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Joined 2026-09-21 Message
Incoming links: Sortedness
Insertion sort 2026-10-13
Insertion sort grows a sorted prefix of an array one entry at a time. In the adjacent-swap version, start with position , compare its current sort key with the preceding sort key, and swap the complete sorting records only when the right sort key is strictly smaller. Continue leftward until no swap is needed or the entry reaches position zero; then process original position . A tie stops the pass. This is the strict comparison and exchange rule in the authored implementation.[ref]
Take , where the subscript labels a complete sorting record and comparisons inspect only its sort key. Starting from index zero, the complete key-comparison trace is:
The loop invariant is that, before pass , positions satisfy sortedness and contain precisely the entries originally in that prefix. Moving the next entry left through strictly larger sort keys extends that prefix, and every swap retains sorting permutation preservation. Equal-key entries never cross, giving stable sorting. Each leftward step decreases the position, and each completed pass increases the prefix length, so the algorithm terminates.
With constant-cost key comparisons and swaps, reverse-ordered distinct sort keys cause comparisons and swaps. Thus worst-case time complexity is in Big O notation. An already sorted array needs only key comparisons for and no swaps. The adjacent-swap version uses constant auxiliary space complexity. Small or nearly sorted arrays can therefore make insertion sort useful despite its quadratic worst case.
Least-significant-digit radix sort 2026-10-18
A least-significant-digit radix sort processes numerical digits from smallest to largest place value. Fix integers and , and take complete sorting records whose sort keys satisfy . Use exactly positions in positional notation with radix , including leading zeros. At pass , apply a stable digit sort to the current array usingThe floor function selects the integer quotient, and the remainder selects the current numerical digit. Move the complete sorting record, including its associated data. There are exactly passes, so this procedure terminates even for an empty array. The radix sort suffix invariant supplies its algorithm correctness argument.
For an original input, let the letters identify the attached data:Here , , and leading zeros only align place values; represents the integer five. The exact pass ledger is:After the units pass, the numerical digits are . The ties retain before and before before . This array is not yet ordered by its whole integer sort keys: still precedes . After the tens pass, the two-position suffixes are . In particular, now precedes : their tens numerical digits differ, so stable sorting does not require retaining their previous order. The hundreds pass produces the final sorted array, with all attached letters preserved.
Three stable digit passes order successively longer suffixes
. Original decimal radix sort ledger. Each card carries an unchanged record label. Orange marks the numerical digit read in the current pass; green marks lower positions already processed. The suffix row beneath each pass shows precisely what is ordered, rather than claiming whole-key sortedness early. Details.
Use binary search. Maintain an inclusive candidate interval
[lo,hi], initially [0,n-1]. Compare the target v once with the middle entry; a ternary comparison returns less than, equal to, or greater than. Equality ends successfully. Otherwise sortedness permits discarding the middle entry and the entire half on the wrong side:lo := 0; hi := n-1;
WHILE lo <= hi DO
mid := lo + (hi-lo) DIV 2;
IF v = S[mid] THEN RETURN TRUE
ELSIF v < S[mid] THEN hi := mid-1
ELSE lo := mid+1
END
END;
RETURN FALSEThe two written tests represent one ternary comparison whose result is then inspected. The loop invariant says that every occurrence of
v, if any, lies in the candidate interval. Initially it is true. The order comparison and sortedness preserve it after either discard. If the interval becomes empty, the loop invariant proves absence.A failed comparison reduces a candidate interval of size to at most . For , at most comparisons are needed; an empty input needs none. HenceThe claim counts comparisons. Indexed access to an array also makes elapsed running time logarithmic in the usual model; a linked list representation would not give the same access cost.
Split the half-open array segment at its midpoint, using one entry for the root:The precondition is . Set . For the child sizes are and , differing by at most one. The same construction in both subtrees proves subtree-size balance at every node. Structural induction also gives the inorder traversal , followed by , followed by , hence precisely ; no sortedness assumption is needed.
PROCEDURE MakeTree(VAR u: ARRAY OF INTEGER;
a, b: INTEGER): Tree;
VAR m: INTEGER; t: Tree;
BEGIN
IF a = b THEN RETURN NIL END;
m := a + (b-a) DIV 2;
NEW(t);
t.label := u[m];
t.left := MakeTree(u, a, m);
t.right := MakeTree(u, m+1, b);
RETURN t
END MakeTree;Exactly records are allocated. Each nonempty recursive call makes two child calls, so there are calls including empty segments. Under constant-cost allocation and reference parameter passing,For positive this is proportional to ; the empty case takes constant time. The call stack has depth.
