Binary Search

Motivation

Linear search

Binary search

VS

O( n )

O( log n )

Features

Linear search

• Linear search can be super expensive!

• Need to be allow to random data structure access (e.g. array)

• Data needs to be sorted

• Linear and not linear data structure access allowed (e.g. linked lists)

• Data doesn't need to be sorted

Binary search

• As fast as the wind!

O( n )

O( log n )

Guess my number

Binary Search

Divide and conquer algorithm

Data needs to be sorted

MID

MID

Find: 6

Implementation

Iterative

function binarySearch(data, target) {
  var low = 0,
      high = data.length - 1,
      mid;

  while (low <= high) {
    mid = Math.floor((low + high) / 2);
    if (data[mid] < target) {
      low = mid + 1;
    } else if (data[mid] > target) {
      high = mid - 1;
    } else {
      return mid;
    }
  }
  return -1;
}

(JavaScript)

Implementation

Recursive

function binarySearchRecursive(data, target, low, high) {
  if (low > high) return -1;

  var mid = Math.floor((low + high) / 2);
  if (data[mid] < target) {
    return binarySearchRecursive(data, target, mid + 1, high);
  } else if (data[mid] > target) {
    return binarySearchRecursive(data, target, low, mid - 1);
  } else {
    return mid;
  }
}

(JavaScript)

Options that we have

• Sort the collection if it is not

• Make sure that the collection is sorted as we add elements to it 

Binary Search Tree

The collection of items is always sorted 

Instead the need of sorting every time we add an element to the collection 

Binary Search Tree

Properties

• The left subtree of a node only contains values that are less than or equal to the node's value

• The right subtree of a node only contains values that are greater than or equal to the node's value

• Both subtrees are binary trees

Warning:  Don't assume that a binary tree is a Binary Search Tree

Binary Search Tree

Binary Search Tree

Operations Complexity

Worst case

Binary tree from a sorted collection

Average

Exercises

Given a sorted array of n integers that has been rotated an unknown number of times, give an O(log n) algorithm that finds an element in the array. You may assume that the array was originally sorted in increasing order.

  EXAMPLE:

    **Input**: find 5 in array (15 16 19 20 25 1 3 4 5 7 10 14)

    **Output**: 8 (the index of 5 in the array)

Exercises

Given a sorted array of strings which is interspersed with empty strings, write a method to find the location of a given string.

find “ball” in [“at”, “”, “”, “”, “ball”, “”, “”, “car”, “”, “”, “dad”, “”, “”] will return 4

find “ballcar” in [“at”, “”, “”, “”, “”, “ball”, “car”, “”, “”, “dad”, “”, “”] will return -1

Exercises

Given a sorted (increasing order) array, write an algorithm to create a binary tree with minimal height.