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Unified Diff: src/pkg/sort/sort.go

Issue 9612044: code review 9612044: sort: provide different stable sort algorithms (Closed)
Patch Set: diff -r 41134e67106d https://code.google.com/p/go/ Created 10 years, 9 months ago
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Index: src/pkg/sort/sort.go
===================================================================
--- a/src/pkg/sort/sort.go
+++ b/src/pkg/sort/sort.go
@@ -283,3 +283,192 @@
// StringsAreSorted tests whether a slice of strings is sorted in increasing order.
func StringsAreSorted(a []string) bool { return IsSorted(StringSlice(a)) }
+
+// Notes on stable sorting:
+// The used algorithms are simple and provable correct on all input and use
+// only logarithmic additional stack space. They perform well if compared
+// experimentaly to other stable in-place sorting algorithms.
+//
+// Remarks on other algoritms evaluated:
+// - GCC's 4.6.3 stable_sort with merge_without_buffer from libstdc++:
+// Not faster.
+// - GCC's __rotate for block rotations: Not faster.
+// - "Practical in-place mergesort" from Jyrki Katajainen, Tomi A. Pasanen
+// and Jukka Teuhola; Nordic Journal of Computing 3,1 (1996), 27-40:
+// The given algorithms are in-place, number of Swap and Assignments
+// grow as n log n but the algorithm is not stable.
+// - "Fast Stable In-Plcae Sorting with O(n) Data Moves" J.I. Munro and
+// V. Raman in Algorithmica (1996) 16, 115-160:
+// This algorithm either needs additional 2n bits or works only if there
+// are enough different elements available to encode some permutations
+// which have to be undone later (so not stable an any input).
+// - All the optimal in-place sorting/merging algorithms I found are either
+// unstable or rely on enough different elements in each step to encode the
+// performed block rearrangements. See also "In-Place Merging Algorithms",
+// Denham Coates-Evely, Department of Computer Science, Kings College,
+// January 2004 and the reverences in there.
+// - Often "optimal" algorithms are optimal in the number of assignments
+// but Interface has only Swap as operation.
+
+// Stable sorts data while keeping the original order of equal elements.
+//
+// It makes one call to data.Len to determine n, O(n*log(n)) calls to
+// data.Less and O(n*log(n)*log(n)) calls to data.Swap.
+func Stable(data Interface) {
+ n := data.Len()
+ blockSize := 20
+ a, b := 0, blockSize
+ for b <= n {
+ insertionSort(data, a, b)
+ a = b
+ b += blockSize
+ }
+ insertionSort(data, a, n)
+
+ for blockSize < n {
+ a, b = 0, 2*blockSize
+ for b <= n {
+ symMerge(data, a, a+blockSize, b)
+ a = b
+ b += 2 * blockSize
+ }
+ symMerge(data, a, a+blockSize, n)
+ blockSize *= 2
+ }
+}
+
+// SymMerge merges the two sorted subsequences data[a:m] and data[m:b] using
+// the SymMerge algorithm from Pok-Son Kim and Arne Kutzner, "Stable Minimum
+// Storage Merging by Symmetric Comparisons", in Susanne Albers and Tomasz
+// Radzik, editors, Algorithms - ESA 2004, volume 3221 of Lecture Notes in
+// Computer Science, pages 714-723. Springer, 2004.
+//
+// Let M = m-a and N = b-n. Wolog M < N.
+// The recursion depth is bound by ceil(log(N+M)).
+// The algorithm needs O(M*log(N/M + 1)) calls to data.Less.
+// The algorithm needs O((M+N)*log(M)) calls to data.Swap.
+//
+// The paper gives O((M+N)*log(M)) as the number of assignments assuming a
+// rotation algorithm wich uses O(M+N+gcd(M+N)) assignments. The argumentation
+// in the paper carries through for Swap operations, especially as the block
+// swapping rotate uses only O(M+N) Swaps.
