The SRFI-32 sort libraries -*- outline -*- Olin Shivers First draft: 1998/10/19 Last update: 2002/7/21 [Todo: del-list-neighbor-dups! vector-copy -> subvector use srfi-23 for reporting errors use srfi-16 for n-aries? Emacs should display this document in outline mode. Say c-h m for instructions on how to move through it by sections (e.g., c-c c-n, c-c c-p). * Table of contents ------------------- Abstract Procedure index Introduction What's wrong with the current state of affairs? Design rules What vs. how Consistency across function signatures Data parameter first, less-than parameter after Ordering, comparison functions & stability All vector operations accept optional subrange parameters Required vs. allowed side-effects Procedure specification Procedure naming and functionality Types of parameters and return values sort-lib - general sorting package Algorithm-specific sorting packages Algorithmic properties Topics to be resolved during discussion phase Porting and optimisation References & Links Acknowledgements Copyright * Abstract ---------- Current Scheme sorting packages are, every one of them, surprisingly bad. I've designed the API for a full-featured sort toolkit, which I propose as a SRFI. The spec comes with 1200 lines of high-quality reference code: tightly written, highly commented, portable code, available for free. Implementors want this code. It's better than what you have. ------------------------------------------------------------------------------- * Procedure index ----------------- list-sorted? vector-sorted? list-merge vector-merge list-sort vector-sort list-stable-sort vector-stable-sort list-delete-neighbor-dups vector-delete-neighbor-dups list-merge! vector-merge! list-sort! vector-sort! list-stable-sort! vector-stable-sort! list-delete-neighbor-dups! vector-delete-neighbor-dups! quick-sort heap-sort insert-sort list-merge-sort vector-merge-sort quick-sort! heap-sort! insert-sort! list-merge-sort! vector-merge-sort! quick-sort3! vector-binary-search vector-binary-search3 ------------------------------------------------------------------------------- * Introduction -------------- As I'll detail below, I wasn't very happy with the state of the Scheme world for sorting and merging lists and vectors. So I have designed and written a fairly comprehensive sorting & merging toolkit. It is - very portable, - much better code than what is currently in Elk, Gambit, Bigloo, Scheme->C, MzScheme, RScheme, Scheme48, MIT Scheme, or slib, and - priced to move: free code. The package includes - Vector insert sort (stable) - Vector heap sort - Vector quick sort (with median-of-3 pivot picking) - Vector merge sort (stable) - Pure and destructive list merge sort (stable) - Stable vector and list merge - Miscellaneous sort-related procedures: Vector and list merging, sorted? predicates, vector binary search, vector and list delete-equal-neighbor procedures. - A general, non-algorithmic set of procedure names for general sorting and merging. Scheme programmers may want to adopt this package. I'd like Scheme implementors to adopt this code and its API -- in fact, the code is a bribe to make it easy for implementors to converge on the suggested API. I mean, you'd really have to be a boor to take this free code I wrote and mutate its interface over to your incompatible, unportable API, wouldn't you? But you could, of course -- it's freely available. More in the spirit of the offering, you could make this API available, and then also write a little module providing your old interface that is defined in terms of this API. "Scheme implementors," in this context, includes slib, which is not a standalone implementation of Scheme, but rather an influential collection of API's and code. The code is tightly bummed. It is clearly written, and commented in my usual voluminous style. This includes notes on porting and implementation-specific optimisations. ------------------------------------------------------------------------------- * What's wrong with the current state of affairs? ------------------------------------------------- It's just amazing to me that in 2002, sorting and merging hasn't been completely put to bed. These are well-understood algorithms, each of them well under a page of code. The straightforward algorithms are basic, core stuff -- sophomore-level. But if you tour the major Scheme implementations out there on the Net, you find badly written code that provides extremely spotty coverage of the algorithm space. One implementation even has a buggy implementation that has been in use for about 20 years. Another has an O(n^2) algorithm... implemented in C for speed. Open source-code is a wonderful thing. In a couple of hours, I was able to download and check the sources of 9 Scheme systems. Here are my notes from the systems I checked. You can skip to the next section if you aren't morbidly curious. slib sorted? vector-or-list < merge list1 list2 < merge! list1 list2 < sort vector-or-list < sort! vector-or-list < Richard O'Keefe's stable list merge sort is right idea, but implemented using gratuitous variable side effects. It also does redundant SET-CDR!s. The vector sort converts to list, merge sorts, then reconverts to vector. This is a bad idea -- non-local pointer chasing bad; vector shuffling good. If you must