Lotto Algorithms - Permutations, Combinations

This thread aims to consolidate preceding contributions and virgin developments, in adjustment to supply efficient and beneficial functions for analyzing Permutations and Combinations.

We find out these tools are used by Lotto-style sport analyzers, and while we personally atmosphere this is an handle whose profitability (in fiscal terms) is as well doubtful from a purely mathematical argument .... :huh:

....... but!! :rolleyes: these cats accept every appropriate to decent algorithms - and they might even constitute lots of lotto capital - (elicit us provided you end, ok? ;) )

Those seeking no sweat "generators" for the works or fragmentary sets of combinations or permutations testament boast easy functions below ("Odometer-style Generators").

Those needing to engage in amassed than that testament thirst for to stare at the slightly and manifold "Index Mapping" functions, which can be used to cause sets however extremely fit more capabilities.

In course to find out the differentiation between the two methods, we call for to catch on the hypothesis of how all practicable combinations (or perms) can be listed in a connatural category categorization called the lexicographical succession.

Here are the combinations of 3 from 5 in lexicographical establishment:

As you can contemplate, it's de facto just the constant numerical category composition, based on the leftmost fields having priority.

Permutations too carry a accustomed lexicographical (sequence) categorization. Here is the adequate listing for K = 4:

By creating a MAPPING overhaul (CombinationToIndex) that assigns an INDEX price to everyone combination or permutation according to its position in the all-inclusive record, we can enact identical combinations by easy numbers.

With an inverse advantage (IndexToCombination) that returns the combination for a inured index reward, we can then constitute any subset of all imaginable combinations from a liable starting mark agnate this:

....

....

These are analogous odometers, you care for a table with a "revolve" for everyone component in the locate. You derivation it at "1 2 3 4 5 ...", and then you "rotate" it with successive calls to a NextCombination advantage which advances the wheels to the close position.

Debug.Print ThisCombination ' returns in fashion reading

NextCombiterritory 50, 6 ' advance to ensuing position

The rules for rotating the wheels are besides heterogeneous than for an odometer (which allows duplicate values), on the other hand there are wonderfully incomplex combinatorial algorithms for doing both Perms and Combs.

All VB functions described testament be presented below, however antecedent we'll stare at the advantages and disadvantages of the 2 approaches.

The odometer reaching is both concise and efficient, so it wins the "fashion" award, and it too can knob VAST games (prize a million balls from a billion!).

However, for analytical purposes, there is a fatal flaw - you can't confess "distance" on this odometer - you enjoy no concept where in the string you are from the contemporary settings.

So the odometer can't confess you the come back to questions prize this:

What's the inaugural combination in the moment half of the unabridged (50, 6) case?

To reply these requires the index mapping functions. The cardinal reinstate is approximately 11 million, as it turns elsewhere:

And the moment controversy, the nearly purpose?

The mapping functions own us to decrease the entire virtual establish into a lovely universal grid, and intersect it accordingly. Generating sequential data members is slightly less efficient than the Odometer, on the contrary yet a miniature fraction of the value of either printing or otherwise displaying or (worse) storing them all.

Because the Odometer mechanism doesn't wish any measurement of the extent of the abundant feasible combination allot, it can attempt with games of gelastic magnitude.

Momentarily you can't determine that with index mapping functions through you simply can't still store the numbers required (in the action of a 10-from-50000 sport the unit of combinations needs a million or so digits to symbolize!).

However that's individual an artificial restriction, as the criterion material types in VB cede us group of scope to mould all in fashion games of attention, plus extension to expand seamlessly to better games coextensive "40 from 100", truly any pastime for which the complete locate magnitude (numeral of viable combinations) is less than 75 x 10 ^ 27.

This is largely commendable - you can choose any K (cipher of balls to pick) for all values of N up to 99. At N = 100 you can isolated energy to K = 46 (just shy of the finishing border at K=50) . However you'd usually benefit smaller K besides, so that lets you boost N much also - e.g. you can replica 10-from-1000

You'll letter that 75 x 10^27 is rather larger than the maximum FAR-REACHING cost in VB. The index mapping functions adoption Variant (Decimal) variables where required, which gives us access to what are effectively triple-length Longs' (96-bit integers).

We used this way to extend the gamut of the latest assistance that we posted to calculate the unit of Combinations(N, K) 129902

And that's while you'll study some Variant variables in the Index Mapping functions below, however not the Odometer ones.

If you've followed everything above, you should treasure this module self-contained and the servicing usage self-explanatory! ;)

Public Sub SetCombination(ByVal N As Continued, ByVal K As Lingering, ByVal Combn As List)