# Algorithms

An algorithm is a well-defined computational procedure that takes in a set of values as inputs and produces a set of values as outputs. An algorithm is thus a sequence of computational steps that transform input into output.

## Terminology

**Nondecreasing order** = an ordering where the next value is greater than or equal to the previous. Ex.

For values $x(n)$ and $x(n+1),\ x(n+1)>=x(n)$

**Instance** = an instance is a set of inputs to an algorithm that satisfy the input the input requirements.

**Correct** = an algorithm is said to be **correct** if, for every input instance, the computation halts when the correct output has been calculated.

**Convex Hull** = the smallest convex polygon that contains all points in a set of numbers that lie on a plane.

**NP-complete problem** = a problem for which there is currently no efficient algorithm that solves it, neither is there a proof that proves that the algorithm exists.

## Proving an Algorithm

To prove an algorithm works, one way is by induction.

Start with the loop invariant, which is the subset of the input that is acted upon by the algorithm.

Then check for correctness in each of the following parts of the algorithm:

**Initialization** - Is it true prior to the first iteration of the loop?

**Maintenance** - Is it true at the start of the loop and also true at the end of the loop?

**Termination** - Is it true at the end of the loop?

## Analysis

Analyzing an algorithm is predicting how much resources it will use. You have to model the implemented technology.

**Input size** - A measure of the input must be selected that has the most significant impact on the resources required for an operation. For instance in a multiplication algorithm the input size should be number of bits used in the input numbers.

**Running time** - This should be a machine independent number. Number of primitive steps executed.

## Worst-case and average-case analysis

An algorithm's running time is dependent on the input data and there will be a best case, average case and worst case running time. The average case is often roughly as bad as the worst case. For instance with insertion sort.

Order of growth if the rate of growth of the running time. We consider the leading, most significant term of the formula of the worst case running time, called theta notation.

For an algorithm with a running time of $an^2 + bn + c$ the theta notation would be $Θ(n^2)$

## Sorting Algorithms

Here's a breakdown of different sorting algorithms. There's sorting algorithms that sort in place, rely on element comparisons, others that don't.

### Insertion Sort

For an unsorted list A of n length, loop through a growing subset of A by comparing the last element of the subset with each previous element and compare. If greater/smaller swap elements and iterate through again through a subset that is 1 larger than the previous.

1. for j = 1 to n 2. key = A[j] 3. i = j-1; 4. while i > 0 and key < A[i] 5. A[i+1] = A[i] 6. i-- 7. A[i+1] = key

### Selection Sort

Find the smaller number and place it in A[1]. Then find smallest number in A[2..n] and place it in A[2]. Repeat

1. for i = 1 to A.length-1 2. //find the smallest number in A(i to n) 3. cur_smallest_index = i 4. for j = i+1 to A.length 5. if A[j] < A[cur_smallest_index] 6. cur_smallest_index = j 7. A[i] = temp 8. A[i] = A[cur_smallest)index] 9. A[cur_smallest_index] = temp;

### Merge Sort

### Heap Sort

Heap sort is an O(n lg n) sorting algorithm that sorts in place with a constant number of array elements stored outside the array at any time.

Heapsort uses a data structure called a “heap” to manage information.

A heap is a binary-tree that consists of a root, and parent nodes that have up to two child nodes. The math to retreive the the parent, left element or right element of a node is really simple.

**Parent** - $return\lfloor i/2 \rfloor$

**Left** - $return 2*i$

**Right** - $return 2*i+1$

There are two kinds of binary heaps, **max-heaps** and **min-heaps**. In both
kinds the values in the nodes satisfy a **heap property**. In **max-heap** the
**max-heap property** is that for every node $i$ other than the root,

$A[Parent(i)]\geq A[i]$

For **min-heap** the **min-heap property** is that for every node $i$ other than
the root,

$A[Parent(i)]\leq A[i]$

There are a set of operation algorithms that are used in conjuction with heaps. They are as follows:

- Max-Heapify
- Build-Max-Heap
- Heapsort
- Max-Heap-Insert, Heap-Extract-Max, Heap-Increase-Key, Heap-Maximum

#### Max-Heapify

Runs in O(lg n), whose inputs are A and index i into the array. When it is called, Max-Heapify assumes that the binary trees rooted at Left(i) and Right(i) are max-heaps, but that A[i] might be smaller than its children. Max-Heapify lets the A[i] value “float” down so that the subtree rooted at index i obeys the max-heap property.

- Is index left smaller than A.heap-size and A[l] > A[i]?
- If yes, largest = l

- Else, largest = i
- Is index right smaller than A.heap-size and A[r] > A[i]?
- If yes, largest = r

- Else, largest = i
- If largest != i
- exchange A[i] with A[largest]
- Max-Heapify(A, largest)

#### Max-Heap-Insert

Max-Heap Insert. Input: heap array A, with data from 1 to length, element to insert p // stuff the element at the end of the heap array 1. A[length+1] = p // iterate up through the heap 2. int i = length + 1; 3. while i >= 0 // check if the parent node of our inserted element is less than it. 4. if (i/2 < p) // swap the inserted element with its parent 5. A[i] = A[i/2] 6. A[i/2] = p 7. i = i/2 8. else 9. break

#### Build-Max-Heap

Build-Max-Heap uses Max-Heapify to a bottom-up manner to convert an array A[1..n], where n = A.length, in a max-heap. This procedure goes through the non-leaf nodes of the tree ($A[\lfloor n/2\rfloor..1]$) and runs Max-Heapify on each one.