Keep For insertion, conceptually treat index
u[0..len) sorted, with the coupling invariantThe abstract list retains insertion order, while the concrete array retains its sorted multiset. Initially both are empty.PROCEDURE Add(x: INTEGER);
VAR i: INTEGER;
BEGIN
ASSERT(len < N);
i := len;
WHILE i > 0 DO
IF u[i-1] <= x THEN EXIT END;
u[i] := u[i-1];
i := i-1
END;
u[i] := x;
len := len+1
END Add;
PROCEDURE Get(k: INTEGER): INTEGER;
BEGIN
ASSERT((0 <= k) & (k < len));
RETURN u[k]
END Get;i as a hole. If is the old sorted prefix of length , the loop invariant is , and , with every shifted value strictly greater than . The loop variant is . The exit condition gives either no left neighbour or a left neighbour at most ; filling the hole establishes sortedness and adds exactly one occurrence, preserving the coupling invariant.When the new entry is smaller than every old entry, all old values must move in this contiguous sorted-prefix representation. Binary search can reduce comparisons to locate the insertion point, but cannot remove that linear movement cost. Appending a new largest value takes constant time in this implementation.For an ascending sorted array, maintain a candidate interval initially containing every index. Compare the target with the middle entry at . Equality returns ; if the target is smaller, replace by ; if larger, replace by . An empty interval reports absence.
The loop invariant is that any target occurrence still possible lies inside the candidate interval. Sortedness justifies discarding the appropriate half after each unequal comparison. The interval length decreases to at most half its previous length, establishing termination and correctness. After unsuccessful comparisons there are at most candidates, so at most comparisons suffice for . Each iteration uses constant work on a random-access array. Thus binary search takes running time, conventionally written for nontrivial input sizes. Empty input is handled in constant time.
Sort the two halves ascending, reverse the second, and concatenate to form a bitonic sequence:For a power-of-two length greater than one, both halves are smaller powers of two. By induction, the recursive calls return ascending permutations of their respective inputs. Reversal makes the second half descending, so part (e)'s precondition holds and
sort :: Ord a => [a] -> [a]
sort [] = []
sort [x] = [x]
sort xs = merge (sort as ++ reverse (sort bs))
where (as,bs) = halve xsmerge sorts their concatenation without losing entries. This proves both sortedness and sorting permutation preservation. Its recurrence relation is , so summing the levels gives time. Equal keys cause no difficulty; ascending and descending here permit equality. This is a bitonic sorting algorithm.Use a lower-bound search separately in each nondecreasing row:
def searchRows(f: Array[Array[Int]], x: Int): Option[(Int, Int)] = {
val n = f.length
val m = if (n == 0) 0 else f(0).length
if (m == 0) return None
var i = 0
while (i < n) {
var lo = 0
var hi = m
while (lo < hi) {
val mid = lo + (hi - lo) / 2
if (f(i)(mid) < x) lo = mid + 1
else hi = mid
}
if (lo < m && f(i)(lo) == x) return Some((i, lo))
i += 1
}
None
}The outer loop invariant says there is no match in earlier rows. The inner invariant is , with every value before
lo smaller than and every value from hi onward at least . Sortedness justifies each exclusion, and hi - lo strictly decreases. At exit, a match in this row exists exactly when the lower-bound index is inside the row and holds . Failure extends the outer exclusion.This is lower-bound search: return the first position whose value is at least the target, allowing the result to equal the array length. The notation for an empty suffix is vacuously true, so this also covers a target larger than every entry and the empty array. Assume a stable sorted array, ordered nondecreasingly.
def ternarySearch(a: Array[Int], x: Int): Int = {
var lo = 0
var hi = a.length
while (lo < hi) {
val third = (hi - lo) / 3
val m1 = lo + third
val m2 = hi - 1 - third
if (a(m1) >= x) hi = m1
else if (a(m2) < x) lo = m2 + 1
else {
lo = m1 + 1
hi = m2
}
}
lo
}The loop invariant isInitially the excluded prefix and suffix are empty, so both quantified assertions hold. During an iteration, . If , sortedness makes every later value at least , so replacing
hi by m1 preserves the right assertion. If the first test failed and , every earlier value is below , so replacing lo by m2 + 1 preserves the left assertion. Otherwise ; both new exclusions are valid, and lo = m1 + 1; hi = m2 retains exactly the unresolved middle. In that branch , since a single value cannot satisfy both strict and nonstrict inequalities in opposite directions.The nonnegative loop variant
hi - lo strictly decreases. In fact, with old length and , the three possible new lengths are , with the middle branch feasible only when nonnegative; each is at most . At exit lo == hi, and the loop invariant is exactly the desired partition. The result is the unique lower-bound index, including the first equal entry when duplicates occur. This ternary lower-bound search uses comparisons and auxiliary space, in the Big O notation word-operation model. The index formulas also avoid computing an overflowing 2 * length. Radix sort suffix invariant 2026-10-18
After passes, , the array is in nondecreasing order of , the integer represented by its lowest numerical digits. Equivalently it is ordered by the tuple under lexicographic order, with the most significant processed numerical digit first. Among equal suffixes it retains the original order of the complete sorting records.
