Complexity analysis in Points + Some Examples

Author: Muhammad-Magdi

Complexity analysis.txt

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+/*
+  Session : Time Complexity analysis
+  By : Muhammad Magdi
+  On : 20/08/2017
+*/
+
+* What's time Complexity?
+    How does the running time change as the size of input change.
+
+* Running time depends on:
+    1- Processor.
+    2- Read/Write speed to memory.
+    3- 32 bit VS 64 bit.
+    4- INPUT.
+
+* What are the cases your code may face?
+    1- Best case.
+    2- Worst case.
+    3- Average case.
+
+* The notations to represent your code's Complexity:
+    1- Big O Notation     ---->     the upper bound of the time.
+    2- Omega Notation     ---->     the lower bound of the time.
+    3- Theta Notation     ---->     the bound itself.    (Calculated mathematically)
+
+* Rules:
+    1- Running time is the sum of running times of all consecutive blocks.
+    2- Nested loops are multiplied.
+        In general -> Nested repetitive Blocks are multiplied.
+    3- In Conditional statements pick the "Worst case" one.
+    4- Drop Constants (addition, subtraction, multiplication or division).
+    5- Drop all lower order terms.
+
+
+* Some useful Observations:
+  Big O                 Name                    Max n
+-------------------------------------------------------------------------------------------
+  O(1)        ---->     Constant      ---->     1e18      ---->     Math, Observation
+  O(Log(n))   ---->     Logarithmic   ---->     1e18      ---->     Binary Search (lower -upper- bound)
+  O(n)        ---->     Linear        ---->     1e8       ---->     one loop
+  O(n*Log(n)) ---->     LogLinear     ---->     4e5       ---->     Sorting, loop + binary search
+  O(n^2)      ---->     Quadratic     ---->     1e4       ---->     nested loop
+  O(2^n)      ---->     Exponential   ---->     25        ---->     Bitmasks, finding all possible answers
+  O(n!)       ---->     factorial     ---->     11        ---->     finding all permutations
+
+
+int calcSum(int a, int b){
+  int sum = a+b;
+  return sum;
+}
+
+double calcAverage(int a, int b){
+  double avg = (a+b)/2.0;
+  return avg;
+}
+
+bool isAlphabit(char x){
+  return (x>='A' && x<='Z' || x>='a' && x<='z');
+}
+
+
+
+double sumHarmonicSeries(int n){
+  double sum = 0;
+  for(int i = 1 ; i <= n ; ++i){
+    sum += (1.0/i);
+  }
+  return sum;
+}
+
+long long calcSumSegment(int a, int b){
+  long long sum = 0;
+  for(int i = a ; i<=b ; ++i)
+    sum += i;
+  return sum;
+}
+
+int fact(int n){
+  if(!n || n==1)	return 1;
+  return n*fact(n-1);
+}
+
+int power1(int base, int power){
+  if(!power)  return 1;
+  return base*power1(base, power-1);
+}
+
+
+
+bool binarySearch(int val){
+  int lo = 0, hi = n, mid;
+  while(lo<hi){
+    mid = ((lo+hi)>>1);
+    if(A[mid] == val) return 1;
+    if(A[mid] < val)
+      lo = mid+1;
+    else
+      hi = mid-1;
+  }
+  return 0;
+}
+
+int calcLog(int n){
+	int ret = 0;
+	while(n > 1){
+		++ret;
+		n /= 2;
+	}
+  return ret;
+}
+
+int power2(int base, int power){
+  if(!power)  return 1;
+  int sub = power2(base, power>>1);
+  return (power&1? sub*sub*base : sub*sub);
+}
+
+
+
+for(int i = 0 ; i < (1<<n) ; ++i){
+  //some O(1) operations
+}
+/*
+8 4 2 1
+0 0 0 1
+0 0 1 0
+0 1 0 0
+1 0 0 0
+
+1<<0 = 2^0 = 1
+1<<1 = 2^1 = 2
+1<<2 = 2^2 = 4
+1<<n = 2^n
+*/
+
+int fib(int n){
+  if(!n || n==1)	return n;
+  return fib(n-1)+fib(n-2);
+}
+
+
+
+for(int i = 0 ; i < (1<<n) ; ++i){
+  for(int i = 0 ; i < n ; ++i){
+    //some constant order statements go here
+  }
+}
+
+void searchArray(){
+  for(int i = 0 ; i < n ; ++i){
+    if(binarySearch(B[i]))
+      puts("Found");
+    else
+      puts("Not Found");
+  }
+}
+
+bool isComp[1008];
+void sieve(){
+  isComp[0] = isComp[1] = 1;
+  for(int i = 2 ; i < n ; ++i){
+    if(!isComp[i]){
+      for(int j = i ; j < n ; j+=i)
+        isComp[i] = 1;
+    }
+  }
+}
+
+
+
+void printPermutations(string s){
+  sort(s.begin(), s.end());
+	do {
+		cout << s << endl;
+	} while(next_permutation(s.begin(), s.end()));
+}