Longest increasing subsequence

The Longest Increasing Subsequence problem is to find the longest increasing subsequence of a given sequence. It also reduces to a Graph Theory problem of finding the longest path in a Directed acyclic graph.

Overview
Formally, the problem is as follows:

Given a sequence $$a_1, a_2, \ldots, a_n$$, find the largest subset such that for every $$i < j$$, $$a_i < a_j$$.

Longest Common Subsequence
A simple way of finding the longest increasing subsequence is to use the Longest Common Subsequence (Dynamic Programming) algorithm.


 * 1) Make a sorted copy of the sequence $$A$$, denoted as $$B$$. $$O(n \lg(n) )$$ time.
 * 2) Use Longest Common Subsequence on with $$A$$ and $$B$$. $$O(n^2)$$ time.

Dynamic Programming
There is a straight-forward Dynamic Programming solution in $$O(n^2)$$ time. Though this is asymptotically equivalent to the Longest Common Subsequence version of the solution, the constant is lower, as there is less overhead.

Let A be our sequence $$a_1,a_2,\ldots,a_n$$. Define $$q_k$$ as the length of the longest increasing subsequence of A, subject to the constraint that the subsequence must end on the element $$a_k$$. The longest increasing subsequence of A must end on some element of A, so that we can find its length by searching for the maximum value of q. All that remains is to find out the values $$q_k$$.

But $$q_k$$ can be found recursively, as follows: consider the set $$S_k$$ of all $$i < k$$ such that $$a_i < a_k$$. If this set is null, then all of the elements that come before $$a_k$$ are greater than it, which forces $$q_k = 1$$. Otherwise, if $$S_k$$ is not null, then q has some distribution over $$S_k$$. By the general contract of q, if we maximize q over $$S_k$$, we get the length of the longest increasing subsequence in $$S_k$$; we can append $$a_k$$ to this sequence, to get that:


 * $$q_k = max(q_j | j \isin S_k) + 1$$

If the actual subsequence is desired, it can be found in $$O(n)$$ further steps by moving backward through the q-array, or else by implementing the q-array as a set of stacks, so that the above "+ 1" is accomplished by "pushing" $$a_k$$ into a copy of the maximum-length stack seen so far.

Some pseudo-code for finding the length of the longest increasing subsequence: function lis_length( a ) n := a.length q := new Array(n) for k from 1 to n:       max := 0; for j from 1 to k-1, if a[k] > a[j]: if q[j] > max, then set max = q[j]. q[k] := max + 1; max := 0 for i from 1 to k-1: if q[i] > max, then set max = q[i]. return max;

Faster Algorithm
There's also an $$O(n \log{n})$$ solution based on some observations. Let $$A_{i,j}$$ be the smallest possible tail out of all increasing subsequences of length $$j$$ using elements $$a_1, a_2, a_3, \ldots, a_i$$.

Observe that, for any particular $$i$$, $$A_{i,1} < A_{i,2} < \ldots < A_{i,j}$$. This suggests that if we want the longest subsequence that ends with $$a_{i+1}$$, we only need to look for a j such that $$A_{i,j} < a_{i+1} <= A_{i,j+1}$$ and the length will be $$j+1$$.

Notice that in this case, $$A_{i+1,j+1}$$ will be equal to $$a_{i+1}$$, and all $$A_{i+1, k}$$ will be equal to $$A_{i,k}$$ for $$k \ne j + 1$$.

Furthermore, there is at most one difference between the set $$A_i$$ and the set $$A_{i+1}$$, which is caused by this search.

Since $$A$$ is always ordered in increasing order, and the operation does not change this ordering, we can do a binary search for every single $$a_1, a_2, \ldots, a_n$$.

Implementation

 * C
 * C++ ($$O(n \log n)$$ algorithm - output sensitive - $$O(n \log k)$$)