+func symMerge(data Interface, a, m, b int) {
+ if a >= m || m >= b {
+ return
+ }
+
+ mid := a + (b-a)/2
+ n := mid + m
+ start := 0
+ if m > mid {
+ start = n - b
+ r, p := mid, n-1
+ for start < r {
+ c := start + (r-start)/2
+ if !data.Less(p-c, c) {
+ start = c + 1
+ } else {
+ r = c
+ }
+ }
+ } else {
+ start = a
+ r, p := m, n-1
+ for start < r {
+ c := start + (r-start)/2
+ if !data.Less(p-c, c) {
+ start = c + 1
+ } else {
+ r = c
+ }
+ }
+ }
+ end := n - start
+ rotate(data, start, m, end)
+ symMerge(data, a, start, mid)
+ symMerge(data, mid, end, b)
+}
+
+// Rotate two consecutives blocks u = data[a:m] and v = data[m:b] in data:
+// Data of the form 'x u v y' is changed to 'x v u y'.
+// Rotate performs at most b-a many calls to data.Swap.
+func rotate(data Interface, a, m, b int) {
+ i := m - a
+ if i == 0 {
+ return
+ }
+ j := b - m
+ if j == 0 {
+ return
+ }
+
+ if i == j {
+ swapRange(data, a, m, i)
+ return
+ }
+
+ p := a + i
+ for i != j {
+ if i > j {
+ swapRange(data, p-i, p, j)
+ i -= j
+ } else {
+ swapRange(data, p-i, p+j-i, i)
+ j -= i
+ }
+ }
+ swapRange(data, p-i, p, i)
+}
+
+/*
+Complexity of Stable Sorting
+
+
+Complexity of block swapping rotation
+
+Each Swap puts one new element into its correct, final position.
+Elements which reach their final position are no longer moved.
+Thus block swapping rotation needs |u|+|v| calls to Swaps.
+This is best possible as each element might need a move.
+
+Pay attention when comparing to other optimal algorithms which
+typically count the number of assignments instead of swaps:
+E.g. the optimal algorithm of Dudzinski and Dydek for in-place
+rotations uses O(u + v + gcd(u,v)) assignments which is
+better than our O(3 * (u+v)) as gcd(u,v) <= u.
+
+
+Stable sorting by SymMerge and BlockSwap rotations
+
+SymMerg complexity for same size input M = N:
+Calls to Less: O(M*log(N/M+1)) = O(N*log(2)) = O(N)
+Calls to Swap: O((M+N)*log(M)) = O(2*N*log(N)) = O(N*log(N))
+
+(The following argument does not fuzz over a missing -1 or
+other stuff which does not impact the final result).
+
+Let n = data.Len(). Assume n = 2^k.
+
+Plain merge sort performs log(n) = k iterations.
+On iteration i the algorithm merges 2^(k-i) blocks, each of size 2^i.
+
+Thus iteration i of merge sort performs:
+Calls to Less O(2^(k-i) * 2^i) = O(2^k) = O(2^log(n)) = O(n)
+Calls to Swap O(2^(k-i) * 2^i * log(2^i)) = O(2^k * i) = O(n*i)
+
+In total k = log(n) iterations are performed; so in total:
+Calls to Less O(log(n) * n)
+Calls to Swap O(n + 2*n + 3*n + ... + (k-1)*n + k*n)
+ = O((k/2) * k * n) = O(n * k^2) = O(n * log^2(n))
+
+
+Above results should generalize to arbitrary n = 2^k + p
+and should not be influenced by the initial insertion sort phase:
+Insertion sort is O(n^2) on Swap and Less, thus O(bs^2) per block of
+size bs at n/bs blocks: O(bs*n) Swaps and Less during insertion sort.
+Merge sort iterations start at i = log(bs). With t = log(bs) constant:
+Calls to Less O((log(n)-t) * n + bs*n) = O(log(n)*n + (bs-t)*n)
+ = O(n * log(n))
+Calls to Swap O(n * log^2(n) - (t^2+t)/2*n) = O(n * log^2(n))
+
+*/
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