allocate temp storage, might as well allocate a temp vector and use vector merge sort. MIT Scheme sort! vector < merge-sort! vector < quick-sort! vector < sort vector-or-list < merge-sort vector-or-list < quick-sort vector-or-list < Naive vector quicksort: loser, for worst-case performance reasons. List sort by "list->vector; quicksort; vector->list," hence also loser. A clever stable vector merge sort, albeit not very bummed. Scheme 48 & T sort-list list < sort-list! list < list-merge! list1 list2 < Bob Nix's implementation of online merge-sort, written in the early 80's. Conses unnecessary bookkeeping structure, which isn't necessary with a proper recursive formulation. Also, does redundant SET-CDR!s. No vector sort. Also, has a bug -- is claimed to be a stable sort, but isn't! To see this, get the S48 code, and try (define (my< x y) (< (abs x) (abs y))) (list-merge! (list 0 2) (list -2) my<) ; -> (0 2 -2) (list-merge! (list 2) (list 0 -2) my<) ; -> (0 -2 2) This could be fixed very easily, but it isn't worth it given the other problems with the algorithm. RScheme vector-sort! vector < sort collection < Good basic implementation of vector heapsort, which has O(n lg n) worst-case time. Code ugly, needs tuning. List sort by "list->vector; sort; vector->list." Nothing for stable sorting. MzScheme quicksort lis < mergesort alox < Sorts lists with (list->vector; quicksort; vector->list) -- but the core quicksort is not available for vector sorting. Nothing for stable sorting. Quicksort picks pivot naively, inducing O(n^2) worse-case behaviour on a fairly common case: an already-sorted list. Bigloo, STK sort vector-or-list < Uses an O(n^2) algorithm... implemented in C for speed. Hmm. (See runtime/Ieee/vector.scm and runtime/Clib/cvector.c) Gambit sort-list list < Nothing for vectors. Simple, slow, unstable merge sort for lists. Elk Another naive quicksort. Lists handled by converting to vector. sort vector-or-list < sort! vector-or-list < Chez Scheme merge < list1 list2 merge! < list1 list2 sort < list sort! < list These are stable. I have not seen the source code. Common Lisp sort sequence < [key] stable-sort sequence < [key] merge result-type sequence1 sequence2 < [key] The sort procedures are allowed, but not required, to be destructive. SML/NJ sort: ('a*'a -> bool) -> 'a list -> 'a list "Smooth applicative merge sort," which is stable. There is also a highly bummed quicksort for vectors. The right solution: Implement a full toolbox of carefully written standard sort routines. Having the source of all these above-cited Schemes available for study made life a lot easier writing this code. I appreciate the authors making their source available under such open terms. ------------------------------------------------------------------------------- * Design rules -------------- ** What vs. how =============== There are two different interfaces: "what" (simple) & "how" (detailed). - Simple: you specify semantics: datatype (list or vector), mutability, and stability. - Detailed: you specify the actual algorithm (quick, heap, insert, merge). Different algorithms have different properties, both semantic & pragmatic, so these exports are necessary. It is necessarily the case that the specifications of these procedures make statements about execution "pragmatics." For example, the sole distinction between heap sort and quick sort -- both of which are provided by this library -- is one of execution time, which is not a "semantic" distinction. Similar resource-use statements are made about "iterative" procedures, meaning that they can execute on input of arbitrary size in a constant number of stack frames. ** Consistency across function signatures ========================================= The two interfaces share common function signatures wherever possible, to facilitate switching a given call from one procedure to another. ** Less-than parameter first, data parameter after ================================================== These procedures uniformly observe the following parameter order: the data to be sorted comes after the comparison function. That is, we write (sort < lis) not (sort lis <). With the sole exception of Chez Scheme, this is the exact opposite of every sort function out there in current use in the Scheme world. (See the summary of related APIs above.) However, it is consistent with common practice across Scheme libraries in general to put the ordering function first -- the "operation currying" convention. (E.g., consider FOR-EACH or MAP or FIND.) The original draft of this SRFI used the data-first/comparison-last convention for backwards compatibility -- a decision I made with internal misgivings. Happily, however, the overwhelming response from the discussion phase supported "cleaning up" this issue and re-converging the parameter order with the general Scheme "op currying" convention. So the original decision was inverted in favor of the comparison-first/data-last convention. ** Ordering, comparison functions & stability ============================================= These routines take a < comparison function, not a <= comparison function, and they sort into increasing order. The difference between a < spec and a <= spec comes up in three places: - the definition of an ordered or sorted data set, - the definition of a stable