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# Read e-book Dynamics of Algorithms

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## Introduction to Dynamic Programming 1 Tutorials & Notes | Algorithms | HackerEarth

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### Robot dynamics algorithms

Links to the MAPLE implementation of the dynamic programming approach may be found among the external links. Let us say there was a checker that could start at any square on the first rank i. That is, a checker on 1,3 can move to 2,2 , 2,3 or 2,4. This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions to subproblems. Let us define a function q i, j as.

Starting at rank n and descending to rank 1 , we compute the value of this function for all the squares at each successive rank. Picking the square that holds the minimum value at each rank gives us the shortest path between rank n and rank 1. The function q i, j is equal to the minimum cost to get to any of the three squares below it since those are the only squares that can reach it plus c i, j.

For instance:. The first line of this equation deals with a board modeled as squares indexed on 1 at the lowest bound and n at the highest bound. The second line specifies what happens at the last rank; providing a base case. The third line, the recursion, is the important part. It represents the A,B,C,D terms in the example. In the following pseudocode, n is the size of the board, c i, j is the cost function, and min returns the minimum of a number of values:.

This function only computes the path cost, not the actual path.

## Rigid Body Dynamics Algorithms

We discuss the actual path below. This, like the Fibonacci-numbers example, is horribly slow because it too exhibits the overlapping sub-problems attribute. That is, it recomputes the same path costs over and over. However, we can compute it much faster in a bottom-up fashion if we store path costs in a two-dimensional array q[i, j] rather than using a function. This avoids recomputation; all the values needed for array q[i, j] are computed ahead of time only once. Precomputed values for i,j are simply looked-up whenever needed. We also need to know what the actual shortest path is. To do this, we use another array p[i, j] ; a predecessor array.

This array records the path to any square s. The predecessor of s is modeled as an offset relative to the index in q[i, j] of the precomputed path cost of s. To reconstruct the complete path, we lookup the predecessor of s , then the predecessor of that square, then the predecessor of that square, and so on recursively, until we reach the starting square. Consider the following code:. In genetics , sequence alignment is an important application where dynamic programming is essential. Each operation has an associated cost, and the goal is to find the sequence of edits with the lowest total cost.

The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:. The partial alignments can be tabulated in a matrix, where cell i,j contains the cost of the optimal alignment of A[ The cost in cell i,j can be calculated by adding the cost of the relevant operations to the cost of its neighboring cells, and selecting the optimum. Different variants exist, see Smith—Waterman algorithm and Needleman—Wunsch algorithm. The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod.

The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:. The dynamic programming solution consists of solving the functional equation.

Then it can be shown that . An interactive online facility is available for experimentation with this model as well as with other versions of this puzzle e. However, there is an even faster solution that involves a different parametrization of the problem:. Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices.

Therefore, our task is to multiply matrices A 1 , A 2 ,. As we know from basic linear algebra, matrix multiplication is not commutative, but is associative; and we can multiply only two matrices at a time. So, we can multiply this chain of matrices in many different ways, for example:. There are numerous ways to multiply this chain of matrices. They will all produce the same final result, however they will take more or less time to compute, based on which particular matrices are multiplied.

Real-time optimization algorithms for dynamic walking, running, and manipulating robots

For example, let us multiply matrices A, B and C. Obviously, the second way is faster, and we should multiply the matrices using that arrangement of parenthesis.

Therefore, our conclusion is that the order of parenthesis matters, and that our task is to find the optimal order of parenthesis. At this point, we have several choices, one of which is to design a dynamic programming algorithm that will split the problem into overlapping problems and calculate the optimal arrangement of parenthesis.

The dynamic programming solution is presented below. Let's call m[i,j] the minimum number of scalar multiplications needed to multiply a chain of matrices from matrix i to matrix j i. This formula can be coded as shown below, where input parameter "chain" is the chain of matrices, i. So far, we have calculated values for all possible m [ i , j ] , the minimum number of calculations to multiply a chain from matrix i to matrix j , and we have recorded the corresponding "split point" s [ i , j ]. This algorithm will produce "tables" m [, ] and s [, ] that will have entries for all possible values of i and j.

The final solution for the entire chain is m[1, n], with corresponding split at s[1, n]. Unraveling the solution will be recursive, starting from the top and continuing until we reach the base case, i. Therefore, the next step is to actually split the chain, i. Top 50 Dynamic Programming Practice Problems.

### Submission history

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