Create Binary-Search.md

Author: cormacpayne

Notes/Binary-Search.md

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+# Binary Search
+
+## Motivating Problem I
+
+Imagine that you have a phone book and want to find your friend's phone number.
+
+With so many people listed, how do you efficiently search for your friend's last name?
+
+## Motivating Problem II
+
+You ask your friend to think of a number between 1 and 100; you attempt to guess what number they're thinking of, and they will tell you if the number you guessed is bigger or smaller than their number.
+
+You keep doing this until you successfully find their number; the number of guesses you had is your score, so you attempt to minimize your score.
+
+Since there are only 100 numbers, your score should be relatively small, but what if you told your friend to think of a number between 1 and 1,000,000?
+
+How can we quickly find what number they are thinking of?
+
+## Definition
+
+One of the fundamental algorithms in computer science, the *binary search* algorithm is used to quickly find a value in a sorted sequence.
+
+The algorithm works by repeatedly dividing in half the portion of the sequence that could contain the target value, until you've narrowed down the possible locations to just one.
+
+## Time Complexity
+
+Suppose we have a sorted sequence of 32 numbers.
+
+If we randomly guess, the maximum number of guesses that it will take to find the target number will be 32.
+
+If we use binary search, the maximum number of guesses that it will take to find the target number will be 5.
+- Remember that every guess eliminates half of the remaining sequence
+
+Now, suppose we have a sorted sequence of **_N_** numbers.
+
+If we randomly guess, the maximum number of guesses that it will take to find the target number will be **_N_**.
+
+If we use binary search, the maximum number of guesses that it will take to find the target number will be **_log(N)_**.
+
+How much faster is **_log(N)_** than **_N_**?
+
+|    **_N_**    | **_log(N)_** |
+| ------------- | ------------ |
+| 10            | ~3           |
+| 100           | ~7           |
+| 10,000        | ~13          |
+| 1,000,000     | ~20          |
+| 1,000,000,000 | ~30          |
+
+## Pseudocode
+
+Let's say that we want to code the binary search algorithm to find a target value in a sorted array of numbers.
+
+We will use two variables, `lo` and `hi`, to represent the indices of the first and last element of the section of the array that we are currently looking at.
+
+Let's call the index of the middle number in this target section `mid`, which can be found by finding the average of `lo` and `hi`.
+
+There are three possible scenarios:
+
+1. `array[mid] == targetValue`
+
+If the middle number of our current sequence is equal to the target value, we have found what we are looking for and are done
+
+2. `array[mid] < targetValue`
+
+If the middle number of our sequence is less than the target value, then we know that the bottom half of the current sequence can be eliminated.
+
+We can adjust the sequence by setting `lo = mid`
+
+3. `array[mid] > targetValue`
+
+If the middle number of our sequence is greater than the target value, then we know that the top half of the current sequence can be eliminated.
+
+We can adjust the sequence by setting `hi = mid`
+
+Now that we have the three possible scenarios handled, we just need to define when to terminate the loop if we never found our target value.
+
+We want to keep searching while we have some range of numbers to check, so if `lo` ever becomes greater than `hi`, then our target value is not in the array due to the fact that we have run out of numbers to check.
+
+```java
+int lo = 0;
+int hi = N-1;
+
+while (lo <= hi) {
+  mid = lo + (hi -lo) / 2;
+  if (array[mid] == targetValue) break;
+  if (array[mid] < targetValue) lo = mid;
+  if (array[mid] > targetValue) hi = mid;
+}
+```
+
+## Problems
+
+- [Aggressive cows](spoj.com/problems/AGGRCOW)
+- [ABCDEF](spoj.com/problems/ABCDEF)
+- [Cutting Cheese](icpc.kattis.com/problems/cheese)