本文介绍了一种求解最长单调递增子序列问题的动态规划算法，并给出了具体实现思路与步骤。通过定义状态转移方程，最终实现了时间复杂度为 O(n^2) 的解决方案。
Question:
Give an O(n squared)-time algorithm to find the longest monotonically increasing subsequence of a sequence of n numbers.
Answer:
A brute-force approach is enumerate all subsequences of the n numbers and find out a monotonically increasing one with the longest length. This algorithm has a poor exponential running time.
This problem exhibit optimal substructure and overlapping subproblems properties, and is suited for dynamic programming.
Let A denotes the array of n numbers, A[i] is the ith number of the array.
Let P{i,j} denotes problem of finding the longest monotonically increasing subsequence. So the original problem is P{1,n}.
Let S(i,j) denotes the length of longest subsequence of P{i,j}.
Let M(i,j) denotes the largest number in the longest subsequence of P{i,j}. Note for P{i,j}, there may exist several subsequence has the same longest length, M(i,j) should be the smallest one of them.
For P{1,j}, if A[j] is larger than M(1,j-1), then S(1,j) equals S(1,j-1) plus 1. Otherwise, it equals S(1,j-1).
So, we have:
S(1,j) = S(1,j-1) if A[j] <> M[1,j-1]
The complicated thing in this procedure is how to maintain M's value correctly. The idea is if A[j] is larger than M(1,j-1), M(1,j) should be A[j]. If A[j] is smaller than M(1,j-1), M(1,j) should either be M(1,j-1) or A[j] if A[j] is larger than M(1,x) where S(1,x) is less than S(1,j-1).
This equation yields a n squared running time algorithm.
Give an O(n squared)-time algorithm to find the longest monotonically increasing subsequence of a sequence of n numbers.
Answer:
A brute-force approach is enumerate all subsequences of the n numbers and find out a monotonically increasing one with the longest length. This algorithm has a poor exponential running time.
This problem exhibit optimal substructure and overlapping subproblems properties, and is suited for dynamic programming.
Let A denotes the array of n numbers, A[i] is the ith number of the array.
Let P{i,j} denotes problem of finding the longest monotonically increasing subsequence. So the original problem is P{1,n}.
Let S(i,j) denotes the length of longest subsequence of P{i,j}.
Let M(i,j) denotes the largest number in the longest subsequence of P{i,j}. Note for P{i,j}, there may exist several subsequence has the same longest length, M(i,j) should be the smallest one of them.
For P{1,j}, if A[j] is larger than M(1,j-1), then S(1,j) equals S(1,j-1) plus 1. Otherwise, it equals S(1,j-1).
So, we have:
S(1,j) = S(1,j-1) if A[j] <> M[1,j-1]
The complicated thing in this procedure is how to maintain M's value correctly. The idea is if A[j] is larger than M(1,j-1), M(1,j) should be A[j]. If A[j] is smaller than M(1,j-1), M(1,j) should either be M(1,j-1) or A[j] if A[j] is larger than M(1,x) where S(1,x) is less than S(1,j-1).
This equation yields a n squared running time algorithm.
Correction:
Having done some tests, the preceding algorithm failed for this case: "8 9 1 2 3 4".
The recursion can be performed another way. Let S(i) denotes the length of the longest subsequence that ended with item A(i). So, the relationship between a problem and its subproblem can be expressed as:
S(i) = max{ S(k)+1 } (k is between 0 and i -1) that all k satisfies A[k] is less than A[i]
And the final result is the largest one of array S.
Source code for this solution is here:
http://code.google.com/p/rxwen-blog-stuff/source/browse/trunk/algorithm/i2a_ex_15.4-5/ex15_4_5.cpp
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