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Incoming links: Algorithm termination
Algorithm correctness Created 2026-10-13 Updated 2026-10-18
Partial correctness means that, if the algorithm terminates on a permitted input, its output meets the formal specification. Total correctness additionally requires algorithm termination on every permitted input.
A precondition describes permitted starting circumstances; a postcondition describes the required result. Partial correctness says that starting with and finishing implies . Total correctness also guarantees finishing. A loop invariant supplies the connection between these conditions during repeated steps.[ref]
Loop invariants. CS 2112 Fall 2019; Loop invariants, Using loop invariants to show code is correct: Establishment, Preservation, Postcondition, and Termination
Correctness is relative to a formal specification. Returning the first entry of an array correctly implements "return the first entry", but fails "return the maximum" on . A fast wrong answer does not acquire algorithm correctness from its time complexity. Numerical closeness matters only when the formal specification allows an approximation. Software testing and test coverage provide evidence about selected executions; they do not by themselves establish algorithm correctness for every permitted input.
Algorithm termination 2026-10-18
Algorithm termination means that execution finishes after finitely many steps on the inputs under consideration. Specify that domain: the stalled loop in partial correctness terminates for but runs forever for . A successful run on one input does not establish termination on another.
A useful method is a loop variant that is a nonnegative integer at every condition check and strictly decreases whenever the condition is true and the body completes. Also prove that condition evaluation and each body execution terminate. An infinite run would then give an infinite strictly decreasing sequence of nonnegative integers, contradicting their well-founded relation. This establishes algorithm termination without enumerating all executions.
For the summation loop verification, begins at and decreases by exactly one per iteration. The body executes exactly times, including zero times for . Merely bounding a decreasing measure below is insufficient for real numbers: decreases forever while staying positive. The choice of a well-founded relation supplies the essential progress argument; well-founded progress in recursion applies the same idea to recursive calls.
Base case 2026-10-18
A case answered without further recursive calls. Complete base cases are one part of an algorithm termination argument; the other cases must progress in a well-founded relation.
Binary array sum by recursion 2026-10-18
Keep an array unchanged, and sum entries in the half-open integer interval of array indices . The precondition requires integers and exact, terminating integer addition. An empty range returns the empty sum zero; a one-entry range returns that entry. Every longer range splits at its middle array index:For the original array , call . The left half returns , the right half returns , and the original invocation returns . The right half's left endpoint increases from zero to two: what decreases is range length , not every argument. Negative entries affect the answer but do not affect algorithm termination.
S(A, l, r):
require integer indices with 0 <= l <= r <= length(A)
size = r - l
if size == 0:
return 0
if size == 1:
return A[l]
middle = l + floor(size / 2)
left_sum = S(A, l, middle)
right_sum = S(A, middle, r)
return left_sum + right_sum
Seven calls compute an array sum with only three frames active at once
. Original binary tree of invocations for , with integer intervals labeling ranges and upward arrows labeling returned sums. Small numbers give the order in which invocations begin when the left child runs first. The highlighted graph path contains the three active stack frames while the entry is read; the result from the completed left sibling is retained by its parent. Details.
For range length , the children have lengths and , both positive and strictly less than . The floor function chooses the split; the ceiling function gives the other length. Algorithm termination follows by this natural number measure. For algorithm correctness, apply mathematical induction on : the two base cases return the specified sum; for a larger range, the two disjoint child ranges cover every original array index exactly once, so addition of their correct sums gives the required sum.
For , the invocation tree is a full binary tree with leaves, one per array entry. If is its number of nonleaf vertices, counting edges gives , so . Thus the total invocation count is , and there are combining additions. Here gives seven calls and three combining additions. For , there is one call and no combining addition; for , there is also one call and no combining addition, but no entry leaf, so the formula does not apply.
The call stack holds the active branch, rather than the entire invocation tree. Execute the left child completely before the right child, retaining the left result in the parent stack frame. The maximum number of active stack frames is for , and one for . Balanced splitting therefore needs frames for , compared with the frames of the factorial recursive process. Assuming constant-cost array access, index operations and addition, its time complexity is and its auxiliary space complexity is machine words for , excluding the stored input. Pass the same array with different endpoints, rather than copying slices. These bounds require all stored indices and partial sums to fit in a constant number of machine words; exact unbounded integer addition and storage require a separate bit cost analysis.
