Planar Tile Layout Problems

In this section, the centralized and distributed algorithms are evaluated by their ability to solve select planar tile layout problems. The problem is an arrangement of tiles to cover a unit square area within a larger bounded region such that the tiles do not mutually overlap (see Figure 6.4). The tiles are arranged using translations in the two-dimensional space; tile rotations are not allowed. If the tiles were to mutually overlap the arrangement would incur a penalty directly proportional to the area of overlap. Though the tiles may not exceed the layout bound, they may overlap the shaded region, but then such a layout would incur a penalty directly proportional

to the area of overlap.

Unit area in which tiles must be laid out

1.0

-1.0 1.0

Layout penalty region Layout bound

Fig. 6.4 Tile layout problem.

In the centralized algorithm implementation, the positions of all the tiles can be changed simultaneously, while in the distributed algorithm im-plementation each node controls the position of a unique tile, and it is assumed that internode communications are delay-prone. The representa-tion used is a concatenarepresenta-tion of the real encoded Cartesian coordinates of an anchor point of each tile, and each search variable has a range of [—1,1].

Each complete tile-set arrangement is scored directly proportional to the percentage of area covered; the objective being its maximization.

Figure 6.5 shows three tile layouts (A, B, C) of respectively increasing search difficulty, which serve as test cases. For each of the test cases the maximum cover is achieved when the tiles do not overlap and fit exactly within the unit area shown in Figure 6.4. For layout-A an exact fit cor-responding to the maximum cover has two alternatives (tile positions can be swapped), while for layouts B and C the maximum cover is achieved

Performance Evaluation Based on Ideal Objectives 85

only for unique respective tile-set configurations. Due to the nature of the

tile-l Tile-2 Tile-1 Tile-2

Layout-A Layout-B

I lie 1

i ue-4

• - —

—

Tile-2

rile-3

Layout-C

Fig. 6.5 Exact tile layouts for the three test cases.

representation used, layouts A and B correspond to four-variable search problems, while layout-C corresponds to an eight-variable search problem.

The search space corresponding to layout-A has optima aligned with the search coordinates, so for this problem the distributed algorithm does not require any randomization in the selection of new iterates. However, for dis-tributed searches corresponding to layouts B and C many competing and unfavorably aligned optima occur requiring randomization in the selection of iterates.

The simulation parameters for the respective searches corresponding to layouts A and B are: Network delay d = 10, Local access delay I = 1,

Number of nodes p = 2, Coordination frequency (for the distributed algo-rithm) / = 0.1, and Population sizes: mc = 200, rrid = 50. The simulation parameters for the search corresponding to layout-C are: Network delay d = 10, Local access delay 1 = 1, Number of nodes p = 4, Coordination frequency (for the distributed algorithm) / = 0.1, and Population sizes:

mc = 4000, rrid = 100. Gaussian mutation with parent-child competition introduces stochastic variation in the samples. A large population size is used for the centralized search corresponding to tile layout-C primarily be-cause of the presence of many competing local optima, and in the absence of problem specific heuristics, a large population size is a means for main-taining diversity as the search matures. The distributed algorithm on the other hand does not require large population sizes because randomization in selection of global iterates introduces the necessary search diversity. Fig-ure 6.6 compares the average (and standard deviation) performance over 20 trials of the evolution of the mean of the population distribution of the centralized algorithm, and the evolution of the global iterates of the dis-tributed algorithm with respect to search generations. Figure 6.7 shows the same comparison with respect to time requirements of the respective algo-rithms. The distributed search corresponding to this problem consistently achieves the global optimum, albeit in more generations (for the harder tile layouts-B and C), due to randomization in the selection of global iterates.

These performance figures reinforce the observation that the distributed computation has a better time-performance advantage when network com-munication delays are considered.

6.3.1 Discussion

In each of the three tile layout problems, a node controls the position of a unique tile, and relies on information from the other nodes for computing the fitness of each of its alternatives. The fitness of any alternative at a given node is a conditional fitness, in other words, local fitness computa-tions are conditioned upon the locacomputa-tions of representative tiles from each of the other nodes. Alternatively, the nodes by design have to cooperate to solve the global problem. One might argue that if the global fitness space is, for purposes of illustration, similar to a quadratic with no correlation between search variables, or sets of search variables, and each node con-trols one such set, then it would be possible to solve the problem with a one-step communication following local search at each of the nodes. When there is any significant correlation among the partitioned search variables,

Performance Evaluation Based on Ideal Objectives 87

Fig. 6.6 Convergence performance over 20 trials of the centralized and distributed al-gorithms applied to the tile layout test cases.

communication is integral to the problem solving.

A selection of the three tile geometries is conditioned upon such an ob-servation, and the three problems A, B, and C correspond to increasingly coupled (epistasis) problems due to the introduction of notches in their tile geometries. Tile layout-A with two rectangular tiles has lower epistasis than the other problems. In this problem, each of the two tiles can be fixed a priori at one of two extreme positions, and the other free tile may move independently to the globally optimum position. However, in any other non-extreme position, the two tiles have to move in unison (via communi-cation) to seek the global optimum (optima in this case). Tile layout-B is a further complication due to the introduction of a significant notch, and un-less each tile is placed exactly in its respective extreme position, the other tile cannot seek the global optimum without communication. Tile layout-C is clearly the most complicated with a puzzle nature to it, and the only way for a tile to seek the global optimum on its own is if all other tiles are positioned a priori in their respective extreme positions. Therefore, for each of the three tile layout problems, a low epistasis situation will arise only very remotely based on the a priori starting positions of the tiles. When

0.2

Fig. 6.7 Time performance over 20 trials of the centralized and distributed algorithms applied to the tile layout test cases.

the tile positions are randomly initialized, as in this presentation, there will be multiple optima encountered (by construction of the problems), and low epistasis is not a concern.

Further, the experimental investigation reveals that especially for Tile layouts B and C, the distributed algorithm does not reliably converge to the global optimum without explicit randomization in the communication, showing that these are not trivial problems for a distributed algorithm. In other words, unless there is randomization introduced in the selection of a global state, there is a good chance of convergence to a local optimum via distributed local hill climbing.