CS 411 Fall 2025  >  Outline for October 24, 2025

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CS 411 Fall 2025
Outline
for October 24, 2025

Outline

Dynamic Programming [L Ch 8 intro]

• Dynamic Programming
  • What It Is
    • Our last time-space trade-off, but worth a whole chapter.
    • Comes out of operations research; not originally a computing term; “programming” refers to planning.
    • Some problems have overlapping subproblems. So solving subproblems involves common work (sub-subproblems solved more than once).
    • To save time, solve each subproblem (or sub-subproblem, or ...) just once, and then save the answer for later use.
    • Originally formulated only in the context of optimization.
  • Variations
    • Bottom-up. The classical dynamic-programming approach. Solve subproblems from smallest to largest, saving results for each as we go.
    • Top-down. Do not solve subproblems before results are needed. Computation is driven by recursive calls.
      • Needs memory function: function that, if called a second time with same arguments, does not need to recompute, since it can look up saved results.
      • This technique is also called memoizing, particularly in the context of functional programming.
    • We look at bottom-up methods now and top-down methods later.
  • Optimization
    • In an optimization problem, we seek the best (largest, smallest, cheapest, etc.) example of something.
    • Feasible solution: something that fits our constraints. We need to be able to measure the “goodness” of a feasible solutions.
    • Objective function: a function that takes a feasible solutions and returns a number indicating how good it is.
    • Optimal solution: feasible solution with the best value of the objective function (largest, if we are doing maximization, or smallest, if we are doing minimization). Best value is the optimum.
    • Example: finding shortest path between two vertices \(x\), \(y\) in a graph. Feasible solutions are paths from \(x\) to \(y\). Objective function is length of path. We are doing minimization. Optimal solution is path with least length.
  • Principle of Optimality
    • To use DP on an optimization problem we need the Principle of Optimality to be true. This says that an optimal solution is composed of optimal solutions to subproblems.
    • Example 1. The principle holds for the above shortest-path problem. If an \(x,y\)-path passes through vertex \(v\), then the portion of the path from \(x\) to \(v\) is a shortest \(x,v\)-path, and similarly for the portion from \(v\) to \(y\).
    • Example 2. The principle does not holds for the longest-path problem.

Dynamic-Programming Examples [L 8.1]

• Coin Row
  • Problem: given row of coins, find greatest value formed by taking some of them, with the restriction that we can never take two consecutive coins.
  • Analysis: \(n\) is number of coins. Basic ops are integer operations.
  • Brute-force solution: recursive. Exponential time.
  • Bottom-up dynamic-programming solution: Find small solutions, work way up. Greatly increases speed: linear time.