Factorial recursion and returning frames 2026-10-18
For a natural number , the factorial satisfies and for . The following description includes the zero base case explicitly; a negative or nonintegral input is outside its precondition:For , the arguments on descent are . Each positive-argument invocation postpones one multiplication until its recursive call returns. At the deepest point, the call stack has five stack frames: four suspended callers and the active zero base case. The invocation at zero returns to the invocation at one, rather than finishing the entire computation. Returns then giveA stack frame retains the caller's argument and where to resume. When the invocation at three receives , it uses its own , obtains , and returns that value to the invocation at four. The recursive calls share a definition, but their local variables belong to separate invocations.
F(n):
require n is an integer and n >= 0
if n == 0:
return 1
child_result = F(n - 1)
return n * child_result
The natural number measure decreases by one on every recursive call, so algorithm termination follows. For algorithm correctness, use mathematical induction: the zero base case returns ; assuming the child's answer is , the caller returns . This also explains why keeping the caller's matters.
There are exactly invocations and multiplications, including one invocation and no multiplications when . Ordinary execution of this recursive process uses at most active stack frames, hence frames for in Big O notation. If each test, call, subtraction and multiplication is counted as constant cost, its time complexity is . That is an arithmetic-operation model: exact arbitrary-length integer multiplication has growing cost, and neither the factorial output nor every stack frame's numeric contents occupies a universally fixed number of bits.
Finite termination of K-means clustering 2026-10-25
For a finite dataset, alternate nearest-centre assignment and replacement of each centre by the centroid of its assigned observations. Keep the current assignment whenever an observation is tied between nearest centres. A changed assignment then strictly decreases the within-cluster sum of squared Euclidean distances; updating centres cannot increase it by the centroid sum-of-squares identity.
After each centre update, the centres are determined by the current partition of a set of observation indices. There are only finitely many such partitions, so a strictly improving sequence cannot revisit one and must stop. A convenient exact stopping test is to check after each update whether every observation is still assigned to a nearest centre. This proof is for algorithm termination; it does not guarantee a global minimum.
Empty clusters need an explicit convention. To retain exactly nonempty clusters, transfer an observation from a cluster with at least two members to each empty cluster, making it a singleton centred at that observation. This cannot increase the objective: the transferred observation now has squared distance zero, and no other contribution changes before centre updating. A deterministic choice is an eligible observation farthest from its currently assigned centre, with ties resolved by index. After the first round, an empty cluster can arise only during a strictly improving reassignment, so these repairs preserve the finite-termination argument.
Greedy Egyptian fraction expansion 2026-10-25
For a positive proper residual fraction , with , choose , append the unit fraction , and replace the residual by in lowest terms. This is a greedy algorithm: the chosen unit fraction is the largest one not exceeding the residual.
The inequalities imply . Thus the numerator of the reduced residual strictly decreases while positive, proving algorithm termination by a nonnegative-integer loop variant. The loop invariant is that the accumulated sum plus the residual is the original rational number. Since and , the new residual is less than . If it remains positive, its next chosen denominator is therefore strictly greater than ; this expansion also satisfies the distinct-denominator convention. The proof does not bound the denominators by the input's fixed-width range. Arbitrary-precision arithmetic, checked conversion, or a different representable expansion can therefore be necessary in code.
Linear probing 2026-10-18
An open addressing probe rule that visits modulo , using modular arithmetic, until a matching record or an appropriate stopping slot is found.[ref]
Lecture 10: Hashing III: Open Addressing. 6.006 Introduction to Algorithms, Fall 2011 by Massachusetts Institute of Technology, 6.006 course staff; Lecture 10: Hashing III: Open Addressing, p. 3, Linear Probing formula
For capacity , the hash table probe numbered , starting at zero, inspects the array indexThe hash function gives the first array index. Each subsequent hash table probe moves one slot forward, wrapping from to by modular arithmetic. Keep the hash function and key equality fixed while hash table records are stored.
Use three distinguishable slot states: occupied by a live hash table record; never used since initialization or rebuilding, written
E; or deleted, written D, a tombstone in open addressing. Lookup compares occupied hash table keys using key equality, stops at E, and continues past D.[ref]5.2 LinearHashTable: Linear Probing. Open Data Structures: An Introduction by Pat Morin; Section 5.2, three entry types and find: nil stops lookup, del does not
For an original example, take an eight-slot array, integer hash table keys, and the hash function . Start with all slots
E and insert the hash table records in that order. The insertion hash table probe traces, expressed as array indices, areThe resulting array holds hash table keys at array indices , with E in the other four slots. The first three hash table keys have the same initial array index, a hash collision; the fourth begins at but joins the same linear probing cluster. Lookup of examines array indices , rejecting before returning . Lookup of absent examines , including the stopping E: five hash table probes for a hash table load factor .
