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// Copyright 2017, The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE.md file.
// Package diff implements an algorithm for producing edit-scripts.
// The edit-script is a sequence of operations needed to transform one list
// of symbols into another (or vice-versa). The edits allowed are insertions,
// deletions, and modifications. The summation of all edits is called the
// Levenshtein distance as this problem is well-known in computer science.
//
// This package prioritizes performance over accuracy. That is, the run time
// is more important than obtaining a minimal Levenshtein distance.
package diff
// EditType represents a single operation within an edit-script.
type EditType uint8
const (
// Identity indicates that a symbol pair is identical in both list X and Y.
Identity EditType = iota
// UniqueX indicates that a symbol only exists in X and not Y.
UniqueX
// UniqueY indicates that a symbol only exists in Y and not X.
UniqueY
// Modified indicates that a symbol pair is a modification of each other.
Modified
)
// EditScript represents the series of differences between two lists.
type EditScript []EditType
// String returns a human-readable string representing the edit-script where
// Identity, UniqueX, UniqueY, and Modified are represented by the
// '.', 'X', 'Y', and 'M' characters, respectively.
func (es EditScript) String() string {
b := make([]byte, len(es))
for i, e := range es {
switch e {
case Identity:
b[i] = '.'
case UniqueX:
b[i] = 'X'
case UniqueY:
b[i] = 'Y'
case Modified:
b[i] = 'M'
default:
panic("invalid edit-type")
}
}
return string(b)
}
// stats returns a histogram of the number of each type of edit operation.
func (es EditScript) stats() (s struct{ NI, NX, NY, NM int }) {
for _, e := range es {
switch e {
case Identity:
s.NI++
case UniqueX:
s.NX++
case UniqueY:
s.NY++
case Modified:
s.NM++
default:
panic("invalid edit-type")
}
}
return
}
// Dist is the Levenshtein distance and is guaranteed to be 0 if and only if
// lists X and Y are equal.
func (es EditScript) Dist() int { return len(es) - es.stats().NI }
// LenX is the length of the X list.
func (es EditScript) LenX() int { return len(es) - es.stats().NY }
// LenY is the length of the Y list.
func (es EditScript) LenY() int { return len(es) - es.stats().NX }
// EqualFunc reports whether the symbols at indexes ix and iy are equal.
// When called by Difference, the index is guaranteed to be within nx and ny.
type EqualFunc func(ix int, iy int) Result
// Result is the result of comparison.
// NumSame is the number of sub-elements that are equal.
// NumDiff is the number of sub-elements that are not equal.
type Result struct{ NumSame, NumDiff int }
// BoolResult returns a Result that is either Equal or not Equal.
func BoolResult(b bool) Result {
if b {
return Result{NumSame: 1} // Equal, Similar
} else {
return Result{NumDiff: 2} // Not Equal, not Similar
}
}
// Equal indicates whether the symbols are equal. Two symbols are equal
// if and only if NumDiff == 0. If Equal, then they are also Similar.
func (r Result) Equal() bool { return r.NumDiff == 0 }
// Similar indicates whether two symbols are similar and may be represented
// by using the Modified type. As a special case, we consider binary comparisons
// (i.e., those that return Result{1, 0} or Result{0, 1}) to be similar.
//
// The exact ratio of NumSame to NumDiff to determine similarity may change.
func (r Result) Similar() bool {
// Use NumSame+1 to offset NumSame so that binary comparisons are similar.
return r.NumSame+1 >= r.NumDiff
}
// Difference reports whether two lists of lengths nx and ny are equal
// given the definition of equality provided as f.
//
// This function returns an edit-script, which is a sequence of operations
// needed to convert one list into the other. The following invariants for
// the edit-script are maintained:
// • eq == (es.Dist()==0)
// • nx == es.LenX()
// • ny == es.LenY()
//
// This algorithm is not guaranteed to be an optimal solution (i.e., one that
// produces an edit-script with a minimal Levenshtein distance). This algorithm
// favors performance over optimality. The exact output is not guaranteed to
// be stable and may change over time.
func Difference(nx, ny int, f EqualFunc) (es EditScript) {
// This algorithm is based on traversing what is known as an "edit-graph".
