What is the exact time complexity of this code ?
I know its exponential but what kind of exponential like 2^n , sqrt(n) ^ sqrt(n) etc.
If attach some proof, that would be great.
https://www.geeksforgeeks.org/minimum-number-of-squares-whose-sum-equals-to-given-number-n/
class squares {
// Returns count of minimum squares that sum to n
static int getMinSquares(int n)
{
// base cases
if (n <= 3)
return n;
// getMinSquares rest of the table using recursive
// formula
int res = n; // Maximum squares required is
// n (1*1 + 1*1 + ..)
// Go through all smaller numbers
// to recursively find minimum
for (int x = 1; x <= n; x++) {
int temp = x * x;
if (temp > n)
break;
else
res = Math.min(res, 1 + getMinSquares(n - temp));
}
return res;
}
public static void main(String args[])
{
System.out.println(getMinSquares(6));
}
}
In my opinion, since each for loop is calling the same recursion for sqrt(n) number of time and each call is for (n - x*x)) ~ n...
So it should be n ^ sqrt(n).
Is this answer correct ?
The recurrence relation for that function is
T(n) = sum from i=1 to i=sqrt(n) of T(n - i*i)
which is bounded above by
T(n) = sqrt(n) * T(n-1)
as each term in the sum above will be at most T(n-1) and there are sqrt(n) of them. T(n) = sqrt(n) * T(n-1) is O( sqrt(n)^n ). I'm sure there is some clever way to get a better bound, but this function looks like it is exponential.
Related
void f2(int n)
{
if (n<=1)
return;
g2(n, n/3);
}
void g2(int n, int m)
{
int i=1;
while (m < n) {
m += i;
i++;
}
f2(n/2);
}
I tried alot to calculate the time complexity and got it wrong, I would really appreciate it if someone could help me on how to approach these programs. (The answer is O(sqrt(n)).
The following explanation can be simplified, but I tried to be as much scrupulous as possible.
Sum of arithmetic progression
First of all lets talk about complixity of the following loop (note the m=0):
int m=0;
int i=1;
while (m < n) {
m += i;
i++;
}
Invariant of the loop is: after ith iteration m == 1+2+...+i == (1+i)*i/2. So the loop stops when the following condition is met:
which is equavalent to
Big O of the left and right parts are equal and both equal to O(i), so O(i)=O(sqrt(n)) is the complexity of the loop.
Complexity of the loop inside g2
The loop inside the g2 is equavalent to the following loop:
int n_modified = n - m;
m = 0;
int i=1;
while (m < n_modified) {
m += i;
i++;
}
which complexity is O(sqrt(n-m)) as we've shown in the previous section.
Complexity of the f2
Now lets get overall formula for complexity of the f2 function. Its complexity is essentially the same as complexity of the g2(n, n/3) call. It consists of two parts: complexity of the g2's loop and complexitiy of the recursion. That is, the formula is
This can be simplified and estimated (factoring and sum of geometric progression):
which gives us the final answer: the complexity of f2 is O(sqrt(n)).
Background Information: I solved the N-Queens problem with the C# algorithm below, which returns the total number of solutions given the board of size n x n. It works, but I do not understand why this would be O(n!) time complexity, or if it is a different time complexity. I am also unsure of the space used in the recursion stack (but am aware of the extra space used in the boolean jagged array). I cannot seem to wrap my mind around understanding the time and space complexity of such solutions. Having this understanding would be especially useful during technical interviews, for complexity analysis without the ability to run code.
Preliminary Investigation: I have read several SO posts where the author directly asks the community to provide the time and space complexity of their algorithms. Rather than doing the same and asking for the quick and easy answers, I would like to understand how to calculate the time and space complexity of backtracking algorithms so that I can do so moving forward.
I have also read in numerous locations within and outside of SO that generally, recursive backtracking algorithms are O(n!) time complexity since at each of the n iterations, you look at one less item: n, then n - 1, then n - 2, ... 1. However, I have not found any explanation as to why this is the case. I also have not found any explanation for the space complexity of such algorithms.