The loop invariant has a simple mathematical induction proof. Before any pass, for every sort key, so the original array satisfies it. Suppose it holds after passes. The next stable digit sort puts smaller groups first and preserves the existing order within each tie. Sincethe new array is ordered by . Equal new suffixes have equal current numerical digits and equal old suffixes, so their original order is also preserved. Every pass is a permutation of complete sorting records, establishing sorting permutation preservation. At termination , the assumption gives : the invariant becomes whole-key sortedness, together with stable sorting.
Sortedness 2026-10-13
For an output sequence of length , sortedness meansunder the chosen total order on sort keys. Its transitivity also gives whenever . An empty sequence and a one-entry sequence satisfy sortedness without any adjacent comparison. For longer sequences, checking adjacent pairs suffices; checking only the first and last sort keys does not. For example, passes an endpoint comparison but fails sortedness at . Sortedness alone says nothing about which input entries survived.
Sorting algorithm 2026-10-13
A sorting algorithm rearranges a finite sequence into nondecreasing order under a chosen total order on its keys. Its output must contain every input entry exactly once, including repetitions: producing ordered values is insufficient if entries are lost. Merge sort is one such algorithm; stable sorting adds a requirement about equal keys.
Sorting creates reusable structure: equal sort keys become adjacent, and a sorted array supports binary search. However, an algorithm correctness check must establish both sortedness and sorting permutation preservation. Replacing every entry by the smallest key can produce ordered keys while destroying the original data. Stable sorting imposes a third condition when equal sort keys occur: retain their arrival order.
For example, take complete sorting records , with the first component as the sort key and the letter as associated data. A correct output is . The sort keys are ordered, every complete sorting record remains, and the equal-key pairs keep before and before . The output satisfies sortedness and sorting permutation preservation, but fails stable sorting. The output has the right multiset of sort keys but loses and duplicates ; it fails sorting permutation preservation of complete sorting records.
To express stable sorting precisely, label every input occurrence by its original position, even when two complete sorting records have identical contents. Let be the output position of the occurrence originally at position . Besides sortedness and sorting permutation preservation, requireHere is the sort key function. Sorting by the pair under lexicographic order gives the same order as stable sorting by : the original position breaks ties without discarding associated data. For a two-field ordering, first apply stable sorting by the secondary sort key, then apply stable sorting by the primary sort key. The second pass retains the secondary order within each group of equal primary sort keys. For instance, first becomes by letter, then by number.
Stable digit sort 2026-10-18
A stable digit sort applies stable sorting to complete sorting records using one specified numerical digit of their encoded sort keys. Counting sort can supply it; stable record placement in counting sort moves records without discarding their associated data. Stability concerns equal current numerical digits, even when the full sort keys differ. This is the information-preservation step in least-significant-digit radix sort.[ref]
11.2 Counting Sort and Radix Sort. Open Data Structures: An Introduction by Pat Morin; §11.2.1, associated-data and stability paragraphs; §11.2.2, least-significant passes and correctness paragraph after Figure 11.8
An original two-record counterexample exposes the need for the tie rule. Start with at width three. A units pass gives . Both tens numerical digits are four. A deliberately unstable tens pass may reverse their tie and return : it is sorted by the tens numerical digit, yet its two-position suffixes are decreasing. The final stable hundreds pass sees two zeros and keeps this wrong order. Sorting permutation preservation still holds, but sortedness fails. Stability in the tens pass would have retained the units information.
With counting sort as the stable digit sort, each pass initializes counters, counts current numerical digits, constructs their prefix sums, and places complete sorting records. Thus each pass costs , and passes cost in Big O notation. Reuse the counters and two arrays of record references between passes: auxiliary space complexity is , rather than . Assume constant-cost numerical digit extraction, counter arithmetic, array indexing and record-reference movement; the bound counts these operations, not the cost of copying arbitrarily large payloads or performing arbitrary-precision integer arithmetic.
The benefit is avoiding a counter for every possible whole sort key: width has possibilities, while each pass needs only counters. A larger radix reduces the number of passes for a fixed key range but enlarges the counter array. The comparison sorting lower bound does not apply to these direct array indices; they reveal more than a two-way comparison. Nor does fixing make disappear: growing key width still increases work.
