CS252 Algorithms
Wednesday, 8 May 2024

+ Questions?

+ Today
- How I think about dynamic programming

+ "Programming" is not our usual programming
- Optimization problem solving techniques
- Linear programming, nonlinear, integer, ...
- Search space, an "objective function" that you
    want to maximize or minimize, and constraints

+ Fibonacci numbers
- recursion is the natural formulation
- recursion is trouble
- memoization: I remember what it means by thinking
    of Fibonacci

+ My general approach to DP problems
- State the problem
    - Inputs
    - Outputs
        - What are we optimizing?
- Try a couple small instances by hand to get a feel for the problem
- Search space
    - What is it?
    - How big is it? (and thus, how long will brute force take?)
- Pause for notation
    - S = search space
    - V : S --> R+ (a "value" for each element of the search space)
    - s_opt and v_opt = V(s_opt) -- the optimal value and
        an/the element s_opt in S that yields it
- Come up with a parameterized characterization of the problem of finding v_opt
    recursively. (This takes practice.) Here, we're trying to find just v_opt,
    not s_opt.
- Show a recursive decomposition of the parameterized problem; include
    base case(s). Here, you're expressing your full solution in terms of the
    solutions of smaller but similar problems.
- Note that solving by straight-up recursion will typically have
    the Fibonacci exponential runtime problem
- Put together a table and solve for v_opt from the bottom up
- Analyze the runtime of your algorithm
- Now that you can compute v_opt, can you enhance the algorithm
    to help find s_opt? (This is often called "back-tracing", and you have
    seen a similar issue when we looked at BFS/DFS and Dijkstra.)


+ Wire-cutting (also called rod-cutting)

- The problem
    - Given a price p[j] you can charge for a piece of
      wire of length j, for j = 1,...,m
    - Find the maximum revenue you can earn by cutting up
        (or not) a piece of wire of length n (which we assume is <= m)
    - Secondary goal: where should I make the cuts?

- Play around

    p[0...6] = 0  2  3  8  9  12  13

- Search space? Search space size?
- What are we optimizing? (v_opt & s_opt)

- Characterize problem as a function to be optimized
- Write a recursive decomposition of the function

- Build up subproblem solutions
    - we could do it top-down (memoized)
    - but we'll generally do it bottom-up (dynamic programming)

- Complexity?

- Backtracing
    - Retrieving the cuts that maximize revenue


+ Other problems
- Minimum edit distance (Levenshtein)
- Weighted interval scheduling
- Knapsack
-