vegetation age. Adventive, wind-dispersed, and annual species were over-represented in the seed banks compared with the regional evergreen forest-dominated flora. These proportions are expected to decline as succession to forest gradually occurs

Evaluating alignment of student and tutor perspectives on feedback on language learning assignments

Feedback alignment is key to the effectiveness of formative feedback, but often tutors can only guess whether their feedback is consistent with students’ needs and expectations. This study aimed to identify areas of potential misalignment by bringing together self-reported data and feedback analysis. Two parallel surveys were conducted with 736 students and 96 tutors from a distance learning undergraduate programme in Language Studies to compare their self-reported attitudes and behaviours. Survey data were then triangulated through the analysis of 216 marked assignments using the feedback analysis categorisation tool (FACT) according to the orientation (focus on a weakness or a strength) and depth (layers of scaffolding) of feedback comments. Misalignment was most apparent in relation to feedback on strengths and mutual assumptions regarding feedback ownership. The findings support the need for a more dialogic approach to feedback and confirm the value of feedback analysis as a means of evaluating feedback alignment for sustainable assessment

Bernard Suits' Utopia of Gameplay: A Critical Analysis

In this dissertation, I integrate the published and newly available unpublished works of Bernard Suits to arrive at an original, holistic interpretation of his corpus. I identify, analyze, and resolve inconsistencies in his position which have not been previously critiqued in depth. Centrally, I provide a critical analysis of Suits’ relatively obscure utopian thesis: his argument that the ‘ideal of existence’ for humankind is a utopia of gameplay. More specifically, I demonstrate that Suits’ utopian thesis fails on its own grounds because his utopian vision—the thought experiment upon which his utopian thesis rests for its plausibility—requires that humanity enter a post-instrumental phase of culture unimaginable from our current species-perspective. The nature of utopian gameplay is obscured behind this cultural gap, and thus we pre-utopians have no rational reason to accept Suits’ assertion that it instantiates the ideal of existence. Finally, I sympathetically rehabilitate Suits’ utopian thesis along perfectionist lines as the utopian game design thesis, and show that its main value lies in its role as a regulative ideal, offering a unique set of normative recommendations for our current gaming practices