Obfuscation of Abstract Data-Types - Rowan

CHAPTER 4. SPLITTING HEADACHES 69
4.2.1 Array-Split properties
In Section 4.1.3 we defined two splits: asp and b(k). Can these be used for
arrays? Yes, in fact, we can use exactly the same definitions as before and the
alternating split matches the array-split defined in Equation (1.4).
When using an array, we may have properties of the array which we might
want to preserve. Two such properties are monotonicity and completeness. An
array A is monotone increasing if
(∀x :: N;y :: N) • (0 ≤ x < y < |A|) ⇒ A[x] ≤ A[y]
We can define monotone decreasing similarly and we define strictly monotone
increasing (decreasing) by replacing ≤ (≥) in the consequent by < (>). Monotonic
arrays are sufficient for binary searches.
We say an array A is complete, if
(∀i :: N) • (0 ≤ i < |A|) ⇒ A[i] ≠ Null
We now state a result (without proof) about the splits asp and b(k).
Suppose for an array A that
and
A ❀ 〈P 0 ,P 1 〉 asp
A ❀ 〈Q 0 ,Q 1 〉 b(k)
Then (i) if A is monotone increasing (decreasing) then P 0 , P 1 , Q 0 , Q 1 are
all monotone increasing (decreasing) and (ii) if A is complete then so are P 0 ,
P 1 , Q 0 , Q 1 .
Note that the converse of (i) is not always true — consider
[3, 2, 5, 6] ❀ 〈[3, 5], [2, 6]〉 asp
and [3, 5, 2, 6] ❀
〈[3, 5], [2, 6]〉 b2
The arrays [3, 5] and [2, 6] are both monotone increasing but the arrays [3, 2, 5, 6]
and [3, 5, 2, 6] are not.
It is not always the case that splits preserve completeness and monotonicity.
For example, let us consider the following array-split. Let us suppose that we
have an array of length N and let sp = (ch, F) with
ch = (λi.i mod 2)
and F = {f 0 ,f 1 } where
f 0 = (λi.(N − i) div 2)
and f 1 = (λi.(i div 2) + 1)
With this split, taking N = 6:
[0, 1, 2, 3, 4, 5] ❀ 〈[Null, 4, 2, 0], [Null, 1, 3, 5]〉 sp
and so this split does not preserve completeness or monotonicity.