Geodesic
Path Reconstruction in Barning--Hall Tree
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 95-102

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In 1963 F. J. M. Barning and in 1970 A. Hall gave a systematic procedure to generate all primitive Pythagorean triples (PPTs) with help of multiplication of the minimal Pythagorean triple $[3,4,5]$ considered as a vector by matrix series where matrices are taken from a prescribed 3-set of unimodular matrices. This provides a structure of an infinite ternary rooted labeled tree. In this article we establish an algorithm that reconstructs a tree path leading from the root to the primitive Pythagorean triple of interest. Because a triple can lie deeply then efficiency of the algorithm is quite important. The established algorithm has polynomial time complexity with respect to the input length relating to a “size” of PPT.
Keywords: pythagorean triples, Barning–Hall tree, triple generators, path reconstruction, algorithm efficiency. 
@article{VNGU_2012_12_3_a7,
     author = {P. G. Emelyanov},
     title = {Path {Reconstruction} in {Barning--Hall} {Tree}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {95--102},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a7/}
}
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P. G. Emelyanov. Path Reconstruction in Barning--Hall Tree. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 95-102. http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a7/