Bound lookup to at most hash table probes, even if no
E remains. For algorithm termination, the first array indices are distinct: equality of and implies that divides by modular arithmetic, and forces . They therefore cover the entire finite set of slots exactly once. If no occupied hash table key matches and no E is encountered, exhausting these hash table probes establishes absence; wrapping around again supplies no new information.[ref]Lecture 10: Hashing III: Open Addressing. 6.006 Introduction to Algorithms, Fall 2011 by Massachusetts Institute of Technology, 6.006 course staff; Lecture 10: Hashing III: Open Addressing, pp. 1–2, probe permutation requirement and Search loop with m trials
The attraction is a compact array and a simple hash table probe rule. Consecutive slots can benefit from spatial locality, although actual memory cache behavior depends on representation and hardware. Correctness rests on preserving the search route; speed depends on how long that route becomes. The linear probing cluster below explains why even a moderate hash table load factor gives no universal constant lookup bound.
Loop variant 2026-10-18
A decreasing measure used to prove algorithm termination. A nonnegative integer loop variant must strictly decrease after every completed body whose condition was true; the body itself must terminate.[ref]
Loop invariants. CS 2112 Fall 2019; Loop invariants, Using loop invariants to show code is correct: Termination and the decrementing function
The prefix maximum correctness example uses . It counts unread entries and is independent of whether the new entry changes the maximum. Ties therefore cannot stall progress: increases even when stays the same. An integer bounded below cannot strictly decrease indefinitely, which supplies the missing algorithm termination argument.
Control flow and four proof obligations for the prefix maximum correctness example
. Initialization, preservation, and exit establish partial correctness; the loop variant establishes algorithm termination and hence total correctness. The loop invariant is restored after both variable assignments. The input array stays unchanged throughout. Details.
Matrix-chain split reconstruction 2026-10-18
Extend the original example by a matrix with matrix dimensions , so . The Catalan number counts all possible groupings. Their scalar multiplication costs areThe optimum is , with cost . Enumeration makes a useful check here, but dynamic programming reaches the same answer by solving ten integer intervals, including the four single-matrix base cases:For example, . The three final-split candidates for the full matrix product supply a compact certificate:The first candidate already uses the better grouping of ; the worse need not be reconsidered at the top level. This is how optimal substructure lets the table represent all five possibilities with only three final candidates.
Read , which separates from . Then separates from , and separates from . This recursion stops at every , returning the corresponding matrix name. Combining the returned expressions recovers , rather than only its cost. Every recursive call uses a shorter integer interval, establishing algorithm termination; a tied minimum may choose any attaining split. The recovered binary tree has leaves and internal nodes, so reconstruction takes steps. Its attraction is the separation between an enormous family of groupings and a small set of reusable dynamic programming states.
Partial correctness Created 2026-10-18 Updated 2026-10-24
Partial correctness means that every terminating execution begun in a program state satisfying the precondition finishes in a program state satisfying the postcondition . For a sequential program , the usual Hoare triple notation isHere the notation expresses partial correctness; it makes no promise that finishes. The precondition and postcondition describe permitted starting circumstances and required final circumstances, rather than tests performed automatically by the program.[ref]
An Axiomatic Basis for Computer Programming by Tony Hoare. Hoare, 1969, §3 Program Execution, p. 577, and §4 General Reservations, pp. 578–579; the paper instead labels the precondition P, program Q and result R
For a sharp example, take the precondition and the following sequential program over mathematical integers:The condition remains true, so execution never reaches the instruction after the loop. Consequently its partial correctness claim holds for the postcondition , and even for the impossible postcondition : there is no terminating execution with a wrong final result. This is vacuous truth, not evidence that the loop computes a sum. The same program fails total correctness for this permitted input.
i = 0
s = 0
while i < n:
i = iThe distinction is useful in a proof: first show that an answer, whenever produced, is right; then establish algorithm termination. The summation loop verification supplies both arguments for a loop that actually progresses. Its finite trace illustrates one execution, while its loop invariant covers every permitted input.
Sort the distinct observations as . For ordered centres , a point is closer to than to precisely when it lies to the left of their midpoint. Thus each Voronoi cell is an interval, with either choice allowed for an endpoint tie. Consequently, after assignment, each nonempty cluster is a contiguous block of the sorted observations.
The arithmetic means of disjoint ordered blocks are strictly ordered, so updating centres preserves this order. The empty-cluster repair in part (b) preserves contiguous blocks too: an observation farthest from its assigned centre lies at an endpoint of its block, and removing that endpoint makes a new singleton block. In one dimension with distinct observations, choosing distinct observations as initial centres makes the first assignment nonempty.
