# Computational Complexity

For n=16 there already are %x1x2x3x4x5x6x7x8x9x10x11x 12x13x14x15 = 653,837,184,000 alternatives to check, which at a rate of one million routes per second takes about 7.5 days; and one single extra city would raise the waiting time to four months. Figure 7. So for three cities, there is only one possible route. The amount of resources consumed by an algorithm in relation to the number of inputs is called ‘computational complexity’. Virtual dice are tossed at some steps of the algorithm to draw decisions, but this is also part of the predefined recipe. Parametric Grasshopper model
Parametric models describe the relations between different parts of a model as a graph where each node defines a (geometric) entity. Shigeru Ban, Heasly Nine Bridges Golf Club, Yeoju, South Korea, 2010
To allow for continuous girders in all three directions, they are split into five layers with two lap joints at every crossing. Girders are created on every projected grid line. An algorithm just has to generate all possible permutations of the listed cities, calculate the respective route lengths and find the shortest one. If we assume that neither starting city nor travelling direction matter, those three routes are effectively the same. The only problem is that for n cities the number of permutations accounts to %x(n-1)!, a term that grows by the factorial (that is, the product of all positive integers
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Figure 8. So it is reasonable to save the projected beam edges in the model, as long as they are updated when the reference surface changes. П = 9 r = 20,160

n = 10 = 181,440

n — 11 r= 1,814,400

Figure 13. The Travelling Salesman Problem
For three cities A, B and C, the six permutations would be [ABC], [ACB], [BCA], [BAC], [CBA] and [CAB]. Figure 11. Since the number of cities is finite, so is the number of routes that can therefore be tested in finite time by a deterministic algorithm. Notably the encoding of an individual’s properties into a genome, the recombination of genomes during reproduction, and the selection based on a quantifiable fitness measure have to be formally well defined and unambiguous. Shigeru Ban, Heasly Nine Bridges Golf Club, Yeoju, South Korea, 2010
The timber roof structure is defined by a regular tri-fold grid that is vertically projected to a curved surface. Even though many parts are similar, 467 individual components with over 2,000 different joints had to be described in detail. Their orientation follows the surface, rendering them curved and twisted. Schematic view of a genetic algorithm (GA)
Evolutionary methods seem to find surprisingly good results in vast solution spaces by chance, but they are based on completely deterministic algorithms. A striking example is the so-called ‘Travelling Salesman Problem’ of finding the shortest route through all cities on a given list. This was only possible by formally describing the whole structure in a parametric system that automatically generated the detailed models from a reference surface and some numerical parameters. The girders intersect at almost 7,500 crossing points. Reducing a circle
A circle can be unambiguously described by three points. Buying faster processors will only help momentarily; building leaner models and applying smarter methods is a much more sustainable approach. terminate and present final result:

create initial population

less than or equal to a number) of the list length. In particular, computational simulations like finite element analysis (FEA) and computational fluid dynamics (CFD) are not easily scalable, which makes it practically impossible to simulate large models in reasonable time, for example to use the results as fitness measures for evolutionary optimisation. But that changes quickly for n>3. The complete roof contains some 3,500 curved timber components with almost 15,000 lap joints.

Updated: 30.10.2014 — 11:25