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Incoming links: Integer
The probability generating function of a nonnegative integer-valued random variable isusing also at . It is always defined for , and indeed the power series converges absolutely for all real or complex with , becauseIt may be defined beyond the unit disk where this power series converges absolutely; such a larger domain depends on the distribution. In particular and .
For parts (b)–(e), use array indices as tree handles. This also makes the shared storage in part (d) type-correct: both pending handles and output markers occupy integer cells. Zero denotes a leaf of a rooted tree; a positive handle denotes a fork whose children are in Initialize
nodes[t-1]. Only finite acyclic structures are admitted. An allocated fork's children are themselves valid handles.CONST MaxForks = 100000;
TYPE Tree = INTEGER;
Node = RECORD left, right: Tree END;
Arena = ARRAY MaxForks OF Node;
VAR nodes: Arena;
used: INTEGER;
PROCEDURE NewFork(left, right: Tree): Tree;
BEGIN
ASSERT(used < MaxForks);
nodes[used].left := left; nodes[used].right := right;
INC(used);
RETURN used
END NewFork;
PROCEDURE Preorder(t: Tree; VAR out: ARRAY OF INTEGER;
VAR count: INTEGER);
PROCEDURE Visit(p: Tree);
BEGIN
IF p = 0 THEN
out[count] := 0; INC(count)
ELSE
out[count] := 1; INC(count);
Visit(nodes[p-1].left); Visit(nodes[p-1].right)
END
END Visit;
BEGIN
count := 0; Visit(t)
END Preorder;used := 0 before constructing any trees; NewFork allocates within this arena. The fixed capacity is an implementation parameter and may be replaced by a dynamically allocated arena. The returned prefix out[0..count) encodes Boolean values by and . This encoding is part of the chosen representation, not a conversion of a pointer into a Boolean value.The recursion invariant is that
Visit(p) appends exactly the preorder traversal of the subtree represented by p. If the full binary tree has forks, it has leaves; henceEach occurrence is visited once, using linear time complexity and a call stack bounded by the height of a binary tree. The arena is the input representation, not an extra copy made by the traversal.Let be the final output length, the number of emitted markers, and the number of pending handles. Each pending subtree contributes at least one future marker, soStore output from the low end and the stack from the high end of the same array. A pop frees one cell before the next write. For a fork, the two pushed children each account for at least one of the remaining markers. Thus the two regions never overlap.The loop invariant includes
PROCEDURE PreorderShared(t: Tree; VAR out: ARRAY OF INTEGER;
VAR count: INTEGER);
VAR top, cap: INTEGER; p: Tree;
BEGIN
cap := LEN(out); count := 0;
top := cap-1; out[top] := t;
WHILE top < cap DO
p := out[top]; INC(top);
IF p = 0 THEN
out[count] := 0; INC(count)
ELSE
out[count] := 1; INC(count);
DEC(top); out[top] := nodes[p-1].right;
DEC(top); out[top] := nodes[p-1].left
END
END
END PreorderShared;count <= top, with pending handles in out[top..cap) and final markers in out[0..count). The preceding accounting proves every intermediate write also respects array bounds. This uses the integer handle representation chosen in part (b). Simply putting a pointer into a genuinely Boolean-typed cell would not be valid Oberon.An Oberon pointer refers to a record containing an integer label and two subtree references:Use
TYPE
Tree = POINTER TO Node;
Node = RECORD
left, right: Tree;
label: INTEGER
END;NIL for the empty binary tree, including either absent child. The representation invariant requires a finite, acyclic binary tree with disjoint left and right subtrees. In particular, shared nodes would represent a different structure and invalidate destructive traversal arguments.Use binary search on an interval with unequal endpoint values. The array need not be sorted:The loop invariant isIt holds initially by the given endpoint inequality, which also implies . When , . If , replacing maintains unequal endpoints. Otherwise , so replacing maintains them. This does not require monotonicity or even just two possible values.
l := 0; r := n;
WHILE r-l > 1 DO
m := l + (r-l) DIV 2;
IF b[l] # b[m] THEN r := m ELSE l := m END
END;
k := l;The positive integer width is a loop variant, reduced to at most its old half rounded up. On exit , givingThere are at most iterations. The additive constant handles .