The one-dimensional K-means iteration bound now follows by counting block partitions. Choosing the gaps between consecutive nonempty blocks givespossibilities. By the strict-decrease argument in part (b), no such partition can recur before algorithm termination. Counting a step as an assignment, any empty-cluster repairs, and a centre update, with the stability check performed after that update, givesFor , one update to the overall arithmetic mean suffices. Arbitrary changes between tied centres would invalidate the strict-decrease argument, which is why the stable tie convention in part (b) matters.
A divide-and-conquer algorithm splits an instance into smaller instances of the same kind, solves them recursively, and combines their answers. The smaller instances should be sufficiently smaller to ensure algorithm termination; the combination cost determines whether splitting saves work. The two halves of merge sort illustrate this separation.
Dynamic programming identifies a collection of subproblems and dependencies among them, solves each distinct subproblem once, and stores its answer. A table in dependency order or memoization prevents repeated evaluation of overlapping subproblems. For optimization, an optimal substructure argument justifies constructing an optimal answer from optimal smaller answers.
A greedy algorithm repeatedly makes a locally best feasible choice and commits to it. It avoids exploring alternative continuations. Algorithm correctness therefore needs a greedy-choice property, often proved by replacing the first choice of an optimal solution with the greedy choice without worsening it.
An Oberon recursive procedure can implement the defining recurrence relation directly. Its precondition is ; arithmetic is assumed to remain in the representable integer range.
PROCEDURE Fib(n: INTEGER): INTEGER;
VAR result: INTEGER;
BEGIN
IF n <= 2 THEN
result := 1
ELSE
result := Fib(n-1) + Fib(n-2)
END;
RETURN result
END Fib;Correctness follows by strong induction: the two base inputs give , and for the returned sum is . On each recursive branch the positive integer argument strictly decreases, while recursion stops at or . The well-ordering principle therefore proves algorithm termination.The method repeats many subproblems and has exponential time complexity; termination alone does not make it efficient.
In tail recursion, each recursive call is a tail call: its result is returned directly, with no suspended addition or other work. A compiler can then reuse an activation record, giving constant stack use. This is a possible optimization, not a guarantee made by the source syntax. The direct Fibonacci number procedure in part (a) must retain information while waiting to add two recursive results.
Extend the Fibonacci numbers with and use the suggested quantity . For ,while . This gives the Oberon procedures
PROCEDURE G(n, a, b: INTEGER): INTEGER;
VAR result: INTEGER;
BEGIN
IF n = 1 THEN
result := a
ELSE
result := G(n-1, a+b, a)
END;
RETURN result
END G;
PROCEDURE FibTail(n: INTEGER): INTEGER;
BEGIN
RETURN G(n, 1, 0)
END FibTail;The assignment of the recursive result does not perform arithmetic after the call; it merely passes that result to
RETURN, so the call is in tail recursion. A dialect permitting an immediate RETURN G(...) in the branch expresses the same control flow. The decreasing argument proves algorithm termination, and the displayed identity proves correctness by mathematical induction. ThusThere are recursive steps, giving arithmetic operations. Stack use is with tail call elimination and without it. With unbounded integers, the growing Fibonacci numbers also make arithmetic costs grow; the linear bound counts arithmetic operations.Represent the two recursively split arrays by adjacent half-open ranges of the original array. This avoids allocating a fresh pair at every call. The caller supplies a distinct work array with at least as many elements as
a; only that work array is used for merging. The Oberon code isPROCEDURE SortRange(VAR a, work: ARRAY OF INTEGER;
l, r: INTEGER);
VAR m, i, j, k: INTEGER;
BEGIN
IF r-l > 1 THEN
m := l + (r-l) DIV 2;
SortRange(a, work, l, m);
SortRange(a, work, m, r);
i := l; j := m; k := l;
WHILE (i < m) & (j < r) DO
IF a[i] <= a[j] THEN
work[k] := a[i]; i := i+1
ELSE
work[k] := a[j]; j := j+1
END;
k := k+1
END;
WHILE i < m DO
work[k] := a[i]; i := i+1; k := k+1
END;
WHILE j < r DO
work[k] := a[j]; j := j+1; k := k+1
END;
k := l;
WHILE k < r DO
a[k] := work[k]; k := k+1
END
END
END SortRange;
PROCEDURE MergeSort(VAR a, work: ARRAY OF INTEGER);
BEGIN
SortRange(a, work, 0, LEN(a))
END MergeSort;For a range of length , the left length is and the right length is , so as required. Both are positive and smaller than , giving the recursive loop variant analogue for algorithm termination. Empty ranges are also harmless, although the question supplies a nonempty array.
