Constraints have played an important role in the construction of GUIs, where they are mainly used to define the layout of the widgets. Resizing behavior is very important in GUIs because areas have domain specific parameters such as form the resizing of windows. If linear objective function is used and window is resized then error is not distributed equally. To distribute the error equally, a quadratic objective function is introduced. Different algorithms are widely used for solving linear constraints and quadratic problems in a variety of different scientific areas. The linear relxation, Kaczmarz, direct and linear programming methods are common methods for solving linear constraints for GUI layout. The interior point and active set methods are most commonly used techniques to solve quadratic programming problems. Current constraint solvers designed for GUI layout do not use interior point methods for solving a quadratic objective function subject to linear equality and inequality constraints. In this paper, performance aspects and the convergence speed of interior point and active set methods are compared along with one most commonly used linear programming method when they are implemented for graphical user interface layout. The performance and convergence of the proposed algorithms are evaluated empirically using randomly generated UI layout specifications of various sizes. The results show that the interior point algorithms perform significantly better than the Simplex method and QOCA-solver, which uses the active set method implementation for solving quadratic optimization.
Most of the research related to GUI layout deals with the various algorithms for solving constraint hierarchies. Research related to constraint based UI layout has provided results in the form of tools @cite_7 @cite_19 and algorithms @cite_10 @cite_5 for specific tasks. The latest work @cite_0 on constraint based GUIs uses a quadratic solving strategy which they find better than linear solving strategies. They @cite_0 implemented the active set method for solving a quadratic objective function subject to some linear constraints. Baraf @cite_14 presents a quadratic optimization algorithm for solving linear constraints in modelling physical systems. QOCA @cite_5 uses the active set algorithm for solving quadratic programming problem for graphical user interface layout.
@cite_14 @cite_7 @cite_0 @cite_19 @cite_5 @cite_10
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Text entry user interfaces have been a bottleneck of non traditional computing devices. One of the promising methods is the virtual keyboard on touch screens. Various layouts have been manually designed to replace the dominant QWERTY layout. This paper presents two computerized quantitative design techniques to search for the optimal virtual keyboard. The first technique simulated the dynamics of a keyboard with “digraph springs” between keys, which produced a “Hooke’s” keyboard with 41.6 wpm performance. The second technique used a Metropolis random walk algorithm guided by a “Fitts energy” objective function, which produced a “Metropolis” keyboard with 43.1 wpm performance. The paper also models and evaluates the perfo rmance of four existing keyboard layouts. We corrected erroneous estimates in the literature and predicted the performance of QWERTY, CHUBON, FITALY, OPTI to be in the neighborhood of 30, 33, 36 and 38 wpm respectively. Our best design was 40 faster than QWERTY and 10 faster than OPTI, illustrating the advantage of quantitative user interface design techniques based on models of human performance over traditional trial and error designs guided by heuristics. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We describe an object-oriented constraint solving toolkit, QOCA, designed for interactive graphical applications. It has a simple yet powerful interface based on the metric space model for constraint manipulation. Currently QOCA supports linear arithmetic constraints and two different metrics: the square of the Euclidean distance and the Manhattan distance. It provides three solvers, all of which rely on keeping the constraints in solved form and relies on novel algorithms for efficient resolving of constraints during direct manipulation. We provide a thorough empirical evaluation of QOCA, both of the interface design and the speed of constraint solving.