Maintain an integer The precondition is the abstract capacity condition. Appending at the first unused array index establishes in constant running time.
len, initially zero, with len = length xs:PROCEDURE Add(x: INTEGER);
BEGIN
ASSERT(len < N);
u[len] := x;
len := len+1
END Add;If
get(k) first applies a comparison sort to all entries, the comparison sorting lower bound gives at least comparisons in the worst case. A merge sort or heapsort meets that order, after which indexing costs constant time:This is a logarithmic factor worse than linear. The qualification to comparison sorting is necessary: merely saying that keys are integers does not prohibit counting sort for a bounded small range or radix sort under suitable machine-key assumptions. The printed assertion of an unavoidable superlinear bound needs the comparison model.Represent a multiset by a pure multiplicity function. Every operation returns a new function describing updated counts:An empty instance is
class ContraMSet[-T](f: T => Int) {
def count(x: T): Int = f(x)
def insert(x: T): ContraMSet[T] =
new ContraMSet[T]((y: T) => f(y) + (if (y == x) 1 else 0))
def delete(x: T): ContraMSet[T] =
new ContraMSet[T]((y: T) => {
val old = f(y)
if (y == x && old > 0) old - 1 else old
})
def union[U <: T](that: ContraMSet[U]): ContraMSet[U] =
new ContraMSet[U]((y: U) => this.count(y) + that.count(y))
}new ContraMSet[T]((_: T) => 0). The constructor's precondition is that f is pure and gives a nonnegative integer count, with finite support of a multiset if only finite multisets are intended. Nonnegativity and the support condition are preserved by each operation. Integer overflow must be excluded for this fixed-width count interface.A contravariant type parameter makes
ContraMSet[T] a consumer of values of type T. Thus a counter able to answer queries for every animal can also answer queries restricted to cats. It does not expose or enumerate stored T values. The return type ContraMSet[T] is legal: its outer covariant result position and the class's contravariant argument position together place negatively.For
union, the argument itself is in a negative position, while its ContraMSet argument would reverse that sign. Using ContraMSet[T] directly as the parameter type would therefore put positively and fail the variance-position rule. Introduce : both operands can count every U, and their sum is a ContraMSet[U]. This is a restriction to a common query domain; operations do not promise to preserve counts outside the narrowed domain.[ref]Scala 2.11 basic declarations and definitions specification. Type Parameters, Variance Annotations; bounds and sign changes
Pointwise evaluation givesNo old function is modified, so updates preserve immutable object. Constructing each wrapper takes constant work, but a later
count may traverse a chain of previously created programming closures; its running time need not be constant in the history length. This functional representation trades enumeration and direct storage of occurrences for composable counting behavior.We prove the contrapositive. If , Euclidean division gives or for an integer . Squaring gives respectively or . Both have remainder upon division by , so . Taking the contrapositive yields .
For a proof by contradiction, suppose for positive integers in lowest terms, so . Squaring gives , and part (a) implies . Write ; then , hence , and part (a) again implies . This contradicts the coprimality of . Therefore is an irrational number.
For each fixed positive integer , part (i) gives . This meets big O notation with constant and threshold . True: for fixed . The constant may depend on ; no claim uniform in a varying is required.
Use the symmetric tree fold equations for the other algebraic data type:The most general type signatures, using the same schematic variable names independently for each function, areFor
foldZag (fZag,eZag) (fZig,eZig) Nal = eZag
foldZag (fZag,eZag) (fZig,eZig) (Cans x z) =
fZag x (foldZig (fZig,eZig) (fZag,eZag) z)foldZig :: (a -> s -> r, r) -> (b -> r -> s, s)
-> Zig a b -> r
foldZag :: (a -> s -> r, r) -> (b -> r -> s, s)
-> Zag a b -> rfoldZig, the first empty value has type r, so the first combining function must return r. Its label argument has type a. The recursively folded tail has some independent result type s, giving a -> s -> r. The second combining function consumes a b label and an r result and returns s, giving b -> r -> s. The second empty value has type s. Swapping the two handler pairs and the label types makes the recursive call well-typed. The identical argument applies to foldZag.The two result types need not coincide. For example, one handler can return an integer and the other a list; forcing both results to have one type would lose parametric polymorphism allowed by the definitions.