sorting algorithm, and - correctness of quicksort. + We say that a data set (a list or vector) is *sorted* or *ordered* if it contains no adjacent pair of values ... X Y ... such that Y < X. In other words, scanning across the data never takes a "downwards" step. If you use a <= procedure where these algorithms expect a < procedure, you may not get the answers you expect. For example, the LIST-SORTED? function will return false if you pass it a <= comparison function and an ordered list containing adjacent equal elements. + A "stable" sort is one that preserves the pre-existing order of equal elements. Suppose, for example, that we sort a list of numbers by comparing their absolute values, i.e., using comparison function (lambda (x y) (< (abs x) (abs y))) If we sort a list that contains both 3 and -3: ... 3 ... -3 ... then a stable sort is an algorithm that will not swap the order of these two elements, that is, the answer is guaranteed to to look like ... 3 -3 ... not ... -3 3 ... Choosing < for the comparison function instead of <= affects how stability is coded. Given an adjacent pair X Y, (< y x) means "Y should be moved in front of X" -- otherwise, leave things as they are. So using a <= function where a < function is expected will *invert* stability. This is due to the definition of equality, given a < comparator: (and (not (< x y)) (not (< y x))) The definition is rather different, given a <= comparator: (and (<= x y) (<= y x)) + A "stable" merge is one that reliably favors one of its data sets when equal items appear in both data sets. *All merge operations in this library are stable*, breaking ties between data sets in favor of the first data set -- elements of the first list come before equal elements in the second list. So, if we are merging two lists of numbers ordered by absolute value, the stable merge operation LIST-MERGE (list-merge (lambda (x y) (< (abs x) (abs y))) '(0 -2 4 8 -10) '(-1 3 -4 7)) reliably places the 4 of the first list before the equal-comparing -4 of the second list: (0 -1 -2 4 -4 7 8 -10) + Some sort algorithms will *not work correctly* if given a <= when they expect a < comparison (or vice-versa). For example, violating quicksort's spec may cause it to produce wrong answers, diverge, raise an error, or do some fourth thing. To see why, consider the left-scan part of the standard quicksort partition step: (let ((i (let scan ((i i)) (if (elt< (vector-ref v i) pivot) (scan (+ i 1)) i)))) ...) Consider applying this loop to a vector of all zeroes (hence, PIVOT, as well, is zero), but erroneously using <= for the ELT< function. The loop will scan right off the end of the vector, producing a vector-index error. The guarantee that the scan loop will terminate before running off the end of the vector depends critically upon ELT< performing as a true, irreflexive < relation. Running off the end of the vector is only one of a variety of possibly ways to lose -- other, variant implementations of quicksort can, instead, loop forever on some data sets if ELT< is a <= predicate. In short, if your comparison function F answers true to (F x x), then - using a stable sorting or merging algorithm will not give you a stable sort or merge, - LIST-SORTED? may surprise you, and - quicksort may fail in a variety of possible ways. Note that you can synthesize a < function from a <= function with (lambda (x y) (not (<= y x))) if need be. Precise definitions give sharp edges to tools, but require care in use. "Measure twice, cut once." I have adopted the choice of < from Common Lisp. One would assume the definers of Common Lisp had a good reason for adopting < instead of <=, but canvassing several of the principal actors in the definition process has turned up no better reason than "an arbitrary but consistent choice." At minimum, then, this SRFI extends the coverage of that consistent choice. ** All vector operations accept optional subrange parameters ============================================================ The vector operations specified below all take optional START/END arguments indicating a selected subrange of a vector's elements. If a START parameter or START/END parameter pair is given to such a procedure, they must be exact, non-negative integers, such that 0 <= START <= END <= (VECTOR-LENGTH V) where V is the related vector parameter. If not specified, they default to 0 and the length of the vector, respectively. They are interpreted to select the range [START,END), that is, all elements from index START (inclusive) up to, but not including, index END. ** Required vs. allowed side-effects ==================================== LIST-SORT! and LIST-STABLE-SORT! are allowed, but not required, to alter their arguments' cons cells to construct the result list. This is consistent with the what-not-how character of the group of procedures to which they belong (the "sort-lib" package). The LIST-DELETE-NEIGHBOR-DUPS!, LIST-MERGE! and LIST-MERGE-SORT! procedures, on the other hand, provide specific algorithms, and, as such, explicitly commit to the use of side-effects on their input lists in order to guarantee their key algorithmic properties (e.g., linear-time operation, constant-space stack use). ------------------------------------------------------------------------------- * Procedure specification ------------------------- The procedures are split into several packages. In a Scheme system that has a module or package system, these procedures should be contained in modules named as follows: Package name Functionality ------------ ------------- sort-lib General sorting for lists & vectors sorted?