Question: Can someone please explain the step-by-step problem-solving approach to identify time and space complexities of recursive backtracking algorithms such as these?
public class Solution {
public int NumWays { get; set; }
public int TotalNQueens(int n) {
if (n <= 0)
{
return 0;
}
NumWays = 0;
bool[][] board = new bool[n][];
for (int i = 0; i < board.Length; i++)
{
board[i] = new bool[n];
}
Solve(n, board, 0);
return NumWays;
}
private void Solve(int n, bool[][] board, int row)
{
if (row == n)
{
// Terminate since we've hit the bottom of the board
NumWays++;
return;
}
for (int col = 0; col < n; col++)
{
if (CanPlaceQueen(board, row, col))
{
board[row][col] = true; // Place queen
Solve(n, board, row + 1);
board[row][col] = false; // Remove queen
}
}
}
private bool CanPlaceQueen(bool[][] board, int row, int col)
{
// We only need to check diagonal-up-left, diagonal-up-right, and straight up.
// this is because we should not have a queen in a later row anywhere, and we should not have a queen in the same row
for (int i = 1; i <= row; i++)
{
if (row - i >= 0 && board[row - i][col]) return false;
if (col - i >= 0 && board[row - i][col - i]) return false;
if (col + i < board[0].Length && board[row - i][col + i]) return false;
}
return true;
}
}
First of all, it's definitely not true that recursive backtracking algorithms are all in O(n!): of course it depends on the algorithm, and it could well be worse. Having said that, the general approach is to write down a recurrence relation for the time complexity T(n), and then try to solve it or at least characterize its asymptotic behaviour.
Step 1: Make the question precise
Are we interested in the worst-case, best-case or average-case? What are the input parameters?
In this example, let us assume we want to analyze the worst-case behaviour, and the relevant input parameter is n in the Solve method.
In recursive algorithms, it is useful (though not always possible) to find a parameter that starts off with the value of the input parameter and then decreases with every recursive call until it reaches the base case.
In this example, we can define k = n - row. So with every recursive call, k is decremented starting from n down to 0.
Step 2: Annotate and strip down the code
No we look at the code, strip it down to just the relevant bits and annotate it with complexities.
We can boil your example down to the following:
private void Solve(int n, bool[][] board, int row)
{
if (row == n) // base case
{
[...] // O(1)
return;
}
for (...) // loop n times
{
if (CanPlaceQueen(board, row, col)) // O(k)
{
[...] // O(1)
Solve(n, board, row + 1); // recurse on k - 1 = n - (row + 1)
[...] // O(1)
}
}
}
Step 3: Write down the recurrence relation
The recurrence relation for this example can be read off directly from the code:
T(0) = 1 // base case
T(k) = k * // loop n times
(O(k) + // if (CanPlaceQueen(...))
T(k-1)) // Solve(n, board, row + 1)
= k T(k-1) + O(k)
Step 4: Solve the recurrence relation
For this step, it is useful to know a few general forms of recurrence relations and their solutions. The relation above is of the general form
T(n) = n T(n-1) + f(n)
which has the exact solution
T(n) = n!(T(0) + Sum { f(i)/i!, for i = 1..n })
which we can easily prove by induction:
T(n) = n T(n-1) + f(n) // by def.
= n((n-1)!(T(0) + Sum { f(i)/i!, for i = 1..n-1 })) + f(n) // by ind. hypo.
= n!(T(0) + Sum { f(i)/i!, for i = 1..n-1 }) + f(n)/n!)
= n!(T(0) + Sum { f(i)/i!, for i = 1..n }) // qed
Now, we don't need the exact solution; we just need the asymptotic behaviour when n approaches infinity.
So let's look at the infinite series
Sum { f(i)/i!, for i = 1..infinity }
In our case, f(n) = O(n), but let's look at the more general case where f(n) is an arbitary polynomial in n (because it will turn out that it really doesn't matter). It is easy to see that the series converges, using the ratio test:
L = lim { | (f(n+1)/(n+1)!) / (f(n)/n!) |, for n -> infinity }
= lim { | f(n+1) / (f(n)(n+1)) |, for n -> infinity }
= 0 // if f is a polynomial
< 1, and hence the series converges
Therefore, for n -> infinity,
T(n) -> n!(T(0) + Sum { f(i)/i!, for i = 1..infinity })
= T(0) n!, if f is a polynomial
Step 5: The result
Since the limit of T(n) is T(0) n!, we can write
T(n) ∈ Θ(n!)
which is a tight bound on the worst-case complexity of your algorithm.
