Testing Addresses for Property Match.

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Algorithm for address search describes the strategy used by the generic address map search and mentions the worst case of testing for a match against an arbitrary multi-dimensional sparse array which is self-interleaved as can be defined quite easily in DDL/DMP.  This page describes the algorithm used for that test.

calculation

Testing address A for a match with property P.  For P the base address is specified, then for each array dimension in P there is an increment and element count.  Does A match an element of P?

A property has an arbitrary number n of dimensions Dₖ for 0 ≤ k ≤ n we call these D₀, D₁ ...  Dₖ ...  Dₙ each of which is specified by an increment I₀ ...  Iₖ ...  Iₙ and a count.  For the algorithms here we will use the range limit instead of the count.  R (= count - 1) thus the maximum inclusive value of property address P due to dimension i is Pmax_i = Pbase + Rₖ.Iₖ.

A generic member of the array P is Px where

(1)  Px = Pbase + x₀.I₀ + x₁.I₁ + ... + xₙ.Iₙ

For n = 0 we have a single point property and the test is trivial

(2)  A ≡ Pbase (or A - Pbase ≡ 0)

For n = 1 we have a linear array, but if increment I₀ > 1 then there are holes so we need to know whether (A - Pbase) is an integer multiple of I₀.  The simplest test is to use the modulus

(3) (A - Pbase) mod I₀ ≡ 0

For higher dimensions testing becomes more complex.  The range of P is given by

(4)  Pbase ≤ P ≤ Pbase + I₀.R₀ + I₁.R₁ + ... + Iₙ.Rₙ

Now recalling (1) we want to test A ≡ Px ? but we do not know the values for x₀, x₁ etc.  Iterating through all x₀, x₁, x₂ etc is not practicable for large arrays.  In most cases increments and counts are chosen such that there is no overlap between dimensions (in a two dimensional case this is the same as saying that one row does not interleave with the next or previous).  However there is no guarantee that this isn’t the case.  What we do know though provides limits on x₀ — xₙ.

First by subtracting Pbase from all calculations we can simplify so

(5)  Ao = A - Pbase

Assume A is in P then for some valid set of x₀ — xₙ

(6)  Ao ≡ x₀.I₀ + x₁.I₁ + ... + xₙ.Iₙ

Solving for an individual xₖ

(7)  xₖ.Iₖ ≡ Ao - (x₀.I₀ + x₁.I₁ + ... [miss out xₖ.Iₖ] ... + xₙ.Iₙ)

The largest value possible of xₖ for a given Ao is when all other x are zero and the bracketed term in (7) becomes zero

(8)  xₖ_largest ≡ Ao/Iₖ

but we know that xₖ ≤ Rₖ so for a given Ao and for each dimension the top limit for xₖ we will call Tₖ

(9)  Tₖ = minimum(Rₖ, Ao/Iₖ)

We can now calculate Tₖ for each dimension.  Now using (7) again what is the smallest for xₖ? this will occur when all other x are at their maximum, but we have now calculated those maxima T₀ — Tₙ.  So

(10)  xₖ_smallest = (Ao - (T₀.I₀ + .. [miss out Tₖ.Iₖ] .. + Tₙ.Iₙ))
                    ----------------------------------------------
                                         Iₖ

but we also know that xₖ ≥ 0 so this gives us the bottom limit on xₖ which we call Bₖ

(14)  Bₖ = maximum(0, xₖ_smallest)

We have now established for a test value A, a restricted range of indexes which we should test for each dimension to see whether A is in P.  We can choose one arbitrary dimension, then iterating over all possible values from Bₖ to Tₖ for all the other dimensions, rearranging (7) and using the modulus test for integral factors our test becomes

(15) (Ao - (x₀.I₀ .. [miss kth term] .. xₙ.Iₙ)) mod Iₖ ≡ 0

repeated for all possible xₖ: Bₖ ≤ xₖ ≤ Tₖ

The process of calculating Bₖ and Tₖ for each dimension may seem onerous but in realistic arrays it hugely reduces the search space.  However, it is possible to optimize further

If the array is non-overlapping, the value of Bₖ and Tₖ come out identical for the outermost dimension (the one with the largest increment Iₖ) incdicating that this dimension need not be iterated at all.  This must be so because in a non-overlapping array there is only one possible span of the lesser dimensions within which Ao could occur.  Having established this value, we now know that we can fix it and update the calculations for the remaining dimensions but with the fixed value for the outermost.  If the array is overlapping (self-interleaved), we may find the outermost dimension does not reduce to a single span to be searched, but nevertheless, substituting the values for Tₖ and Bₖ into the equations above reduces the range to search within other dimensions further.

Generalising this principle, if the dimensions are sorted so they are tested in order from outermost (largest increment) to innermost (smallest increment), instead of the declaration order which is arbitrary, the search reduces to the smallest possible range.

The parser builds a single table (a 1 dimensional array) in which each each entry E defines a region within the space by a low address and a high address E_lo and E_hi (inclusive values).
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