02b: Algorithms: Big-O

ICS211, Fall 2012
Dr. Zach

(switch view)

Status

Comparing Algorithms

Comparing Implementations

Comparing Algorithms: Number of steps involved?

Comparing Algorithms: Worst-case growth rate!

Example (variant of p.81)

for (int i = 0; i < n; i++) {
  for (int j = 0; j < n; j++) {
    //statement 1
    //statement 2
  }
}
for (int k = 0; k < n; k++) {
  //statement 1
  //statement 2
  //...
  //statement 5
}
//25 more statements

Time to run = T(n) = n*n*2 + n*5 + 25 = 2n2 + 5n + 25
(assuming each statement takes 1 tick to run)

Big-O Defined

Big-O Definition Example

(Note we could have selected f(n) = n4 here instead. Valid, but convention (especially in this class) is to choose the simplest and lowest-order f(n) that you can.)

Big-O Definition Revisited

Informal Approach

Summary: Big-O tells us the algorithm won't get any worse than a given f(n), no matter how many more processed elements are added.

Big-O Example: Search (p.78)

/** Returns index of target found in x array, or -1 if not found */
public static int search(int[] x, int target) {
  for (int i = 0; i < x.length; i++) {
    if (x[i] == target) {
      return i;
    }
  }
  return -1;  //looped through whole array and didn't find target
}

Big-O Example: Box

/** Prints a box of #s of width * height.  */
public static void drawBox(int width, int height) {
  for (int row = 0; row < height; row++) {
    for (int col = 0; col < width; col++) {
      System.out.print('#');
    }
    System.out.println();
  }
}

Big-O Example: Many boxes

/** Prints 5 boxes of the given size.  */
public static void drawBoxes(int size) {
  for (int i = 0; i < 5; i++) {
    drawBox(size, size);
    System.out.println();
  }
}

Big-O Example: Many boxes variant

/** Prints the given number of boxes.  */
public static void drawBoxes(int many) {
  for (int i = 0; i < many; i++) {
    drawBox(10, 10);
    System.out.println();
  }
}

Big-O Example: Find duplicates

/** Returns whether there are any repeated values in the given array. */
public static boolean findDuplicates(int[] nums) {
  for (int i = 0; i < nums.length; i++) {
    for (int j = i + 1; j < nums.length; j++) {
      if (nums[i] == nums[j]) {
        return true;
      }
    }
  }
  return false;
}

Lessons from Examples

Some Big-O Limitations: vs reality

Java Review

Arrays and reading from a file

Arrays

int[] nums = {5, 6, 90};
int[] ages = new int[10];
int size = nums.length;

nums[1] = 60;
for (int i = 0; i < nums.length; i++) {  //use .length, not 3 here
  nums[i]++;
}
for (int n : nums) {
  System.out.println(n);
}

Should already know this (esp. after Fri's lab).

Arrays of Arrays (Multi-dimensional)

char[][] ticTacToe = new char[3][3];
tictactoe[1][1] = 'X';
char[] middleRow = ticTacToe[1];
System.out.println(middleRow[1]);

int[][] matrix = new int[4][];  //sub-arrays can be different size if created later
for (int i = 0; i < matrix.length; i++) {
  matrix[i] = new int[i + 1];
}
matrix[1][1] = 9;

Command Line Arguments

public class Hello {
  public static void main(String[] args) {
    //...
  }
}

Cmd line args examples


Reading from a file

import java.util.Scanner;
import java.io.File;
import java.io.IOException;

public class PrintFile {
  /** Prints to the screen the contents of the file named input.txt. */
  public static void main(String[] args) {
    try {
      Scanner file = new Scanner(new File("input.txt"));
      while (file.hasNextLine()) {
        String line = file.nextLine();
        System.out.println(line);
      }
      file.close();
    }catch (IOException e) {
      System.out.println("Could not read from file: " + e.getMessage());
    }
  }
}

Reading from a file details

Reading from a file: Alternative

import java.io.IOException;
import java.io.FileReader;
import java.io.BufferedReader;

public class Hello {
  /** Prints to the screen the contents of the file named input.txt. */
  public static void main(String[] args) {
    try {
      BufferedReader file = new BufferedReader(new FileReader("input.txt"));
      String line = file.readLine();
      while (line != null) {
        System.out.println(line);
        line = file.readLine();
      }
    }catch (IOException e) {
      System.out.println("Could not read from file: " + e.getMessage());
    }
  }
}

Interesting aside

String line = file.readLine();  //initializer
while (line != null) {          //condition
  System.out.println(line);
  line = file.readLine();       //"increment"
}

Those are the components of a for loop... 

for (String line = file.readLine(); line != null; line = file.readLine()) {
  System.out.println(line);
}

Could have done something like this with Scanner too.

For next time...