In addition, we've proven that it doesn't matter how much work you do within the for-loop in adddition to the recursive calls, as long as it's polynomial, the complexity stays Θ(n!) (for this form of recurrence relations). (In bold because there are lots of SO answers that get this wrong.)
For a similar analysis with a different form of recurrence relation, see here.
Update
I made a mistake in the annotation of the code (I'll leave it because it is still instructive). Actually, both the loop and the work done within the loop do not depend on k = n - row but on the initial value n (let's call it n0 to make it clear).
So the recurrence relation becomes
T(k) = n0 T(k-1) + n0
for which the exact solution is
T(k) = n0^k (T(0) + Sum { n0^(1-i), for i = 1..k })
But since initially n0 = k, we have
T(k) = k^k (T(0) + Sum { n0^(1-i), for i = 1..k })
∈ Θ(k^k)
which is a bit worse than Θ(k!).
public void function2(long input) {
long s = 0;
for (long i = 1; i < input * input; i++){
for(long j = 1; j < i * i; j++){
s++;
}
}
}
l'm pretty certain that the time complexity of this function is n^3, however if someone could provide a line by line explanation of this, that would be great.
First of all, you need to define what n is if you write something like O(n^3), otherwise it doesn't make any sense. Let's say n is the value (as opposed to e.g. the bit-length) of input, so n = input.
The outer loop has k iterations, where k = n^2. The inner loop has 1^2, 2^2, 3^2, ... up to k^2 iterations, so summing up everything you get O(k^3) iterations (since the sum of the p-th powers of the first m integers is always O(m^(p+1))).
Hence the overall time complexity is O(n^6).
I am trying to find the tightest upper bound for the following upper bound. However, I am not able to get the correct answer. The algorithm is as follows:
public staticintrecursiveloopy(int n){
for(int i = 0; i < n; i++) {
for(int j = 0; j < n; j++) {
System.out.println("Hello.");
}
} if(n <= 2) {
return 1;
} else if(n % 2 == 0) {
return(staticintrecursiveloopy(n+1));
} else{
return(staticintrecursiveloopy(n-2));
}
}
I tried to draw out the recursion tree for this. I know that for each run of the algorithm the time complexity will be O(n2) plus the time taken for each of the recursive calls. Also, the recursion tree will have n levels. I then calculated the total time taken for each level:
For the first level, the time taken will be n2. For the second level, since there are two recursive calls, the time taken will be 2n2. For the third level, the time taken will be 4n 2 and so on until n becomes <= 2.
Thus, the time complexity should be n2 * (1 + 2 + 4 + .... + 2n). 1 + 2 + 4 + .... + 2n is a geometric sequence and its sum is equal to 2n - 1.Thus, the total time complexity should be O(2nn2). However, the answer says O(n3). What am I doing wrong?
Consider the below fragment
for(int i = 0; i < n; i++) {
for(int j = 0; j < n; j++) {
System.out.println("Hello.");
}
}
This doesn't need any introduction and is O(n2)
Now consider the below fragment
if(n <= 2) {
return 1;
} else if(n % 2 == 0) {
return(staticintrecursiveloopy(n+1));
} else {
return(staticintrecursiveloopy(n-2));
}
How many times do you think this fragment will be executed?
If n%2 == 0 then the method staticintrecursiveloopy will be executed 1 extra time. Otherwise it goes about decresing it by 2, thus it'll be executed n/2 times (or (n+1)/2 if you include the other condition).
Thus the total number of times the method staticintrecursiveloopy will be executed is roughly n/2 which when expressed in terms of complexity becomes O(n).
And the method staticintrecursiveloopy calls a part with complexity O(n2) in each iteration, thus the total time complexity becomes
O(n) * O(n2) = O(n3).
Can someone give me example of an algorithm that has square root(n) time complexity. What does square root time complexity even mean?
Square root time complexity means that the algorithm requires O(N^(1/2)) evaluations where the size of input is N.
