Space Efficient LCS
by Isai Damier, Android Engineer @ Google

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#=======================================================================
# Author: Isai Damier
# Title: Longest Common Subsequence
# Project: geekviewpoint
# Package: algorithms
#
# Statement:
# Given two sequences, find the longest common subsequence
# between them.
#
# Time Complexity: O(n*m)
#  where m and n are the size of the input lists, respectively
#
# Sample Input: awaited, alpine
# Sample Output: aie
#
# DEFINITION OF SUBSEQUENCE:
# A sequence is a particular order in which related objects follow
# each other (e.g. DNA, Fibonacci). A sub-sequence is a sequence
# obtained by omitting some of the elements of a larger sequence.
#
# SEQUENCE       SUBSEQUENCE     OMISSION
# [3,1,2,5,4]     [1,2]            3,5,4
# [3,1,2,5,4]     [3,1,4]          2,5
#
# SEQUENCE       NOT SUBSEQUENCE   REASON
# [3,1,2,5,4]     [4,2,5]           4 should follow 5
#
# STRATEGY:
# An easy way to find a longest common subsequence of characters between
# two words is to first track the lengths of all the common sequences
# and then from those lengths pick a maximum; finally, from that
# maximum length, trace out the actual longest common subsequence
# between the two words.
#
# Let's use a table C (i.e. list C) to track the lengths of all common
# sub-sequences between the two input words. Naturally, C starts with
# all cells set to zero. Then to fill up C, we walk thru the two words,
# comparing characters. Each time we find a match, we mark the
# corresponding cell in C as one plus whatever maximum we had seen so
# far for that particular sub-sequence.
#
# When we are done filling up C, we find a maximum length z in C. Then
# using the indices of z, we trace that actual sequence and return it
# as the answer.
#
# The ensuing code shows the details of how we fill C and then from C
# extract the actual LCS.
#
#        ~~~~    ~~~~    ~~~~    ~~~~    ~~~~    ~~~~    ~~~~
#
# The remaining paragraphs contain details that you may not particularly
# care for. You may skip them and go right to the code.
#
# Finding the LCS of two sequences X and Y, i.e. LCS(X,Y), is similar
# in structure to finding the Fibonacci of some number n, i.e. fib(n).
# Both problems exhibit optimal substructures. Which is to say,
# for instance, the solution to fib(5) contains the solution to fib(4)
# which in turn contains fib(3), and so on. Similarly, the solution
# of LCS(X,Y) contains the solution of LCS(X-1,Y), which in turn
# contains the solution of LCS(X-1,Y-1), and so on. Whenever a problem
# exhibits "optimal substructure", we can use recursion to solve it:
#
#             _
#            | fib(n-1)+fib(n-2)    if n > 1
#   fib(n) = | n                    if n == 1 or n == 0
#            |_
#
# For the LCS recurrence relation, a few notes are necessary:
# Let C(i,j) be the length of the prefix of LCS(X,Y) given by
#
#   C(i,j) = |LCS(X[1..i],Y[1..j])|
#
# so that if the length of X is m and the length of Y is n,
#
#   C(m,n) = |LCS(X,Y)|
#
# Then the recurrence relation for filling C is
#             _
#            | C(i-1,j-1)+1            if X(i) == Y(j)
#   C(i,j) = | max[C(i-1,j),C(i,j-1)]  otherwise
#            |_
#
# In addition to "optimal substructures", the LCS(X,Y) like the fib(n)
# contains overlapping subproblems.  What we mean is this: In solving
# f(5), we meet f(2) both as a subproblem of f(4) and as a subproblem
# of f(3).
#
#             fib(5)
#           /        \
#       fib(4)        fib(3)       :fib(5)=fib(4)+fib(3)
#       /    \       /     \
#   fib(3) fib(2)  fib(2)  fib(1)  :fib(5)=fib(3)+fib(2)+fib(2)+fib(1)
#
# Whenever a problem exhibits "overlapping subproblems", if you use
# recursion as a strategy you will end up solving the same subproblems
# more than once. Therefore, it makes more sense to track subproblems
# using a table so that once you solve a subproblem, you can simply
# lookup it's solution next time you need.
#
# CONCLUSION:
# In the foregoing "strategy" section, we essentially described
# dynamic programming.
#
# Dynamic Programming: (aka Dynamic Tabling) is a strategy for solving
#   problems, which exhibit optimal substructures and overlapping
#   subproblems, by using a table to track the solution of subproblems
#   that have already been solved so as not to solve a subproblem more
#   than once. This tracking or note taking is usually referred to as
#   memoization, meaning writing in a memo.
#=======================================================================
  
 
#=======================================================================
# Space Complexity: O(min(n,m))
#  where m and n are the size of the input lists, respectively.
#
# There are a number of reduced space LCS algorithms. This is just
# one approach.
#=======================================================================
def spaceEfficientLCS_length( A, B ):
    if len( A ) > len( B ):
      return LCS_length( A, B )
    return LCS_length( B, A );
 
 
def LCS_length( B, A ):
    m, n = len( A ) , len( B )
    T = [[0 for y in range( n + 1 )] for x in range( 2 )]
 
    for i in range( m - 1, -1, -1 ):
      ndx = i & 1; # odd or even
      for j in range( n - 1, -1, -1 ):
        if A[i] == B[j]:
          T[ndx][j] = 1 + T[1 - ndx][j + 1]
        else:
          T[ndx][j] = max( T[1 - ndx][j], T[ndx][j + 1] )
 
    return T[0][0]