Evaluate the supplied function once at each positive integer, retaining its largest value so far. In particular do not initialize the maximum to zero: might be negative everywhere.At the loop head the loop invariant is ,
PROCEDURE record(f, emit);
VAR n, best, y: INTEGER;
BEGIN
best := f(1);
emit(1);
n := 2;
WHILE TRUE DO
y := f(n);
IF y > best THEN
emit(n);
best := y
END;
n := n + 1
END
END record;best , and precisely the record values with argument less than have been emitted, in increasing argument order. The first argument is a record because its comparison set is empty. The strict test correctly excludes ties. Updating best only on a strict increase preserves the loop invariant.Exactly one call of
f is made per processed argument. For an arbitrary supplied function, an unexamined argument can have a value larger than all previous values, so skipping its evaluation cannot be correct in general. This assumes is a total, deterministic function; the requested infinite computation ignores integer overflow.Use the Euclidean algorithm with floor quotient and nonnegative remainder, including for a negative numerator:For , Euclidean division gives with . If , the singleton
PROCEDURE rat2cf(n, d: INTEGER): fraction;
VAR q, r: INTEGER; p: fraction;
BEGIN
q := n DIV d; r := n MOD d;
NEW(p); p.head := q; p.tail := NIL;
IF r # 0 THEN p.tail := rat2cf(d, r) END;
RETURN p
END rat2cf;[q] is exact. Otherwiseso the recursive tail represents . Its numerator is positive and greater than its denominator, so every tail head is a positive integer. The new denominator is strictly smaller than , proving algorithm termination by descent through the positive integers.The returned list represents exactly . For example gives
[-3,1,2]; truncating negative division toward zero would violate the required positivity of the tail. There are Euclidean algorithm steps, with output allocation and recursive stack depth linear in the number of generated nodes.Append each quotient directly to the output linked list, retaining a pointer to its last node. The nodes are required output; auxiliary variables remain simple integers and pointers.At the start of each iteration,
PROCEDURE rat2cf(n, d: INTEGER): fraction;
VAR first, last, p: fraction; q, r: INTEGER;
BEGIN
first := NIL; last := NIL;
REPEAT
q := n DIV d; r := n MOD d;
NEW(p); p.head := q; p.tail := NIL;
IF first = NIL THEN first := p
ELSE last.tail := p
END;
last := p;
n := d; d := r
UNTIL d = 0;
RETURN first
END rat2cf;d is positive, and the output prefix consists exactly of the quotients already obtained by the Euclidean algorithm. last points to its final node, or both first and last are null when the prefix is empty. The current ratio n/d is the unexpanded tail of the original continued fraction. The quotient/remainder step extends that prefix by the same value as the recursive version, then replaces the unexpanded ratio by its reciprocal residual tail. If the remainder is zero, no tail remains and the loop stops before another division.The loop variant is the positive current denominator, which strictly decreases after each nonfinal iteration. The input denominator is positive, so at least one node is allocated, including for zero or an integral ratio. This is the same exact conversion with auxiliary space, excluding the returned list, and no input mutation.
Use a Scala For example,
object for the singleton pattern, and replace Performance with a version that reports to it. This code plugs into the source's Equipment, GymEquipment, and Functionality definitions; Prediction can remain an independent observer once its displayed string line break is made executable.object GymStandings {
private var scores = Map.empty[GymEquipment, Int]
def record(e: GymEquipment, calories: Int): Unit = {
scores = scores.updated(e, calories)
}
def remove(e: GymEquipment): Unit = {
scores = scores - e
}
def leaders: List[(GymEquipment, Int)] = {
if (scores.isEmpty) Nil
else {
val best = scores.values.max
scores.toList.filter { case (_, score) => score == best }
.sortBy { case (equipment, _) => equipment.name }
}
}
}
class Performance(e: GymEquipment) extends Functionality {
private def calories: Int =
((e.age - e.weight + e.heartRate) * e.time) / 10
e.addFunctionality(this)
GymStandings.record(e, calories)
def update(): Unit = {
val current = calories
GymStandings.record(e, current)
println("Your current performance is: " + current + " calories burned.")
