Lecture 7

Longest common subsequence

Optimal binary search trees

Definition

Given a sequence \(K = (k_1, \cdots, k_n)\) of \(n\) distinct keys in sorted order, and \(n + 1\) dummy keys, \(d_0,\cdots d_n\) for values not in \(K\), we wish to build a binary search tree.

\(d_i\) represents all values between \(k_i\) and \(k_{i + 1}\)

For each key \(k_i\), we have a probability \(p_i\) that a serach will be for \(k_i\); and for each dummy key \(d_i\), we have a probability \(q_i\)

\[\sum_{i = 1}^n p_i + \sum_{i = 0}^{n} q_i = 1\]

Expected cost of a search in a binary search tree T

\[T = 1 + \sum_{i = 1}^n depth_T(k_i) \ p_i + \sum_{i = 0}^n depth_T(d_i) \ q_i\]

The optimal structure

Obs

The subtree of the optimal binary search tree is optimal.

The translation equation

Define \(e[i,j]\) as the expected cost of searching an optimal binary search tree.

Notions

\(w(i,j)\) the sum of probabilities:

\[w(i,j) = \sum_{k=i}^j p_k + \sum_{k = i-1}^{j} q_k\]

We have the translation equation

\[e(i, j) = e(i, r - 1) + e(r + 1, j) + w(i,j) \quad \text{for some } r \in (i, j)\]

Time complexity

This is a \(O(n^3)\) solution. A tiny optimization can make the complexity to \(O(n^2)\).

1974, A. V. Aho, J. E. Hopcroft, and J. D. Ullman

Greedy Algorithm

Activity-selection

Question

Suppose we have a set of \(n\) activities, activity \(a_i\) has a given start time \(s_i\) and a end time \(f_i\).

The Goal is to find the number of maximum-size of mutually compatible activities.

Theorem

Consider any \(S_k \neq \emptyset\), Let \(a_m\) be the activity in \(S_k\) with the earliest finish time.

Then \(a_m\) is used in some maximum-size subset of mutually compatible activities of \(S_k\)

Proof

Suppose that \(A_k\) is a maximum-size subset, construst a new optimal solution \(A_k'\) containing \(a_m\)