-lib Sorted predicates for lists & vectors list-merge-sort-lib List merge sort vector-merge-sort-lib Vector merge sort vector-heap-sort-lib Vector heap sort vector-quick-sort-lib Vector quick sort vector-insert-sort-lib Vector insertion sort delndup-lib List and vector delete neighbor duplicates binsearch-lib Vector binary search A Scheme system without a module system should provide all of the bindings defined in all of these modules as components of the "SRFI-32" package. Note that there is no "list insert sort" package, as you might as well always use list merge sort. The reference implementation's destructive list merge sort will do fewer SET-CDR!s than a destructive insert sort. ** Procedure naming and functionality ===================================== Almost all of the procedures described below are variants of two basic operations: sorting and merging. These procedures are consistently named by composing a set of basic lexemes to indicate what they do. Lexeme Meaning ------ ------- "sort" The procedure sorts its input data set by some < comparison function. "merge" The procedure merges two ordered data sets into a single ordered result. "stable" This lexeme indicates that the sort is a stable one. "vector" The procedure operates upon vectors. "list" The procedure operates upon lists. "!" Procedures that end in "!" are allowed, and sometimes required, to reuse their input storage to construct their answer. ** Types of parameters and return values ======================================== In the procedures specified below, - A LIS parameter is a list; - A V parameter is a vector; - A < or = parameter is a procedure accepting two arguments taken from the specified procedure's data set(s), and returning a boolean; - START and END parameters are exact, non-negative integers that serve as vector indices selecting a subrange of some associated vector. When specified, they must satisfy the relation 0 <= start <= end <= (vector-length v) where V is the associated vector. Passing values to procedures with these parameters that do not satisfy these types is an error. If a procedure is said to return "unspecified," this means that nothing at all is said about what the procedure returns, not even the number of return values. Such a procedure is not even required to be consistent from call to call in the nature or number of its return values. It is simply required to return a value (or values) that may be passed to a command continuation, e.g. as the value of an expression appearing as a non-terminal subform of a BEGIN expression. Note that in R5RS, this restricts such a procedure to returning a single value; non-R5RS systems may not even provide this restriction. ** sort-lib - general sorting package ===================================== This library provides basic sorting and merging functionality suitable for general programming. The procedures are named by their semantic properties, i.e., what they do to the data (sort, stable sort, merge, and so forth). Procedure Suggested algorithm ------------------------------------------------------------------------- list-sorted? < lis -> boolean list-merge < lis1 lis2 -> list list-merge! < lis1 lis2 -> list list-sort < lis -> list (vector heap or quick) list-sort! < lis -> list (list merge sort) list-stable-sort < lis -> list (vector merge sort) list-stable-sort! < lis -> list (list merge sort) list-delete-neighbor-dups = lis -> list list-delete-neighbor-dups! = lis -> list vector-sorted? < v [start end] -> boolean vector-merge < v1 v2 [start1 end1 start2 end2] -> vector vector-merge! < v v1 v2 [start start1 end1 start2 end2] -> unspecified vector-sort < v [start end] -> vector (heap or quick sort) vector-sort! < v [start end] -> unspecified (heap or quick sort) vector-stable-sort < v [start end] -> vector (vector merge sort) vector-stable-sort! < v [start end] -> unspecified (vector merge sort) vector-delete-neighbor-dups = v [start end] -> vector vector-delete-neighbor-dups! = target source [t-start s-start s-end] -> t-end LIST-SORTED? and VECTOR-SORTED? return true if their input list or vector is in sorted order, as determined by their < comparison parameter. All four merge operations are stable: an element of the initial list LIS1 or vector V1 will come before an equal-comparing element in the second list LIS2 or vector V2 in the result. The procedures LIST-MERGE LIST-SORT LIST-STABLE-SORT LIST-DELETE-NEIGHBOR-DUPS do not alter their inputs and are allowed to return a value that shares a common tail with a list argument. The procedures LIST-SORT! LIST-STABLE-SORT! are "linear update" operators -- they are allowed, but not required, to alter the cons cells of their arguments to produce their results. On the other hand, the procedures LIST-DELETE-NEIGHBOR-DUPS! LIST-MERGE! make only a single, iterative, linear-time pass over their argument lists, using SET-CDR!s to rearrange the cells of the lists into the final result -- they work "in place." Hence, any cons cell appearing in the result must have