Kaczmarz Algorithm with Soft Constraints for User Interface Layout

Abstract

ABSTRACT. The Kaczmarz method is an iterative method for solving large systems of equations that projects iterates orthogonally onto the solution space of each equation. In contrast to direct methods such as Gaussian elimination or QRfactorization, this algorithm is efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the Kaczmarz method as described in the literature has its limitations: it considers only equality constraints and does not support soft constraints, which makes it inapplicable to the UI layout problem.In this paper we extend the Kaczmarz method for solving specifications containing soft constraints, using the prioritized IIS detection algorithm. Furthermore, the performance and convergence of the proposed algorithms are evaluated empirically using randomly generated UI layout specifications of various sizes. The results show that these methods offer improvements in performance over standard methods like Matlab's LINPROG, a well-known efficient linear programming solver.

Tiling algebra for constraint-based layout editing

“…In contrast to a linear approach, this leads to unique layouts with certain aesthetic properties [27]. An overview of different solving techniques can be found in other works [1,28,29].…”

Hildreth’s algorithm with applications to soft constraints for user interface layout

“…In each subsystem, we apply the mixed Hildreth's algorithm, both randomly and deterministically, see Algorithm 2. The work [24] also uses prioritized IIS detection method, but with the ORM method. (It is called the Kaczmarz prioritized IIS detection method in [24].)…”

“…The work [24] also uses prioritized IIS detection method, but with the ORM method. (It is called the Kaczmarz prioritized IIS detection method in [24].) Moreover, they do not consider randomized methods or relaxation parameter.…”

“…The results show that both deterministic and randomized mixed Hildreth's algorithm are optimal and efficient in many ways. We observe that our implementation outperforms Matlab's LINPROG linear optimization package [36], LP-Solve [7], the implementation of QRdecomposition of the Apache Commons Math Library [4], and the Kaczmarz prioritized IIS detection method [24]. LP-Solve is a well-known linear programming solver that has been used for UI layout.…”

“…The proposed algorithm was used to solve each of the problems of the test data set and the time was measured. As a reference, all the generated specifications were also solved with an implementation of the cyclic Algorithm 2 (referred as Hildreth algorithm here), the randomized Algorithm 2 (referred as randomized Hildreth's algorithm here), Matlab's LINPROG solver [36], LP-Solve [7], and the Kaczmarz with prioritized IIS detection [24]. LIN-PROG is widely known for its speed, and LP-Solve was previously used to solve UI layout problems [28].…”