}
}new Performance(treadmill) and new Performance(bike) attach two observers; later GymStandings.leaders returns the leading equipment/score pairs. Initial registration records the equipment's present score, even before its first subsequent update. Recording replaces the same equipment's previous score, so repeated notifications do not multiply its contribution. The explicit remove supports taking an equipment object out of the comparison; detaching its observer when its session ends also prevents later re-registration through another notification. In the original single-threaded event model no extra synchronization is needed.All equipment observers report to the same standings software object, and
leaders returns every maximum or Nil if empty. The query takes to find/filter scores for active equipment objects, plus sorting the tied names for deterministic presentation by name; equal names may remain in either order and still denote distinct equipment. This uses the exact supplied integer score formula and its integer division; it does not substitute a different performance metric. A functioning unsubscribe operation, if required, must explicitly correct the source to _functionalities = _functionalities diff List(f).Use repeated squaring, including index zero:The loop invariant is for , with for every stored entry except possibly the initial entry when . Squaring gives , proving preservation. The first failed guard identifies the first larger power outside the range. If , no power with needs storing.
b2[0] := b; top := 0;
WHILE b2[top]*b2[top] <= n DO
b2[top+1] := b2[top]*b2[top];
top := top+1
END;Index zero is deliberately included. The original paper asks for indices but then claims that their products represent every allowable power. Taken without , that claim fails for odd exponents, since every available exponent would be even. The extra initialized entry makes the intended binary expansion argument valid without changing the requested entries.
Let be the largest representable positive integer. The worst case is , requiring indices through . Thus
enough = 5 for signed 32-bit integers, or 6 for signed 64-bit integers, suffices for stored powers that fit; a larger convenient constant is harmless. This counts slots starting at zero and does not reserve a slot for an out-of-range sentinel. The multiplication in the guard can still suffer integer overflow, independently of this array capacity bound.Store a second array For the second loop, write . Immediately before processing , its loop invariant isInitially the failed squaring guard gives ; this also holds when , since then . The test succeeds exactly when . If successful, adding selects that binary digit; otherwise the upper bound tightens to . Either way the invariant holds for the next smaller . At , , so
weight with weight[k] = 2^k. Descending through the binary digits of the answer gives this greedy algorithm:PROCEDURE FastLog(b, n: INTEGER): INTEGER;
VAR b2, weight: ARRAY enough OF INTEGER;
top, k, i, p: INTEGER;
BEGIN
b2[0] := b; weight[0] := 1; top := 0;
WHILE b2[top]*b2[top] <= n DO
b2[top+1] := b2[top]*b2[top];
weight[top+1] := weight[top]+weight[top];
top := top+1
END;
i := 0; p := 1; k := top;
WHILE k >= 0 DO
IF p*b2[k] <= n THEN
p := p*b2[k]; i := i+weight[k]
END;
k := k-1
END;
RETURN i
END FastLog;i = L.There are squarings and descending tests. For fixed , this is arithmetic operations, hence logarithmic in the bit length of , and also within the weaker bound. This counts arithmetic operations on integers as unit cost; bit complexity would also charge for the growing operands. No division is used.
Let be the maximum positive machine integer. A positive product suffers integer overflow when . The vulnerable expressions are
p*b in part (b), b2[top]*b2[top] in the table guard, and p*b2[k] in the descending search. Even a rejected candidate must be evaluated before its comparison with .The table guard becomes dangerous as soon as its current entry exceeds : approximately for signed 32-bit integers, or for signed 64-bit integers. These are thresholds for the intermediate power, not for alone. For example, with 32-bit arithmetic, and reach the stored power and then try its overflowing square . If , even the initial square may overflow for a tiny . In part (b), the final test can overflow when the next power above exceeds ; at , this already happens for .
For positive operands, integer division gives the exact equivalence . Therefore replace the three guards respectively byand perform each multiplication only after its guard succeeds. Every accepted product is then at most . With these guarded products and sufficient table capacity, the algorithms work throughout , . The stored exponents and weights also fit in those machine ranges. Checking for overflow after multiplying would be too late.
p <= n DIV b
b2[top] <= n DIV b2[top]
p <= n DIV b2[k]Separate integer partitions according to whether they contain a part equal to . Removing one such part is a bijection to partitions of whose parts are at most . The alternatives are disjoint, givingUse , because the empty integer partition is counted once, and for , with for . This Oberon implementation avoids negative recursive arguments:The precondition is . Each recursive call reduces , proving algorithm termination. Every positive part of a partition of is at most , so the required one-argument count is . Counts require a sufficiently wide integer representation.
PROCEDURE P(i, n: INTEGER): INTEGER;
VAR answer: INTEGER;
BEGIN
IF n = 0 THEN answer := 1
ELSIF i = 0 THEN answer := 0
ELSIF i > n THEN answer := P(i-1, n)
ELSE answer := P(i-1, n) + P(i, n-i)
END;
RETURN answer
END P;
PROCEDURE Partitions(n: INTEGER): INTEGER;
BEGIN
RETURN P(n, n)
END Partitions;