Stable record placement in counting sort 2026-10-18
Initialize a mutable endpoint array to the inclusive cumulative counts . Scan input array indices . For each complete sorting record with sort key , first decrement , then write at output array index . This is equivalent to writing at the old endpoint minus one and then decrementing the endpoint.[ref]
11.2 Counting Sort and Radix Sort. Open Data Structures: An Introduction by Pat Morin; §11.2.1 Counting Sort, final scan paragraph and stability paragraph after Theorem 11.7
In the worked example, the first input sorting record read is at array index six. Its endpoint changes from to , so the complete sorting record goes to . Next, changes from seven to six and goes to . Writing before the decrement at the old endpoint would put outside its reserved integer interval and try to put beyond the output array. All seven writes are:Each endpoint changes only when a sorting record with its own sort key is placed. Consequently, stays at five through the placements of before dropping to four for .
Here is a complete implementation under the stated input contract; each sorting record is a pair whose first component is its integer sort key:
def counting_sort_records(records, K):
end = [0] * K
output = [None] * len(records)
for key, data in records:
end[key] += 1
for key in range(1, K):
end[key] += end[key - 1]
for i in range(len(records) - 1, -1, -1):
key = records[i][0]
end[key] -= 1
output[end[key]] = records[i]
return outputThe same array
end successively stores frequencies, inclusive cumulative counts, and moving exclusive endpoints; the formulas describe their mathematical values without requiring three stored arrays. The write copies the complete sorting record, including its associated data. In particular, two occurrences of an identical pair still cause two separate writes.For a short algorithm correctness proof, let be the number of already placed sorting records with sort key . A loop invariant isAll integer intervals here mean output array indices. Initially , so each whole reserved integer interval is unfilled. When the next sorting record has sort key , fewer than such occurrences have been placed, giving . Decrementing selects a fresh slot inside its reserved integer interval. At termination, and : every output slot has been filled exactly once with a complete input occurrence. The disjoint, increasing sort key ranges establish sortedness, while the fresh-slot property establishes sorting permutation preservation.
For stable sorting, take two input occurrences with equal sort keys at array indices . The backward scan visits first and puts it farther right in their shared integer interval. When is visited, the endpoint has decreased at least once, so its output position is smaller. Thus their original order survives. A forward scan with this same decrement rule reverses every equal-sort key group: it would put at positions five, four, three. A forward scan can instead be stable if it starts from and fills each integer interval left to right, incrementing after each write. The scan direction and endpoint convention must agree.
Count initialization takes writes, output initialization takes writes, counting takes updates, the prefix sum pass takes updates, and placement takes writes. Hence time complexity and auxiliary storage are in Big O notation, assuming constant-cost sort key access, integer arithmetic, array indexing and complete sorting record movement. Large records can instead be represented by constant-size references.[ref]
11.2 Counting Sort and Radix Sort. Open Data Structures: An Introduction by Pat Morin; §11.2.1 Counting Sort, auxiliary counter and output arrays, loop-cost discussion and Theorem 11.7
For example, three sorting records with sort keys zero, and zero require one million counters if ; unused sort keys still incur initialization and prefix sum work. The bound is therefore linear in the input length only when . Empty input skips the record scans, but this implementation still initializes counters and accumulates their prefix sums. For nonempty input, counting all records and filling their ranges gives a concrete termination argument.
Direct sort key indexing is a different operation from a binary comparison. The comparison sorting lower bound concerns comparison sorts; counting sort uses its restricted integer sort keys as array indices. Its small- speed is particularly useful for categorical numerical labels and for each stable sorting pass of radix sort: the reserved integer intervals turn group sizes into exact destinations.
Ternary lower-bound search 2026-10-24
A lower-bound search that probes two positions to retain one of three subintervals. For unresolved of positive length , put , and . If , retain ; if , retain ; otherwise retain . Sortedness preserves the lower-bound partition. The new interval lengths are , and the feasible middle case has nonnegative length at most . This proves strict progress and comparison cost. Equality must retain the first possible boundary rather than return an arbitrary matching occurrence.
Upper-bound search 2026-10-31
For a sorted array , find the unique boundary with . A binary search maintains , an excluded prefix , and an excluded suffix . At an interior midpoint , replace by when , and otherwise replace by . Sortedness preserves the loop invariant; the width is a decreasing loop variant. Termination at gives the boundary, including and . The time complexity is with constant-cost comparisons and indexed access. Unlike searching for , this formulation does not risk integer overflow at the largest representable key.