As an example for an algorithm which takes O(sqrt(n)) time, Grover's algorithm is one which takes that much time. Grover's algorithm is a quantum algorithm for searching an unsorted database of n entries in O(sqrt(n)) time.
Let us take an example to understand how can we arrive at O(sqrt(N)) runtime complexity, given a problem. This is going to be elaborate, but is interesting to understand. (The following example, in the context for answering this question, is taken from Coding Contest Byte: The Square Root Trick , very interesting problem and interesting trick to arrive at O(sqrt(n)) complexity)
Given A, containing an n elements array, implement a data structure for point updates and range sum queries.
update(i, x)-> A[i] := x (Point Updates Query)
query(lo, hi)-> returns A[lo] + A[lo+1] + .. + A[hi]. (Range Sum Query)
The naive solution uses an array. It takes O(1) time for an update (array-index access) and O(hi - lo) = O(n) for the range sum (iterating from start index to end index and adding up).
A more efficient solution splits the array into length k slices and stores the slice sums in an array S.
The update takes constant time, because we have to update the value for A and the value for the corresponding S. In update(6, 5) we have to change A[6] to 5 which results in changing the value of S1 to keep S up to date.
The range-sum query is interesting. The elements of the first and last slice (partially contained in the queried range) have to be traversed one by one, but for slices completely contained in our range we can use the values in S directly and get a performance boost.
In query(2, 14) we get,
query(2, 14) = A[2] + A[3]+ (A[4] + A[5] + A[6] + A[7]) + (A[8] + A[9] + A[10] + A[11]) + A[12] + A[13] + A[14] ;
query(2, 14) = A[2] + A[3] + S[1] + S[2] + A[12] + A[13] + A[14] ;
query(2, 14) = 0 + 7 + 11 + 9 + 5 + 2 + 0;
query(2, 14) = 34;
The code for update and query is:
def update(S, A, i, k, x):
S[i/k] = S[i/k] - A[i] + x
A[i] = x
def query(S, A, lo, hi, k):
s = 0
i = lo
//Section 1 (Getting sum from Array A itself, starting part)
while (i + 1) % k != 0 and i <= hi:
s += A[i]
i += 1
//Section 2 (Getting sum from Slices directly, intermediary part)
while i + k <= hi:
s += S[i/k]
i += k
//Section 3 (Getting sum from Array A itself, ending part)
while i <= hi:
s += A[i]
i += 1
return s
Let us now determine the complexity.
Each query takes on average
Section 1 takes k/2 time on average. (you might iterate atmost k/2)
Section 2 takes n/k time on average, basically number of slices
Section 3 takes k/2 time on average. (you might iterate atmost k/2)
So, totally, we get k/2 + n/k + k/2 = k + n/k time.
And, this is minimized for k = sqrt(n). sqrt(n) + n/sqrt(n) = 2*sqrt(n)
So we get a O(sqrt(n)) time complexity query.
Prime numbers
As mentioned in some other answers, some basic things related to prime numbers take O(sqrt(n)) time:
Find number of divisors
Find sum of divisors
Find Euler's totient
Below I mention two advanced algorithms which also bear sqrt(n) term in their complexity.
