Lecture 6

Dynamic Programming

最大字段和

Divide and Conquer

\[O(n \log n)\]

Dynamic Programming

• State: The max subsequence sum end at index \(i\), which is called \(f(i)\)
• Transition:
  • \(f(i) = \max (A[i], f(i - 1) + A[i])\)
• Solution:
  • \(\max_{i \in \{ 1, \cdots N\}} f(i)\)

Question

How to get the construction of answer?

Overview

• Dynamic Programming vs Divide and Conquer

  • Sililarities
    • partition
    • combining
  • Differences
    • overlapping subproblems vs no overlapping subproblems
    • DP is typical applied to optimization problems

• Steps of DP

  • Characterize the structure of an optimal solution
  • Recursively define the value of an optimal solution
    • Top Down Approach
  • Compute the value of an optimal solution in a bottom-up fashion
  • Construct an optimal solution from computed information

Rod cutting

Question

Given a rod of length \(n\) inches and a table of prices \(p_i\) for \(i = 1, 2, \cdots ,n\), determine the maximum revenue \(r_n\) obtainable by cutting up the rod and selling the pieces.

Solutions

• Top-down with memoization
• Bottom-up solution

Matrix-chain multiplication

Elements of DP

How to discover optimal substructure

• Make a choice to split the problem into one or more subproblems
• Just assume you are given the choice that leads to an optimal solution
• Given this choice, try to best characterize the resulting space of subproblems
• Show the subproblems chosen are optimal by using a "cut-and-paste" technique

Independence of subproblems

Unweighted longest simple path

• The subproblems in finding the longest simple path are not independent
• independent: The solution to one subproblem does not affect the solution to another subproblem of the same problem.