// See Figure 1 from "An O(ND) Difference Algorithm and Its Variations"
// by Eugene W. Myers. Since D can be as large as N itself, this is
// effectively O(N^2). Unlike the algorithm from that paper, we are not
// interested in the optimal path, but at least some "decent" path.
//
// For example, let X and Y be lists of symbols:
// X = [A B C A B B A]
// Y = [C B A B A C]
//
// The edit-graph can be drawn as the following:
// A B C A B B A
// ┌─────────────┐
// C │_|_|\|_|_|_|_│ 0
// B │_|\|_|_|\|\|_│ 1
// A │\|_|_|\|_|_|\│ 2
// B │_|\|_|_|\|\|_│ 3
// A │\|_|_|\|_|_|\│ 4
// C │ | |\| | | | │ 5
// └─────────────┘ 6
// 0 1 2 3 4 5 6 7
//
// List X is written along the horizontal axis, while list Y is written
// along the vertical axis. At any point on this grid, if the symbol in
// list X matches the corresponding symbol in list Y, then a '\' is drawn.
// The goal of any minimal edit-script algorithm is to find a path from the
// top-left corner to the bottom-right corner, while traveling through the
// fewest horizontal or vertical edges.
// A horizontal edge is equivalent to inserting a symbol from list X.
// A vertical edge is equivalent to inserting a symbol from list Y.
// A diagonal edge is equivalent to a matching symbol between both X and Y.
// Invariants:
// • 0 ≤ fwdPath.X ≤ (fwdFrontier.X, revFrontier.X) ≤ revPath.X ≤ nx
// • 0 ≤ fwdPath.Y ≤ (fwdFrontier.Y, revFrontier.Y) ≤ revPath.Y ≤ ny
//
// In general:
// • fwdFrontier.X < revFrontier.X
// • fwdFrontier.Y < revFrontier.Y
// Unless, it is time for the algorithm to terminate.
fwdPath := path{+1, point{0, 0}, make(EditScript, 0, (nx+ny)/2)}
revPath := path{-1, point{nx, ny}, make(EditScript, 0)}
fwdFrontier := fwdPath.point // Forward search frontier
revFrontier := revPath.point // Reverse search frontier
// Search budget bounds the cost of searching for better paths.
// The longest sequence of non-matching symbols that can be tolerated is
// approximately the square-root of the search budget.
searchBudget := 4 * (nx + ny) // O(n)
// The algorithm below is a greedy, meet-in-the-middle algorithm for
// computing sub-optimal edit-scripts between two lists.
//
// The algorithm is approximately as follows:
// • Searching for differences switches back-and-forth between
// a search that starts at the beginning (the top-left corner), and
// a search that starts at the end (the bottom-right corner). The goal of
// the search is connect with the search from the opposite corner.
// • As we search, we build a path in a greedy manner, where the first
// match seen is added to the path (this is sub-optimal, but provides a
// decent result in practice). When matches are found, we try the next pair
// of symbols in the lists and follow all matches as far as possible.
// • When searching for matches, we search along a diagonal going through
// through the "frontier" point. If no matches are found, we advance the
// frontier towards the opposite corner.
// • This algorithm terminates when either the X coordinates or the
// Y coordinates of the forward and reverse frontier points ever intersect.
//
// This algorithm is correct even if searching only in the forward direction
// or in the reverse direction. We do both because it is commonly observed
// that two lists commonly differ because elements were added to the front
// or end of the other list.
//
// Running the tests with the "cmp_debug" build tag prints a visualization
// of the algorithm running in real-time. This is educational for
// understanding how the algorithm works. See debug_enable.go.
f = debug.Begin(nx, ny, f, &fwdPath.es, &revPath.es)
for {
// Forward search from the beginning.
if fwdFrontier.X >= revFrontier.X || fwdFrontier.Y >= revFrontier.Y || searchBudget == 0 {
break
}
for stop1, stop2, i := false, false, 0; !(stop1 && stop2) && searchBudget > 0; i++ {
// Search in a diagonal pattern for a match.
z := zigzag(i)
p := point{fwdFrontier.X + z, fwdFrontier.Y - z}
switch {
case p.X >= revPath.X || p.Y < fwdPath.Y:
stop1 = true // Hit top-right corner
case p.Y >= revPath.Y || p.X < fwdPath.X:
stop2 = true // Hit bottom-left corner
case f(p.X, p.Y).Equal():
// Match found, so connect the path to this point.
