Space-Time Loop Tiling for Dynamic Programming Codes

We present a new space-time loop tiling approach and demonstrate its application for the generation of parallel tiled code of enhanced locality for three dynamic programming algorithms. The technique envisages that, for each loop nest statement, sub-spaces are first generated so that the intersection of them results in space tiles. Space tiles can be enumerated in lexicographical order or in parallel by using the wave-front technique. Then, within each space tile, time slices are formed, which are enumerated in lexicographical order. Target tiles are represented with multiple time slices within each space tile. We explain the basic idea of space-time loop tiling and then illustrate it by means of an example. Then, we present a formal algorithm and prove its correctness. The algorithm is implemented in the publicly available TRACO compiler. Experimental results demonstrate that parallel codes generated by means of the presented approach outperform closely related manually generated ones or those generated by using affine transformations. The main advantage of code generated by means of the presented approach is its enhanced locality due to splitting each larger space tile into multiple smaller tiles represented with time slices.

On Hyperoctahedral Enumeration System, Application to Signed Permutations

In this paper, we give the denitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use the inversion statistic on this group to code the signed permutations. We show that this correspondence provides a way with which the signed permutations group can be ordered. With this classication scheme, we can nd the r-th signed permutation from a given number r and vice versa without consulting the list in lexicographical order of the elements of the signed permutations group.

Lyndon Words and Christoffel Words

This chapter covers the lexicographical ordering of lower Christoffel words, which is equivalent to the ordering by their slopes (Borel and Laubie). Lower Christoffel words are particular Lyndon words. They are maximum for the lexicographical order among Lyndon words of a given slope (Borel and Laubie). They are, together with the upper Christoffel words, the only unbordered finite Sturmian words (Chuan). They are exactly the Lyndon words which are Sturmian words (Berstel and de Luca). The standard factorization of a lower Christoffel word is obtained by cutting before the smallest lexicographical suffix. Finally, they are exactly the Lyndon words which are equilibrated (Melançon).

Continued Fractions

Basic theory of continued fractions: finite continued fractions (for rational numbers) and infinite continued fractions (for irrational numbers). This also includes computation of the quadratic number with a given periodic continued fraction, conjugate quadratic numbers, and approximation of reals and convergents of continued fractions. The chapter then takes on quadratic bounds for the error term and Legendre’s theorem, and reals having the same expansion up to rank n. Next, it discusses Lagrange number and its characterization as an upper limit, and equivalence of real numbers (equivalent numbers have the same Lagrange number). Finally, it covers ordering real numbers by alternating lexicographical order on continued fractions.