4. A certain string processing language allows the programmer to break a string into two pieces. It costs n units of time to break a string of n characters into two pieces, since this involves copying the old string. A programmer wants to break a string into many pieces, and the order in which the breaks are made can affect the total amount of time used. For example, suppose we wish to break a 20-character string after characters 3, 8, and 10. If the breaks are made in left-to-right order, then the first break costs 20 units of time, the second break costs 17 units of time, and the third break costs 12 units of time, for a total of 49 units. If the breaks are made in right-to-left order, the first break costs 20 units of time, the second break costs 10 units of time, and the third break costs 8 units of time, for a total of only 38 units. Give a dynamic programming algorithm that takes a list of character positions after which to break and determines the cheapest break cost in O(n³) time.

Solutions to a dynamic programming problem have (at minimum) three things: – A recurrence relation

– A brief description of what your recurrence function represents and what each case represents.

– A brief description of the memory element/storage and how it’s filled in.

– Always give complete solutions, not just examples.
– Always declare all your variables, in English. In particular, always describe the specific

- problem your algorithm is supposed to solve. – Never use weak induction.