For the first merge loop, regard the two sorted ranges as fixed input sequences and . They are not overwritten during the merge. A loop invariant is that , , , and
work[l:k) is the stable sorted merge of the consumed prefixes a[l:i) and a[m:j). Every emitted element is at most every remaining element. Taking the smaller head preserves this property, and choosing the left head on equality preserves stable sorting. The loop variant isIt is nonnegative and decreases by one per iteration. On exit at least one range is exhausted.For the second loop, the remaining left range is already sorted and any right remainder is absent whenever that loop runs. Appending the next left element preserves the same merge loop invariant; its loop variant is . For the third loop, the left range is exhausted and the right remainder is appended in order; its loop variant is . On completing both loops,
work[l:r) is the complete sorted merge and .In the final copying loop, the loop invariant is that
a[l:k) equals work[l:k), work[l:r) remains the correct sorted answer, and positions outside a[l:r) are unchanged. Its loop variant is . At exit the whole answer has been copied back. By strong induction on range length, the recursive calls sort their halves, and the merge preserves the original multiset. ThusThe work array must not alias a: overwriting the source while merging would invalidate the merge loop invariant.Maintain all pairs discovered so far. Extend every discovered path by one original binary relation edge, keeping the old pairs first:
transclose :: Eq a => [(a, a)] -> [(a, a)]
transclose r = grow (removedup r)
where
grow s =
let t = removedup (s ++ relcomp r s)
in if length t == length s then s else grow tAfter extensions, the represented binary relation is . This loop invariant follows by distributivity of composition of relations over a union. The old list
s is a duplicate-free prefix of t; therefore equality of lengths is equivalent to equality of the represented sets and signals a fixed point.Let be the finite set of endpoints appearing in the input. No extension introduces a new endpoint, so there are at most distinct pairs. Every nonfinal iteration adds at least one pair, proving algorithm termination. At the fixed point , we have and . Mathematical induction gives for every positive , while the loop invariant says that each discovered pair belongs to some . HenceThis is the transitive closure of a relation, rather than a reflexive closure: diagonal pairs appear only when supported by nonempty cycles. The empty input is handled immediately.
Set . At each iteration with , compute the remainder , then setStop at the first index with and return . The positive second coordinates strictly decrease, so algorithm termination follows from the well-ordering principle. The equality of greatest common divisors established in (ii) is a loop invariant, and at termination it gives
Maintain one array index per input and a candidate value. Cycle through the arrays, counting consecutive agreements with that candidate; when a larger head is encountered, start counting agreements again. The loop invariant has three parts. All indices satisfy ; no discarded entry is common to every input; and every original common value is at least
p is an auxiliary array of indices, not a copy of any input.PROCEDURE commonMany(as, n, m);
VAR p: ARRAY m OF INTEGER;
j, candidate, agree, x: INTEGER;
BEGIN
FOR j := 0 TO m-1 DO
p[j] := 0;
IF n[j] = 0 THEN RETURN None END
END;
candidate := as[0][0];
j := 0; agree := 0;
WHILE agree < m DO
IF p[j] = n[j] THEN RETURN None END;
x := as[j][p[j]];
IF x < candidate THEN
p[j] := p[j] + 1
ELSIF x = candidate THEN
agree := agree + 1;
j := (j + 1) MOD m
ELSE
candidate := x;
agree := 1;
j := (j + 1) MOD m
END
END;
RETURN Some(candidate)
END commonMany;candidate. Additionally, the agree cyclically consecutive arrays immediately preceding index j have heads equal to candidate. These facts are true initially, with an empty agreement run.A head below
candidate is not globally common, by the lower-bound clause, so advancing its pointer is safe. A head above candidate gives a new lower bound: every globally common value lies in this array's unprocessed suffix and so is at least its head. Only this array is yet certified to agree with the new candidate. An equal head extends the run by one. An exhausted suffix proves absence; a run of agreements covers every array exactly once and proves that candidate is common, and the lower-bound clause proves minimality. The exhaustion test precedes each dereference, including after the last entry of a suffix is discarded.For algorithm termination and time complexity, charge comparisons to the pair
(j,p[j]). A fixed entry is discarded at most once, can raise the candidate at most once, and can be counted as an agreement at most once: before the same array is revisited, either the algorithm has finished or the candidate has strictly increased. Thus there are at most three comparisons per input entry, besides the initialization. The result is the smallest common value or None in time and auxiliary space. For , the first head immediately gives the answer.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.