fwdPath.connect(p, f)
fwdPath.append(Identity)
// Follow sequence of matches as far as possible.
for fwdPath.X < revPath.X && fwdPath.Y < revPath.Y {
if !f(fwdPath.X, fwdPath.Y).Equal() {
break
}
fwdPath.append(Identity)
}
fwdFrontier = fwdPath.point
stop1, stop2 = true, true
default:
searchBudget-- // Match not found
}
debug.Update()
}
// Advance the frontier towards reverse point.
if revPath.X-fwdFrontier.X >= revPath.Y-fwdFrontier.Y {
fwdFrontier.X++
} else {
fwdFrontier.Y++
}
// Reverse search from the end.
if fwdFrontier.X >= revFrontier.X || fwdFrontier.Y >= revFrontier.Y || searchBudget == 0 {
break
}
for stop1, stop2, i := false, false, 0; !(stop1 && stop2) && searchBudget > 0; i++ {
// Search in a diagonal pattern for a match.
z := zigzag(i)
p := point{revFrontier.X - z, revFrontier.Y + z}
switch {
case fwdPath.X >= p.X || revPath.Y < p.Y:
stop1 = true // Hit bottom-left corner
case fwdPath.Y >= p.Y || revPath.X < p.X:
stop2 = true // Hit top-right corner
case f(p.X-1, p.Y-1).Equal():
// Match found, so connect the path to this point.
revPath.connect(p, f)
revPath.append(Identity)
// Follow sequence of matches as far as possible.
for fwdPath.X < revPath.X && fwdPath.Y < revPath.Y {
if !f(revPath.X-1, revPath.Y-1).Equal() {
break
}
revPath.append(Identity)
}
revFrontier = revPath.point
stop1, stop2 = true, true
default:
searchBudget-- // Match not found
}
debug.Update()
}
// Advance the frontier towards forward point.
if revFrontier.X-fwdPath.X >= revFrontier.Y-fwdPath.Y {
revFrontier.X--
} else {
revFrontier.Y--
}
}
// Join the forward and reverse paths and then append the reverse path.
fwdPath.connect(revPath.point, f)
for i := len(revPath.es) - 1; i >= 0; i-- {
t := revPath.es[i]
revPath.es = revPath.es[:i]
fwdPath.append(t)
}
debug.Finish()
return fwdPath.es
}
type path struct {
dir int // +1 if forward, -1 if reverse
point // Leading point of the EditScript path
es EditScript
}
// connect appends any necessary Identity, Modified, UniqueX, or UniqueY types
// to the edit-script to connect p.point to dst.
func (p *path) connect(dst point, f EqualFunc) {
if p.dir > 0 {
// Connect in forward direction.
for dst.X > p.X && dst.Y > p.Y {
switch r := f(p.X, p.Y); {
case r.Equal():
p.append(Identity)
case r.Similar():
p.append(Modified)
case dst.X-p.X >= dst.Y-p.Y:
p.append(UniqueX)
default:
p.append(UniqueY)
}
}
for dst.X > p.X {
p.append(UniqueX)
}
for dst.Y > p.Y {
p.append(UniqueY)
}
} else {
// Connect in reverse direction.
for p.X > dst.X && p.Y > dst.Y {
switch r := f(p.X-1, p.Y-1); {
case r.Equal():
p.append(Identity)
case r.Similar():
p.append(Modified)
case p.Y-dst.Y >= p.X-dst.X:
p.append(UniqueY)
default:
p.append(UniqueX)
}
}
for p.X > dst.X {
p.append(UniqueX)
}
for p.Y > dst.Y {
p.append(UniqueY)
}
}
}
func (p *path) append(t EditType) {
p.es = append(p.es, t)
switch t {
case Identity, Modified:
p.add(p.dir, p.dir)
case UniqueX:
p.add(p.dir, 0)
case UniqueY:
p.add(0, p.dir)
}
debug.Update()
}
type point struct{ X, Y int }
func (p *point) add(dx, dy int) { p.X += dx; p.Y += dy }
// zigzag maps a consecutive sequence of integers to a zig-zag sequence.
// [0 1 2 3 4 5 ...] => [0 -1 +1 -2 +2 ...]
func zigzag(x int) int {
if x&1 != 0 {
x = ^x
}
return x >> 1
}