originally appeared in an input. The intent of this iterative-algorithm commitment is to allow the programmer to be sure that if, for example, LIST-MERGE! is asked to merge two ten-million-element lists, the operation will complete without performing some extremely (possibly twenty-million) deep recursion. The vector procedures VECTOR-SORT VECTOR-STABLE-SORT VECTOR-DELETE-NEIGHBOR-DUPS do not alter their inputs, but allocate a fresh vector for their result, of length END - START. The vector procedures VECTOR-SORT! VECTOR-STABLE-SORT! sort their data in-place. (But note that VECTOR-STABLE-SORT! may allocate temporary storage proportional to the size of the input -- I am not aware of O(n lg n) stable vector-sorting algorithms that run in constant space.) VECTOR-MERGE returns a vector of length (END1-START1)+(END2-START2). VECTOR-MERGE! writes its result into vector V, beginning at index START, for indices less than END = START + (END1-START1) + (END2-START2). The target subvector V[start,end) may not overlap either source subvector V1[start1,end1) V2[start2,end2). The ...-DELETE-NEIGHBOR-DUPS-... procedures: These procedures delete adjacent duplicate elements from a list or a vector, using a given element-equality procedure. The first/leftmost element of a run of equal elements is the one that survives. The list or vector is not otherwise disordered. These procedures are linear time -- much faster than the O(n^2) general duplicate-element deletors that do not assume any "bunching" of elements (such as the ones provided by SRFI-1). If you want to delete duplicate elements from a large list or vector, you can sort the elements to bring equal items together, then use one of these procedures, for a total time of O(n lg n). The comparison function = passed to these procedures is always applied (= x y) where X comes before Y in the containing list or vector. - LIST-DELETE-NEIGHBOR-DUPS does not alter its input list; its answer may share storage with the input list. - VECTOR-DELETE-NEIGHBOR-DUPS does not alter its input vector, but rather allocates a fresh vector to hold the result. - LIST-DELETE-NEIGHBOR-DUPS! is permitted, but not required, to mutate its input list in order to construct its answer. - VECTOR-DELETE-NEIGHBOR-DUPS! reuses its input vector to hold the answer, packing its answer into the index range [start,end'), where END' is the non-negative exact integer returned as its value. It returns END' as its result. The vector is not altered outside the range [start,end'). - VECTOR-DELETE-NEIGHBOR-DUPS! scans vector SOURCE in range [S-START,S-END), writing its result to vector TARGET beginning at index T-START. It returns exact, non-negative integer T-END, which indicates that the results of the operation are found in index range [T-START,T-END) of TARGET; elements of TARGET outside this range are unaltered. It is an error for memory cell TARGET[T-START] to be a memory cell in the region SOURCE[1 + S-START, S-END). In a Scheme implementation that does not allow distinct vectors to share storage, this means that one of the following must be true: 1. (not (eq? source target)) 2. t-start not-in [s-start + 1, s-end) - Examples: (list-delete-neighbor-dups = '(1 1 2 7 7 7 0 -2 -2)) => (1 2 7 0 -2) (vector-delete-neighbor-dups = '#(1 1 2 7 7 7 0 -2 -2)) => #(1 2 7 0 -2) (vector-delete-neighbor-dups = '#(1 1 2 7 7 7 0 -2 -2) 3 7) => #(7 0 -2) ;; Result left in v[3,9): (let ((v (vector 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6))) (cons (vector-delete-neighbor-dups! = v 3) v)) => (9 . #(0 0 0 1 2 3 4 5 6 4 4 5 5 6 6)) ** Algorithm-specific sorting packages ====================================== These packages provide more specific sorting functionality, that is, specific committment to particular algorithms that have particular pragmatic consequences (such as memory locality, asymptotic running time) beyond their semantic behaviour (sorting, stable sorting, merging, etc.). Programmers that need a particular algorithm can use one of these packages. sorted?-lib - sorted predicates list-sorted? < lis -> boolean vector-sorted? < v [start end] -> boolean Return #f iff there is an adjacent pair ... X Y ... in the input list or vector such that Y < X. The optional START/END range arguments restrict VECTOR-SORTED? to the indicated subvector. list-merge-sort-lib - list merge sort list-merge-sort < lis -> list list-merge-sort! < lis -> list list-merge lis1 < lis2 -> list list-merge! lis1 < lis2 -> list The sort procedures sort their data using a list merge sort, which is stable. (The reference implementation is, additionally, a "natural" sort. See below for the properties of this algorithm.) The ! procedures are destructive -- they use SET-CDR!s to rearrange the cells of the lists into the proper order. As such, they do not allocate any extra cons cells -- they are "in place" sorts. Additionally, LIST-MERGE! is iterative -- it can operate on arguments of arbitrary size with a constant number of stack frames. The merge operations are stable: an element of LIS1 will come before an equal-comparing element in LIS2 in the result list. vector-merge-sort-lib - vector merge sort vector-merge-sort < v [start end temp] -> vector vector-merge-sort! < v [start end temp] -> unspecified vector-merge < v1 v2 [start1 end1 start2 end2] -> vector