Problem B:Longest Ordered Subsequence
A numeric sequence of ai is ordered if a1 < a2 < ... < aN. Let the subsequence of the given numeric sequence (a1, a2, ..., aN) be any sequence (ai1, ai2, ..., aiK), where 1 <= i1 < i2 < ... < iK <= N. For example, sequence (1, 7, 3, 5, 9, 4, 8) has ordered subsequences, e. g., (1, 7), (3, 4, 8) and many others. All longest ordered subsequences are of length 4, e. g., (1, 3, 5, 8).
We address the problem of predicting a word from previous words in a sample of text. In particular,
we discuss n-gram models based on classes of words. We also discuss several statistical algorithms
for assigning words to classes based on the frequency of their co-occurrence with other words. We
find that we are able to extract classes that have the flavor of either syntactically based groupings
or semantically based groupings, depending on the nature of the underlying statistics.
The combinatorial core of the OVSF code assignment problem
that arises in UMTS is to assign some nodes of a complete binary
tree of height h (the code tree) to n simultaneous connections, such that
no two assigned nodes (codes) are on the same root-to-leaf path. Each
connection requires a code on a specified level. The code can change over
time as long as it is still on the same level. We consider the one-step code
assignment problem: Given an assignment, move the minimum number of
codes to serve a new request. Minn and Siu proposed the so-called DCAalgorithm
to solve the problem optimally. We show that DCA does not
always return an optimal solution, and that the problem is NP-hard.
We give an exact nO(h)-time algorithm, and a polynomial time greedy
algorithm that achieves approximation ratio Θ(h). Finally, we consider
the online code assignment problem for which we derive several results
