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# CEG 221 Lesson 4: Algorithm Development I PowerPoint PPT Presentation

CEG 221 Lesson 4: Algorithm Development I. Mr. David Lippa. Overview. Algorithm Development I What is an algorithm? Why are algorithms important? Examples of Algorithms Introduction to Algorithm Development Example of Flushing Out an Algorithm Questions. What is an Algorithm?.

CEG 221 Lesson 4: Algorithm Development I

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## CEG 221Lesson 4: Algorithm Development I

Mr. David Lippa

### Overview

• Algorithm Development I

• What is an algorithm?

• Why are algorithms important?

• Examples of Algorithms

• Introduction to Algorithm Development

• Example of Flushing Out an Algorithm

• Questions

### What is an Algorithm?

• An algorithm is a high-level set of clear, step-by-step actions taken in order to solve a problem, frequently expressed in English or pseudo code.

• Pseudo code – a way to express program-specific explanations while still in English and at a very high level

• Examples of Algorithms:

• Computing the remaining angles and side in an SAS Triangle

• Computing an integral using rectangle approximation method (RAM)

### Why are algorithms important?

• Algorithms provide a means of expressing the problem to be solved, the solution to that problem, and a general step-by-step way to reach the solution

• Algorithms are expressed in pseudo-code, and can then be easily written in a programming language and verified for correctness.

A

b = 100

B

42°

a = 60

C

### Example: Triangulation with SAS

• Let’s assume we have a triangle and we wish to compute the missing values:

• Start with the mathematical computation if we were to do it manually:

• c = A2 + B2 – 2 ab cos C

• sin A = (a sin C) / c

A = asin(a (sin C) / c )

• B = 180 – A – C (iff A and C are in degrees)

• We’re also assuming that angles are in degrees

### SAS: From Math to Pseudocode

• Now that we have all the math done, we can develop the algorithm’s pseudo code:

• Get the sides and their enclosing angle from the user (in degrees)

• Run the computation from the previous slide

• Convert angle A to degrees prior to computing angle B

• Display results to the user

### Flushing Out Initial Pseudo Code

• Pseudo code breaks up large tasks into single operations that can each be broken up further or expressed by a C primitive

• Example:

• “Get user input” means to prompt the user using “printf”; read the input with “scanf”

• “Convert radians to degrees” is just a bit of math – multiply by 360 and divide by 2π

### Flushing Out Initial Pseudo Code: Hiding the Work

• For a simple operation, like converting from degrees to radians, using functions to hide the work is less important.

• But how would you like to write out 10 or 1000 lines of code to perform a simple task on 1 data value or case? Probably not. That’s why we can hide the work in a function.

• Example: Matrix Multiplication

### Introduction to Algorithm Development

• We’ll get to Matrix Multiplication in a minute.

• Developing this algorithm will help you practice seeing how to take a problem, find a solution, develop an algorithm, flush out the algorithm, and then finally implementing it.

• Keep in mind that “hiding the work” is important – this is crucial to modular design

### Matrix Multiplication:Initial Design

• Break down the math:

• [A] x [B] = [C]  for each element in [C], dot product the rows of [A] with the columns of [B] to get the first value in [C].

• [A] is an m x n matrix and [B] is an n x p matrix

• [C] is an m x p matrix

For each row i in [A]

For each column j in [B]

C[i, j] = i ∙ j

### Matrix Multiplication:Dot Product

• Since dot products are not implemented in C, we have to do it ourselves

• Dot product(X, Y) = X1∙ Y1 + X2∙ Y2 + … + Xn∙ Yn

• There are n pairs that need to be multiplied together

• Multiply the kth value in A’s row by the kth value in B’s column

• Assume every value in C is initialized to 0

For each column k in [A] (or for each row k in [B])

C[i, j] += A[i, k] x B[k, j]

### Matrix Multiplication:Refined Design

• Now we can pull it all together:

• Assumptions:

• Every element of C is initialized to 0.

For each row i in [A]

For each column j in [B]

For each column k in [A]

C[i, j] += A[i, k] x B[k, j]

Since C does not implement accessors [i, j] for 2D arrays allocated dynamically, we must implement it.

Below is a 3 x 4 matrix with 2D and 1D coordinates overlayed.

Examples:

(0, 0) maps onto 0

(1, 0) maps onto 4

(2, 1) maps onto 9

Calculation:

0 * 4 + 0 = 0

1 * 4 + 0 = 4

2 * 4 + 1 = 11

From these values, we can derive that:

C[i, j] = C[i * numCols + j]

0

0,0

1

0,1

2

0,2

3

0,3

4

1,0

5

1,1

6

1,2

7

1,3

8

2,0

9

2,1

10

2,2

11

2,3

### Matrix Multiplication: Accessing a Dynamically Allocated 2D Array

• Code for the “at” function:

int at(int i, int j, int numCols)

{

return i * numCols + j;

}

### Matrix Multiplication:Wrapping Up Pseudo Code

• This pseudo code can now be written into C code that takes:

• 2 Pointers to an array of doubles: [A], [B]

• Numbers of rows & columns of each (m, n, p)

• And allocates memory with dynamic memory allocation to return [C], an m x p matrix that is the result of [A] x [B]

• Now, any time we need to multiply any matrices, we can use and reuse this module.

### Matrix Multiplication: The Code

double* mult(double *pA, int numRowsA, int numColsA,

double *pB, int numRowsB, int numColsB)

{

// allocate memory, set pC to 0

double *pC = malloc(numRowsA * numColsB * sizeof(double));

memset(pC, 0, numRowsA * numColsB * sizeof(double));

for (i = 0; i < numRowsA; i++)

for (j = 0; j < numColsB; j++)

for (k = 0; k < numColsA; i++)

{

pC[at(i, j, numColsB)] =

A[at(i, k, numColsA)] * B[at(k, j, numColsB)];

}

return pC;

}

### Flushing Out the Code

• There are two more steps to algorithm development when you use functions:

• Pointers can be NULL

• What do you get if you multiply a 2 x 3 matrix by a 5 x 7? You can’t!

• This brings us to defining requirements for each function. If those requirements aren’t met, we return an error value, in this case the NULL pointer.

### Matrix Multiplication: Requirements

• Neither matrix can be NULL

• If [A]’s dimensions are m x n and [B]’s dimensions are NOT n x p, bail out because we cannot compute [A] x [B]

• If malloc() fails to allocate memory, bail out

### Matrix Multiplication: Final Code

double* mult(double *pA, int numRowsA, int numColsA,

double *pB, int numRowsB, int numColsB)

{

// (m by n) x (p by q) -> can’t multiply -> bail out

if (numColsA != numRowsB) return NULL;

double *pC = malloc(numRowsA * numColsB * sizeof(double));

// no memory, bad matrices -> bail out

if (pA == NULL || pB == NULL || pC == NULL) return NULL;

memset(pC, 0, numRowsA * numColsB * sizeof(double));

for (i = 0; i < numRowsA; i++)

for (j = 0; j < numColsB; j++)

for (k = 0; k < numColsA; i++)

{

pC[at(i, j, numColsB)] =

A[at(i, k, numColsA)] * B[at(k, j, numColsB)];

}

return pC;

}

### Next Time

• Algorithm Development II

• Review of basic algorithm development