edit distance

编辑距离算法

最新推荐文章于 2024-10-24 21:49:14 发布

原创最新推荐文章于 2024-10-24 21:49:14 发布 · 1.2k 阅读

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本文通过不同实现方式探讨了编辑距离算法，这是一种衡量两个字符串相似度的方法。从简单的动态规划到复杂的状态转移，逐步优化并详细解释了每一步的原理。

做毕业设计的缘故，写了几个edit distance 的小程序。

最常见的edit distance 是用动态规划实现的。所以先写一个矩阵。

class Matrix def initialize(m, n) @m, @n = m, n @ary = Array.new(@m*@n) end def []=(i, j, val) @ary[i*@n+j] = val end def [](i, j) @ary[i*@n+j] end end

但Hal Fulton 写了一个更精致的：

class Matrix def initialize @store = [[]] end def [](a,b) if @store[a]==nil || @store[a][b]==nil return nil else return @store[a][b] end end def []=(a,b,x) @store[a] = [[]] if @store[a]==nil @store[a][b] = x end end

=begin if xi = yj M(i, j) <- M(i-1, j-1) else M(i, j) <- 1+min(M(i-1, j-1), M(i-1, j), M(i, j-1)) end =end x = %w[a b c d e] y = %w[e d c b a] m, n = x.size, y.size M = Matrix.new(m+1, n+1) 0.upto(n) {|j| M[0, j] = j} 1.upto(m) {|i| M[i, 0] = i} 1.upto m do |i| 1.upto n do |j| if x[i-1].eql? y[j-1] M[i, j] = M[i-1, j-1] else M[i, j] = 1 + [M[i-1, j-1], M[i-1, j], M[i, j-1]].min end end end puts M[m, n] # => 4

改进版：

=begin if x[i] = y[j] M(i, j) <- M(i-1, j-1) elsif x[i-1] = y[j] and x[i] = y[j-1] # transposition M(i, j) <- M(i-2, j-2) + 1 # replacement M(i, j) <- M(i-1, j-1) + 1 # insertion M(i, j) <- M(i, j-1) + 1 # deletion M(i, j) <- M(i-1, j) + 1 else # replacement M(i, j) <- M(i-1, j-1) + 1 # insertion M(i, j) <- M(i, j-1) + 1 # deletion M(i, j) <- M(i-1, j) + 1 end =end x = %w[a b c d e] y = %w[e d c b a] m, n = x.size, y.size M = Matrix.new =begin M(-1, j) = M(i, -1) = max(m, n) M(0, j) = j M(i, 0) = i =end 0.upto(n) {|j| M[0, j] = j} 1.upto(m) {|i| M[i, 0] = i} 1.upto m do |i| 1.upto n do |j| if x[i-1] == y[j-1] M[i, j] = M[i-1, j-1] elsif (i >= 2 and j >= 2) and (x[i-2] == y[j-1] and x[i-1] == y[j-2]) M[i, j] = 1 + [M[i-2, j-2], M[i-1, j-1], M[i, j-1], M[i-1, j]].min else M[i, j] = 1 + [M[i-1, j-1], M[i, j-1], M[i-1, j]].min end end end puts M[m, n] # => 4

再改进：

=begin d[i, j] = min( d[i-1, j] + 1, d[i, j-1] + 1, d[i-1, j-1] + cost(A[i]->B[j]), d[i'-1, j'-1] + (i-i'-1) + (j-j'-1) + 1 ) =end x = %w[a b c d e] y = %w[e d c b a] m, n = x.size, y.size M = Matrix.new =begin M(-1, j) = M(i, -1) = max(m, n) M(0, j) = j M(i, 0) = i =end 0.upto(n) {|j| M[0, j] = j} 1.upto(m) {|i| M[i, 0] = i} 1.upto m do |i| 1.upto n do |j| if x[i-1] == y[j-1] M[i, j] = M[i-1, j-1] else min = [m, n].max 1.upto(i-1) do |p| 1.upto(j-1) do |q| if (x[i-1] == y[q-1]) and (x[p-1] == y[j-1]) d = M[p-1, q-1] + (i - p - 1) + (j - q - 1) min = d if min > d end end end M[i, j] = 1 + [min, M[i-1, j-1], M[i, j-1], M[i-1, j]].min end end end puts M[m, n] # => 4

最终版，需要一个四维数组：

class Array4 def initialize @store = [[[[]]]] end def [](a, b, c, d) if @store[a]==nil || @store[a][b]==nil || @store[a][b][c]==nil return nil else return @store[a][b][c][d] end end def []=(a, b, c, d, x) @store[a] = [[[]]] if @store[a]==nil @store[a][b] = [[]] if @store[a][b]==nil @store[a][b][c] = [] if @store[a][b][c]==nil @store[a][b][c][d] = x end end

x = %w[a b c d e] y = %w[e d c b a] m, n = x.size, y.size M = Array4.new upto m do |len| len.upto m do |i| 1.upto n do |j| j.downto 1 do |v| u = i-len+1 if u == i and v == j M[u, i, v, j] = (x[i-1] == y[j-1]) ? 1 : 0 elsif u == i and v != j M[u, i, v, j] = (x[i-1] == y[j-1]) ? j-v : 1 + [j-v, M[u, i, v, j-1]].min elsif u != i and v == j M[u, i, v, j] = (x[i-1] == y[j-1]) ? i-u : 1 + [i-u, M[u, i-1, v, j]].min else # if u != i and v != j if x[u-1] == y[j-1] and x[i-1] == y[v-1] # swap if i - u == 1 M[u, i, v, j] = j-v elsif j - v == 1 M[u, i, v, j] = i-u else M[u, i, v, j] = 1 + [M[u+1, i-1, v+1, j-1], M[u, i-1, v, j], M[u, i, v, j-1], M[u, i-1, v, j-1]].min end else M[u, i, v, j] = (x[i-1] == y[j-1]) ? M[u, i-1, v, j-1] : 1 + [M[u, i-1, v, j], M[u, i, v, j-1], M[u, i-1, v, j-1]].min end end end end end end puts M[1, m, 1, n] # => 2