Advanced Design and Analysis Techniques Part 1 15.1 and 15.2

Techniques -1 This part covers three important techniques for the design and analysis of efficient algorithms: dynamic programming (Chapter 15), greedy algorithms (Chapter 16),and amortized analysis (Chapter 17).

Techniques - 2 Earlier parts have presented other widely applicable techniques, such as divide-and-conquer, randomization, and the solution of recurrences.

Dynamic programming Dynamic programming typically applies to optimization problems in which a set of choices must be made in order to arrive at an optimal solution. Dynamic programming is effective when a given subproblem may arise from more than one partial set of choices; the key technique is to store the solution to each such subproblem in case it should reappear.

Greedy algorithms Like dynamic-programming algorithms, greedy algorithms typically apply to optimization problems in which a set of choices must be made in order to arrive at an optimal solution. The idea of a greedy algorithm is to make each choice in a locally optimal manner.

Dynamic programming -1 Dynamic programming, like the divide-and-conquer method, solves problems by combining the solutions to subproblems. Divide and- conquer algorithms partition the problem into independent subproblems, solve the subproblems recursively, and then combine their solutions to solve the original problem.

Dynamic programming -2 Dynamic programming is applicable when the subproblems are not independent, that is, when subproblems share subsubproblems. A dynamic-programming algorithm solves every subsubproblem just once and then saves its answer in a table, thereby avoiding the work of recomputing the answer every time the subsubproblem is encountered.

Dynamic programming -2 Dynamic programming is typically applied to optimization problems. In such problems there can be many possible solutions. Each solution has a value, and we wish to find a solution with the optimal (minimum or maximum) value. We call such a solution an optimal solution to the problem, as opposed to the optimal solution, since there may be several solutions that achieve the optimal value.

The development of a dynamic-programming algorithm The development of a dynamic-programming algorithm can be broken into a sequence of four steps. 1. Characterize the structure of an optimal solution. 2. Recursively define the value of an optimal solution. 3. Compute the value of an optimal solution in a bottom-up fashion. 4. Construct an optimal solution from computed information.

Assembly-line scheduling

Step 1: The structure of the fastest way through the factory

Step 2: A recursive solution

Step 3: Computing the fastest times

Step 4: Constructing the fastest way through the factory

Matrix-chain multiplication We can multiply two matrices A and B only if they are compatible: the number of columns of A must equal the number of rows of B. If A is a p × q matrix and B is a q × r matrix, the resulting matrix C is a p × r matrix

Counting the number of parenthesizations

Step 1: The structure of an optimal parenthesization Step 2: A recursive solution Step 3: Computing the optimal costs

Step 3: Computing the optimal costs

Step 4: Constructing an optimal solution