Is this Longest Common Subsequence Correct?

问题:

I just wrote this implementation to find out the length of the longest increasing subsequence using dynamic programming. So for input as [10, 22, 9, 33, 21, 50, 41, 60, 80] the LIS is 6 and one of the set is [10, 22, 33, 50, 60, 80].

When I run the below code I get the correct answer as 6 with O(n) complexity. Is it correct?

def lis(a):
    dp_lis     = []
    curr_index = 0
    prev_index = 0

    for i in range(len(a)):
        prev_index = curr_index
        curr_index = i

        print 'if: %d < %d and %d < %d' % (prev_index, curr_index, a[prev_index], a[curr_index])
        if prev_index < curr_index and a[prev_index] < a[curr_index]:
            print '\tadd ELEMENT: ', a[curr_index]
            new_lis = 1 + max(dp_lis)
            dp_lis.append(new_lis)
        else:
            print '\telse ELEMENT: ', a[curr_index]
            dp_lis.append(1)

    print "DP LIST: ", dp_lis
    return max(dp_lis)

if __name__ == '__main__':
    a = [10, 22, 9, 33, 21, 50, 41, 60, 80]
    print lis(a)

回答1:

Use this correct, proven but inefficient implementation of the algorithm to check against your results - it's the standard recursive solution, it doesn't use dynamic programming:

def lis(nums):
    def max_length(i):
        if i == -1:
            return 0
        maxLen, curLen = 0, 0
        for j in xrange(i-1, -1, -1):
            if nums[j] < nums[i]:
                curLen = max_length(j)
                if curLen > maxLen:
                    maxLen = curLen
        return 1 + maxLen
    if not nums:
        return 0
    return max(max_length(x) for x in xrange(len(nums)))

Check to see if your_lis(nums) == my_lis(nums) for as many different-sized input lists with numbers as possible, they should be equal. At some point, for long lists my implementation will be far slower than yours.

As a further comparison point, here's my own optimized dynamic programming solution. It runs in O(n log k) time and O(n) space, returning the actual longest increasing subsequences it finds along the way:

def an_lis(nums):
    table, lis = lis_table(nums), []
    for i in xrange(len(table)):
        lis.append(nums[table[i]])
    return lis

def lis_table(nums):
    if not nums:
        return []
    table, preds = [0], [0] * len(nums)
    for i in xrange(1, len(nums)):
        if nums[table[-1]] < nums[i]:
            preds[i] = table[-1]
            table.append(i)
            continue
        minIdx, maxIdx = 0, len(table)-1
        while minIdx < maxIdx:
            mid = (minIdx + maxIdx) / 2
            if nums[table[mid]] < nums[i]:
                minIdx = mid + 1
            else:
                maxIdx = mid
        if nums[i] < nums[table[minIdx]]:
            if minIdx > 0:
                preds[i] = table[minIdx-1]
            table[minIdx] = i
    current, i = table[-1], len(table)
    while i:
        i -= 1
        table[i], current = current, preds[current]
    return table

回答2:

I implement dynamic programming algorithms fairly often.

I have found that the best way to check for correctness is to write a brute-force version of the algorithm and compare the output with the dynamic programming implementation on small examples.

If the output of the two versions agree, then you have reasonable confidence of correctness.