Matrix-chain multiplication
Abstract model
Matrix $A_i$ has dimensions $p_i\times p_{i+1}$. Every full parenthesization computes the same product, but a split at $k$ inside $A_i\cdots A_j$ costs the two subproblem optima plus $p_i p_{k+1}p_{j+1}$ scalar multiplications. Minimize this cost over all full parenthesizations. The verification case fills an interval-DP table declared inside the function.
Problem statement
Given the compatible dimension sequence $p_0,p_1,\ldots,p_n$, determine the minimum number of scalar multiplications needed to compute $A_0A_1\cdots A_{n-1}$. The matrices themselves need not be multiplied.
For example, dimensions 30 35 15 5 10 20 25 describe six matrices and have
optimal cost 15125.
Constraints
The verification case uses $1\le n\le8$ and $1\le p_i\le100$. It declares
cost[64] locally and uses its first $n^2$ cells. These bounds keep every
cost at most $7,000,000$.
Source
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms, 4th ed., MIT Press, 2022, Section 14.2 (Section 15.2 in the 3rd edition), "Matrix-chain multiplication."