Fundamentals of Dynamic Programming: Concepts, Strategies, and Implementation Steps

1. Basic Concepts Dynamic programming is a process where each decision depends on the current state and leads to a state transition; a sequence of decisions generated in changing states constitutes the multi‑stage optimization process known as dynamic programming.

2. Basic Idea and Strategy Similar to divide-and-conquer, the problem is decomposed into subproblems (stages) solved sequentially; the solution of a previous subproblem provides useful information for the next. Overlapping subproblems are stored in a two‑dimensional array to avoid recomputation. Unlike divide-and-conquer, subproblems are not independent.

3. Situations Where DP Applies A problem is suitable for DP when it satisfies three properties: optimal substructure, no after‑effect (the decision at a stage does not affect previous stages), and overlapping subproblems (though the last is not strictly required).

4. Basic Steps to Solve a DP Problem The typical workflow includes: (1) divide the problem into ordered stages; (2) define states and state variables respecting the no‑after‑effect property; (3) determine decisions and write the state‑transition equations; (4) identify boundary conditions; (5) optionally follow a simplified design process of analyzing optimal‑solution properties, defining recursion, applying memoization (top‑down or bottom‑up), and constructing the optimal solution from computed information.

5. Implementation Remarks The main difficulty lies in the theoretical design of the four steps; once designed, implementation is straightforward. The three essential DP elements are: problem stages, state for each stage, and the recurrence relation linking consecutive stages.

6. DP Algorithm Framework The algorithm can be expressed in pseudo‑code as follows:

for (j=1; j<=m; j=j+1) // first stage
    xn[j] = initial value;

for (i=n-1; i>=1; i=i-1) // other n-1 stages
    for (j=1; j>=f(i); j=j+1) // f(i) depends on i
        xi[j] = max (or min){ g(xi-1[j1:j2]), ..., g(xi-1[jk:jk+1]) };

t = g(x1[j1:j2]); // compute overall optimal solution from sub‑solutions
print(x1[j1]);

for (i=2; i<=n-1; i=i+1) {
    t = t - xi-1[ji];
    for (j=1; j>=f(i); j=j+1)
        if (t == xi[ji])
            break;
}