vector-merge! < v v1 v2 [start start1 end1 start2 end2] -> unspecified The sort procedures sort their data using vector merge sort, which is stable. (The reference implementation is, additionally, a "natural" sort. See below for the properties of this algorithm.) The optional START/END arguments provide for sorting of subranges, and default to 0 and the length of the corresponding vector. Merge-sorting a vector requires the allocation of a temporary "scratch" work vector for the duration of the sort. This scratch vector can be passed in by the client as the optional TEMP argument; if so, the supplied vector must be of size >= END, and will not be altered outside the range [start,end). If not supplied, the sort routines allocate one themselves. The merge operations are stable: an element of V1 will come before an equal-comparing element in V2 in the result vector. VECTOR-MERGE-SORT! leaves its result in V[start,end). VECTOR-MERGE-SORT returns a vector of length END-START. VECTOR-MERGE returns a vector of length (END1-START1)+(END2-START2). VECTOR-MERGE! writes its result into vector V, beginning at index START, for indices less than END = START + (END1-START1) + (END2-START2). The target subvector V[start,end) may not overlap either source subvector V1[start1,end1) V2[start2,end2). vector-heap-sort-lib - vector heap sort heap-sort < v [start end] -> vector heap-sort! < v [start end] -> unspecified These procedures sort their data using heap sort, which is not a stable sorting algorithm. HEAP-SORT returns a vector of length END-START. HEAP-SORT! is in-place, leaving its result in V[start,end). vector-quick-sort-lib - vector quick sort quick-sort < v [start end] -> vector quick-sort! < v [start end] -> unspecified quick-sort3! c v [start end] -> unspecified These procedures sort their data using quick sort, which is not a stable sorting algorithm. QUICK-SORT returns a vector of length END-START. QUICK-SORT! is in-place, leaving its result in V[start,end). QUICK-SORT3! is a variant of quick-sort that takes a three-way comparison function C. C compares a pair of elements and returns an exact integer whose sign indicates their relationship: (c x y) < 0 => x x=y (c x y) > 0 => x>y To help remember the relationship between the sign of the result and the relation, use the function - as the model for C: (- x y) < 0 means that x < y; (- x y) > 0 means that x > y. The extra discrimination provided by the three-way comparison can provide significant speedups when sorting data sets with many duplicates, especially when the comparison function is relatively expensive (e.g., comparing long strings). WARNING: Some sort algorithms, such as insertion sort or heap sort, can tolerate being passed a <= comparison function when they expect a < function -- insertion and merge sort may simply invert stability; and heap sort will run a bit slower, but otherwise produce a correct answer. Quicksort, however, is much more critically sensitive to the distinction between a < and a <= comparison. If QUICK-SORT or QUICK-SORT! expect a < comparison function, and are erroneously given a <= function, they may, depending on implementation, produce an unsorted result, go into an infinite loop, cause a run-time error, occasionally produce a correct result, or do some fifth thing. Implementors may wish to write QUICKSORT3! so that it (a) tests the comparison function (by checking that (c v[start] v[start]) produces false), or (b) is tolerant of an erroneous <= function, or (c) both. Clients of this function, however, should not count on this. vector-insert-sort-lib - vector insertion sort insert-sort < v [start end] -> vector insert-sort! < v [start end] -> unspecified These procedures stably sort their data using insertion sort. INSERT-SORT returns a vector of length END-START. INSERT-SORT! is in-place, leaving its result in V[start,end). delndup-lib - list and vector delete neighbor duplicates list-delete-neighbor-dups = lis -> list list-delete-neighbor-dups! = lis -> list vector-delete-neighbor-dups = v [start end] -> vector vector-delete-neighbor-dups! = v [start end] -> end' These procedures delete adjacent duplicate elements from a list or a vector, using a given element-equality procedure =. The first/leftmost element of a run of equal elements is the one that survives. The list or vector is not otherwise disordered. These procedures are linear time -- much faster than the O(n^2) general duplicate-element deletors that do not assume any "bunching" of elements (such as the ones provided by SRFI-1). If you want to delete duplicate elements from a large list or vector, you can sort the elements to bring equal items together, then use one of these procedures, for a total time of O(n lg n). The comparison function = passed to these procedures is always applied (= x y) where X comes before Y in the containing list or vector. LIST-DELETE-NEIGHBOR-DUPS does not alter its input list; its answer may share storage with the input list. VECTOR-DELETE-NEIGHBOR-DUPS does not alter its input vector, but rather allocates a fresh vector to hold the result. LIST-DELETE-NEIGHBOR-DUPS! is permitted, but not required, to mutate its input list in order to construct its answer. VECTOR-DELETE-NEIGHBOR-DUPS! reuses its input vector to hold the answer, packing its answer into the index range [start,end'), where END' is the non-negative exact integer returned as its value. It returns END' as its result. The vector is not altered outside the range [start,end'). Examples: (list-delete-neighbor-dups = '(1 1 2 7 7 7 0 -2 -2)) => (1 2 7 0 -2) (vector-delete-neighbor-dups = '#(1 1 2 7 7 7 0 -2 -2)) => #(1 2 7 0 -2) (vector-delete-neighbor-dups = '#(1 1 2 7 7 7 0 -2 -2) 3 7) => #(7 0 -2) ;; Result left in v[3,9): (let ((v (vector 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6))) (cons (vector-delete-neighbor-dups! = v 3) v)) => (9 . #(0 0 0 1 2 3 4 5 6 4 4 5 5 6 6)) binsearch-lib - vector binary search lib vector-binary-search elt< elt->key key v [start end] -> integer-or-false vector-binary-search3 c v [start end] -> integer-or-false VECTOR-BINARY-SEARCH searches vector V in range [START,END) (which default to 0 and the length of V, respectively) for an element whose associated key is equal to KEY. The procedure ELT->KEY is used to map an element to its associated key. The elements of the vector are assumed to be ordered by the ELT< relation on these keys. That is, (vector-sorted? (lambda (x y) (elt< (elt->key x) (elt->key y))) v start end) => true An element E of V is a match for KEY if it's neither less nor greater than the key: (and (not (elt< (elt->key e) key)) (not (elt< key (elt->key e)))) If there is such an element, the procedure returns its index in the vector as an exact integer. If there is no such element in the searched range, the procedure returns false. (vector-binary-search < car 4 '#((1 . one) (3 . three) (4 . four) (25 . twenty-five))) => 2 (vector-binary-search < car 7 '#((1 . one) (3 . three) (4 . four) (25 . twenty-five))) => #f VECTOR-BINARY-SEARCH3 is a variant that uses a three-way comparison function C. C compares its parameter to the search key, and returns an exact integer whose sign indicates its relationship to the search key. (c x) < 0 => x < search-key (c x) = 0 => x = search-key (c x) > 0 => x > search-key (vector-binary-search3 (lambda (elt) (- (car elt) 4)) '#((1 . one) (3 . three) (4 . four) (25 . twenty-five))) => 2 Rationale: - Why isn't VECTOR-BINARY-SEARCH's ELT->KEY computation simply absorbed into the < function? It is separated out because the < function is applied twice inside the binary-search inner loop, once with the search key for the first argument and the element key for the second argument, and once, with the reverse argument order. This is not necessary for VECTOR-BINARY-SEARCH3. - When a comparison operation is able to produce a three-way discrimination, the inner loop of the binary search can trim the number of per-iteration comparisons from an average of 1.5 to a guaranteed single comparison per iteration. This can be a significant savings when searching with an expensive comparison operation (e.g., one that uses string compare, sends email, references a database, or queries a network service such as a web server). - Failure is signaled by false (rather than, say, -1) so that searches can be used in conditional forms such as (or (vector-binary-search ...) ...) or (cond ((vector-binary-search ...) => index-consumer) ...) ------------------------------------------------------------------------------- * Algorithmic properties ------------------------ Different sort and merge algorithms have different properties. Choose the algorithm that matches your needs: Vector insert sort Stable, but only suitable for small vectors -- O(n^2). Vector quick sort Not stable. Is fast on average -- O(n lg n) -- but has bad worst-case behaviour. Has good memory locality for big vectors (unlike heap sort). A clever pivot-picking trick (median of three samples) helps avoid worst-case behaviour, but pathological cases can still blow up. Vector heap sort Not stable. Guaranteed fast -- O(n lg n) *worst* case. Poor locality on large vectors. A very reliable workhorse. Vector merge sort Stable. Not in-place -- requires a temporary buffer of equal size. Fast -- O(n lg n) -- and has good memory locality for large vectors. The implementation of vector merge sort provided by this SRFI's reference implementation is, additionally, a "natural" sort, meaning that it exploits existing order in the input data, providing O(n) best case. Destructive list merge sort Stable, fast and in-place (i.e., allocates no new cons cells). "Fast" means O(n lg n) worse-case, and substantially better if the data is already mostly ordered, all the way down to linear time for a completely-ordered input list (i.e., it is a "natural" sort). Note that sorting lists involves chasing pointers through memory, which can be a loser on modern machine architectures because of poor cache & page locality. Pointer *writing*, which is what the SET-CDR!s of a destructive list-sort algorithm do, is even worse, especially if your Scheme has a generational GC -- the writes will thrash the write-barrier. Sorting vectors has inherently better locality. This SRFI's destructive list merge and merge sort implementations are opportunistic -- they avoid redundant SET-CDR!s, and try to take long already-ordered runs of list structure as-is when doing the merges. Pure list merge sort Stable and fast -- O(n lg n) worst-case, and possibly O(n), depending upon the input list (see discussion above). Algorithm Stable? Worst case Average case In-place ------------------------------------------------------ Vector insert Yes O(n^2) O(n^2) Yes Vector quick No O(n^2) O(n lg n) Yes Vector heap No O(n lg n) O(n lg n) Yes Vector merge Yes O(n lg n) O(n lg n) No List merge Yes O(n lg n) O(n lg n) Either ------------------------------------------------------------------------------- * Porting and optimisation -------------------------- This package should be trivial to port. This code is tightly bummed, as far as I can go in portable Scheme. You could speed up the vector code a lot by error-checking the procedure parameters and then shifting over to fixnum-specific arithmetic and dangerous vector-indexing and vector-setting primitives. The comments in the code indicate where the initial error checks would have to be added. There are several (QUOTIENT N 2)'s that could be changed to a fixnum right-shift, as well, in both the list and vector code (SRFI 33 provides such an operator). The code is designed to enable this -- each file usually exports one or two "safe" procedures that end up calling an internal "dangerous" primitive. The little exported cover procedures are where you move the error checks. This should provide *big* speedups. In fact, all the code bumming I've done pretty much disappears in the noise unless you have a good compiler and also can dump the vector-index checks and generic arithmetic -- so I've really just set things up for you to exploit. The optional-arg parsing, defaulting, and error checking is done with a portable R4RS macro. But if your Scheme has a faster mechanism (e.g., Chez), you should definitely port over to it. Note that argument defaulting and error-checking are interleaved -- you don't have to error-check defaulted START/END args to see if they are fixnums that are legal vector indices for the corresponding vector, etc. ------------------------------------------------------------------------------- * References & Links -------------------- This document, in HTML: http://srfi.schemers.org/srfi-32/srfi-32.html [This link may not be valid while the SRFI is in draft form.] This document, in simple text format: http://srfi.schemers.org/srfi-32/srfi-32.txt Archive of SRFI-32 discussion-list email: http://srfi.schemers.org/srfi-32/mail-archive/maillist.html SRFI web site: http://srfi.schemers.org/ [CommonLisp] Common Lisp: the Language Guy L. Steele Jr. (editor). Digital Press, Maynard, Mass., second edition 1990. Available at http://www.elwood.com/alu/table/references.htm#cltl2 The Common Lisp "HyperSpec," produced by Kent Pitman, is essentially the ANSI spec for Common Lisp: http://www.xanalys.com/software_tools/reference/HyperSpec/ [R5RS] Revised^5 Report on the Algorithmic Language Scheme, R. Kelsey, W. Clinger, J. Rees (editors). Higher-Order and Symbolic Computation, Vol. 11, No. 1, September, 1998. and ACM SIGPLAN Notices, Vol. 33, No. 9, October, 1998. Available at http://www.schemers.org/Documents/Standards/ ------------------------------------------------------------------------------- * Acknowledgements ------------------ I thank the authors of the open source I consulted when designing this library, particularly Richard O'Keefe, Donovan Kolby and the MIT Scheme Team. ------------------------------------------------------------------------------- * Copyright ----------- ** SRFI text ============ This document is copyright (C) Olin Shivers (1998, 1999). All Rights Reserved. This document and translations of it may be copied and furnished to others, and derivative works that comment on or otherwise explain it or assist in its implementation may be prepared, copied, published and distributed, in whole or in part, without restriction of any kind, provided that the above copyright notice and this paragraph are included on all such copies and derivative works. However, this document itself may not be modified in any way, such as by removing the copyright notice or references to the Scheme Request For Implementation process or editors, except as needed for the purpose of developing SRFIs in which case the procedures for copyrights defined in the SRFI process must be followed, or as required to translate it into languages other than English. The limited permissions granted above are perpetual and will not be revoked by the authors or their successors or assigns. This document and the information contained herein is provided on an "AS IS" basis and THE AUTHORS AND THE SRFI EDITORS DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. ** Reference implementation =========================== Short summary: no restrictions. While I wrote all of this code myself, I read a lot of code before I began writing. However, all such code is, itself, either open source or public domain, rendering irrelevant any issue of "copyright taint." The natural merge sorts (pure list, destructive list, and vector) are not only my own code, but are implementations of an algorithm of my own devising. They run in O(n lg n) worst case, O(n) best case, and require only a logarithmic number of stack frames. And they are stable. And the destructive-list variant allocates zero cons cells; it simply rearranges the cells of the input list. Hence the reference implementation is Copyright (c) 1998 by Olin Shivers. and made available under the same copyright as the SRFI text (see above).