Algorithms Complexity Evaluation
If we have multiple ways to solve a particular problem statement, and if if need to find out which one is more efficient or less complex then:
So, the question is: How to compare algorithms?
The Big O Notation
Big O Notation is a convenient way to express the worst-case scenario for an algorithm
Thus, Big O Notation will be our definition to figure out better algorithm
Basics
- Complexity measure in machine independent manner
- Any algorithm consists of indivisible process operation or step
- Each step take same amount of time to execute
- Algorithms running time measured in processor operations instead of seconds
- This type of measurement is called DTIME or TIME
- DTIME: Number of computation step a computer would take for a solution using certain algorithm
Time Complexity
Time complexity is a function that represents dependency between input data and time it take to complete all the computations. As we need to find worst case time complexity we need to consider N (towards infinity)as big as possible.
- Rule 1: Always worst Case
- Rule 2: Remove Constants, find the leading term
- Rule 3: Different inputs should have different variables. For consecutive action complexity will be Addition, O(a+b). For nested it would be Multiplications, O(a*b) + for steps in order * for nested steps
- Rule 4: Drop Non-dominant terms
Leading term in bellow example:
N² + 2^n = N²
9*N + log2N+7 = 9*N
N*log2N +log2N = N*log2N
Big O notation
- Big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows.
- The letter O is used because the growth rate of a function is also referred to as the order of the function.
- If f(x) is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
- If f(x) is a product of several factors, any constants (terms in the product that do not depend on x) can be omitted.
In above example T1(N) function has highest growth rate is 47*2^N. Now one may apply the second rule: 47*2^N is a product of 47and 2^N in which the first factor does not depend on N. Omitting this factor results in the simplified form 2^N. Thus, we say that f(N) is a “big O” of 2^N
Complexity Evaluation
Few useful formulas for complexity evaluation
- 1 + 2 + 3 + 4 + … +N = N²/2 = O(N²)
- 2⁰+2¹+2²+….+ 2^n = *2^N - 1 = O(2^n)
- log a N = log a b+log b N = O(log N) (a, b const)
Addition and Multiplication Complexity
Log N complexity
Relation between number of operation and number of element in the array: N: is the number of elements in the Array K: number of operations
So we know 2^k-1 = N
k-1= log2 N
K = log2N +1
Since T(N) = k
Then, T(N) = log2 N+1
T(N) = O(log N)
String Complexity Evaluation
1. Compare String
- Comparing two String to find which one greater string processor need to compare between two arrays of characters.
- In worst case the loop needs to iterate over full length of one array
- So, resulting complexity is O(s1.length) = O(N)
2. String Concatenation
- Strings are immutable in Javascript, ruby, Java and many other language
- This mean every string operation will create a new string in memory
- For every string concatenation (“ab” + “cde”) of length N and K will bellow algorithm and complexity will be O(N + K)
3. Getting the Substring
Complexity of getting a substring is O(end — start) = O(N-K)
If we add or remove one char complexity will be
Concat: O(N+1) = O(N)
Substring: O(N-1) = O(N)
Here is an example which concatenates a single charter N times
4. Recursive Functions Complexity
To evaluate complexity of recursive function
1. Calculate complexity of a single function call
2. Number of recursive call
3. Multiply number of recursive call by single call complexity
function foo(n){
if(n==1)return 1; --------> complexity 1
return n+foo(n-1); --> foo(4) => foo(3) => foo(2) => foo(1)
--> N number of recursive call
};Complexity of this algorithm is O(1) * N = O(N)
function foo(n){
if(n==0)return 1; --------> complexity O(1)
return n+foo(n-1); --> foo(4) => foo(3) => foo(2) => foo(1)
--> N number of recursive call
};- Above function has one recursive call with step -1
- Chain of call will be from N to 0, => N+1 times
- Thus, complexity is O(1) * (N+1) = O(N)
Example:function foo(n){
if(n==0)return; --------> complexity O(1)
foo(n/2) // N/2 number of recursive call
// when an algorithm process half of the elements per iteration, will have long N complexity
};
Complexity of the above algorithm is O(1) + long2 N = O(log N)
function fib(n, cache){ // As no operation depends on N
if(n<=1){return n}; // Single call complexity is O(1)
if (cache[n] == undefined){
cache[n] = fib(n-1, cache) + fib(n-2, cache); // Cache the recursive call result
}
return cache[n];
};function printFib(n){
let cache = [];
for(i=0; i<=n; i++){
const nthFib = fib(i, cache); // fib function executed fro 0 to N
console.log(nthFib);
}
};
For N=5 following call tree will be created
To eavalute complexity for the above algorithm
- First 2 calls have no dependency on N so, O(1)
- Starting from N=2, every call finish in 3 calls\
- So total call will be 1 + 1 + 3*(N-1) = 3N -1 = O( N)
- Therefore, complexity of the algorithm will be O(1)*O(N) = O(N)
Few Interesting Pitfalls
Since Big O Notation tells us about the upper bound of an algorithm ignoring constants and low-order terms. An algorithm may have exact number of steps in the worst case more or less than the Big O notation.
Exact Steps > Big O
An algorithm takes 5 * N steps has complexity of O(N) in the worst case, which is smaller than the exact number of steps.
for(i = 0; i < 5*n; i++ )
k++;Exact Steps < Big O
Another example is, when the number of loops are decreasing by the time. So, first time it will loop N, N-1, N-2, …, till 1. The Big O notation for the following code is O(N³), while the exact number of steps is much less than that.
for (int i = 0; i < n; ++i)
for (int j = i; j < n; ++j)
for (int k = j; k < n; ++k)
cunt++;Space Complexity
Amount of memory required for an algorithm to run in a computer
- Space complexity function is denoted as S
- All Big O notation rule for time complexity holds true for space complexity evaluation too
- For normal function space complexity is O(1)
- For recursive function space complexity S(N) = O(1)*N= O(N)
- For string concatenation O(n²)