Build-Max-Heap(A) 1. A.heap-size = A.length 2. for i = [A.length/2] down to 1 3. Max-Heapify(A,i)

#### Heapsort

Heapsort works by building a max heap and then deconstructing it in order which in turn constructs a sorted array! Super neat and works in O(n lg n) time.

The algorithm works as follows:

Heapsort(A) 1. Build-Max-Heap(A) 2. for i = A.length down to 2 3. Exchange A[1] with A[i] 4. A.heap-length -- 5. Max-Heapify(A)

### Priority Queue

Apparently heapsort is great and all, but quicksort beats it. What people do use is the heap data structure, with its most popular application being a priority queue. A priority queue is a data structure that contains a set of keys sorted by their priority. We use the following procedures when utilizing priority queues.

- Insert
- Find-Max

There are others that you can implement, but will make it difficult to address a
element within the queue^{1)}.

- Delete-Max
- Change-Key

## Divide and conquer

Many algorithms are recursive, they call themselves recursively to deal with closely related sub-problems.

**Divide** - divide the problem into smaller subsets of the big set

**Conquer** - conquer the subproblems

**Combine** - combine the solutions to come up with the main solution of the bigger problem

A recurrence is an equation or inequality that describes a function in terms of its value of smaller inputs.

Methods for solving recurrences:

**substitution method**guess a bound and then use mathematical induction to prove our guess correct**recursion-tree method**convert the recurrance into a tree whose nodes represent the costs incurred at various levels of the recursion.**master method**provides bounds for recurrences of the form $T(n) = aT(n/b) + f(n)$ where $a \geq 1,\ b\ > \ 1$ and $f(n)$ is a given function.

### Sliding Window

An approach for scanning through data is to use a sliding window technique.

For example, take this problem: find the length of the longest substring without repeating characters. Ex. Input: “abcabcbb” output = 3 (abc) and for “vadva” output = 3 (vadv)

To do this we have to scan through the input string while keeping track of the characters used. One such answer is as the following:

${\rm L{\small ONGEST}-S{\small UBSTRING}} (string)$

// our sliding window will be bound by i and j 1. int i = 0; j = 0; max = 0 // this is our hash table that contains a flag for every character 2. bool characterArray[128] = {false} // iterate up starting from i = 0 3. for int j from 0 to < string.length // if we've never seen this character before // and tally our max and update our character hash key 4. if characterArray(string[j]) = false 5. max = (max > (j+1-i)) ? max : (j+1-i) 6. characterArray(string[j]] = true) // if we have seen it before, reset it, and slide i forward one // and try again 7. else 8. characterArray(string[i]] = false) 9. i++, j-- 10. return max

### Maximum subarray

The maximum subarray problem can be solved using divide and conquer.

By breaking up the input array into two subarrays of $n/2$ length as follows:

- A[low..mid]
- A[mid+1..high]

The maximum subarray A[i..j] can only be located in one of 3 places; in the first one, crossing the middle, or the end. To solve this equation we must find the maximum subarray in each of those 3 places and simply pick the biggest one.

**Find-Max-Crossing-Subarray Algorithm**

## Growth of Functions

Asymptotic efficiency is when we use an input size large enough to make only the growth of the running time relevant.

### Θ-notation

Worst case running time of an algorithm.

$$Θ(g(n)) = \{f(n)\ :\ there\ exist\ positive\ constants\ c_1,\ c_2,\ and\ n_0\ such\ that\\\ 0 \le c_1*g(n)\le f(n) \le c_2*g(n)\ for\ all\ n\ \ge\ n_0\}$$

### O-notation

O(g(n)) is defined as the set of functions which grow no faster than g(n).

$$0(g(n))\ =\ \{ f(n)\ :\ there\ exist\ positive\ constants\ c\ and\ n_0\ such\ that\\\ 0\leq f(n) \leq c*g(n)\ for\ all\ n\geq n_0\}.$$

or

$O(g(n))={f(n)\ :\exists\ c>0,\ n_0>0 \ni 0\leq f(n) \leq\ c*g(n) \forall n\geq n_0}$

Example^{2)}:

Using this example of $f(n)=5n$ and $g(n)=n$ we can prove that $O(g(n))=n$.

The definition of $O(g(n))$ is:

$f(n)\in O(g(n))\iff \exists c>0,n_0\ni \forall n \geq n_0,f(n)\leq c*g(n) $

Using this definition we can prove that $f(n)$ is a member of $O(g(n))$:

$n\in O(g(n)) \iff \exists c>0, 5n \leq c*n $

By substituting constant c = 6:

$n\in O(g(n)) \iff \exists 6>0, 5n \leq 6*n$

### Ω-notation

Ω notation provides an **asymptotic lower bound**. For a given function $g(n)$, we denote by $Ω(g(n)$ (pronounced “big-omega of g of n” or “omega of g of n”) the set of functions

$$Ω(g(n))=\{f(n)\ :\ there\ exist\ positive\ constants\ c\ and\ n_0\ such\ that\\\ 0\leq c*g(n)\leq f(n)\ for\ all\ n\geq n_0\}$$.

### o-notation

We use o-notation to denote an upper bound that is not asymptotically tight. We define $o(g(n))$ as the set:

$$o(g(n)) = \{f(n)\ :\ for\ any\ positive\ constant\ c>0,\ there\ exists\ a\ constant\\\ n_0>0\ such\ that\ 0\leq f(n)\ < c*g(n)\ for\ all\ n \geq n_0\}.$$

## Backtracking

Backtracking is an algorithm that looks at all possible solutions for a problem and evaluates each one.