MO's Algorithm
try this problem: Powerful array
My solution:
#include <bits/stdc++.h>
using namespace std;
const int N = 1E6 + 10, k = 500;
struct node {
int l, r, id;
bool operator<(const node &a) {
if(l / k == a.l / k) return r < a.r;
else return l < a.l;
}
} q[N];
long long a[N], cnt[N], ans[N], cur_count;
void add(int pos) {
cur_count += a[pos] * cnt[a[pos]];
++cnt[a[pos]];
cur_count += a[pos] * cnt[a[pos]];
}
void rm(int pos) {
cur_count -= a[pos] * cnt[a[pos]];
--cnt[a[pos]];
cur_count -= a[pos] * cnt[a[pos]];
}
int main() {
int n, t;
cin >> n >> t;
for(int i = 1; i <= n; i++) {
cin >> a[i];
}
for(int i = 0; i < t; i++) {
cin >> q[i].l >> q[i].r;
q[i].id = i;
}
sort(q, q + t);
memset(cnt, 0, sizeof(cnt));
memset(ans, 0, sizeof(ans));
int curl(0), curr(0), l, r;
for(int i = 0; i < t; i++) {
l = q[i].l;
r = q[i].r;
/* This part takes O(n * sqrt(n)) time */
while(curl < l)
rm(curl++);
while(curl > l)
add(--curl);
while(curr > r)
rm(curr--);
while(curr < r)
add(++curr);
ans[q[i].id] = cur_count;
}
for(int i = 0; i < t; i++) {
cout << ans[i] << '\n';
}
return 0;
}
Query Buffering
try this problem: Queries on a Tree
My solution:
#include <bits/stdc++.h>
using namespace std;
const int N = 2e5 + 10, k = 333;
vector<int> t[N], ht;
int tm_, h[N], st[N], nd[N];
inline int hei(int v, int p) {
for(int ch: t[v]) {
if(ch != p) {
h[ch] = h[v] + 1;
hei(ch, v);
}
}
}
inline void tour(int v, int p) {
st[v] = tm_++;
ht.push_back(h[v]);
for(int ch: t[v]) {
if(ch != p) {
tour(ch, v);
}
}
ht.push_back(h[v]);
nd[v] = tm_++;
}
int n, tc[N];
vector<int> loc[N];
long long balance[N];
vector<pair<long long,long long>> buf;
inline long long cbal(int v, int p) {
long long ans = balance[h[v]];
for(int ch: t[v]) {
if(ch != p) {
ans += cbal(ch, v);
}
}
tc[v] += ans;
return ans;
}
inline void bal() {
memset(balance, 0, sizeof(balance));
for(auto arg: buf) {
balance[arg.first] += arg.second;
}
buf.clear();
cbal(1,1);
}
int main() {
int q;
cin >> n >> q;
for(int i = 1; i < n; i++) {
int x, y; cin >> x >> y;
t[x].push_back(y); t[y].push_back(x);
}
hei(1,1);
tour(1,1);
for(int i = 0; i < ht.size(); i++) {
loc[ht[i]].push_back(i);
}
vector<int>::iterator lo, hi;
int x, y, type;
for(int i = 0; i < q; i++) {
cin >> type;
if(type == 1) {
cin >> x >> y;
buf.push_back(make_pair(x,y));
}
else if(type == 2) {
cin >> x;
long long ans(0);
for(auto arg: buf) {
hi = upper_bound(loc[arg.first].begin(), loc[arg.first].end(), nd[x]);
lo = lower_bound(loc[arg.first].begin(), loc[arg.first].end(), st[x]);
ans += arg.second * (hi - lo);
}
cout << tc[x] + ans/2 << '\n';
}
else assert(0);
if(i % k == 0) bal();
}
}
There are many cases.
These are the few problems which can be solved in root(n) complexity [better may be possible also].
Find if a number is prime or not.
Grover's Algorithm: allows search (in quantum context) on unsorted input in time proportional to the square root of the size of the input.link
Factorization of the number.
There are many problems that you will face which will demand use of sqrt(n) complexity algorithm.
As an answer to second part:
sqrt(n) complexity means if the input size to your algorithm is n then there approximately sqrt(n) basic operations ( like **comparison** in case of sorting). Then we can say that the algorithm has sqrt(n) time complexity.
Let's analyze the 3rd problem and it will be clear.
let's n= positive integer. Now there exists 2 positive integer x and y such that
x*y=n;
Now we know that whatever be the value of x and y one of them will be less than sqrt(n). As if both are greater than sqrt(n)
x>sqrt(n) y>sqrt(n) then x*y>sqrt(n)*sqrt(n) => n>n--->contradiction.
So if we check 2 to sqrt(n) then we will have all the factors considered ( 1 and n are trivial factors).
Code snippet:
int n;
cin>>n;
print 1,n;
for(int i=2;i<=sqrt(n);i++) // or for(int i=2;i*i<=n;i++)
if((n%i)==0)
cout<<i<<" ";
Note: You might think that not considering the duplicate we can also achieve the above behaviour by looping from 1 to n. Yes that's possible but who wants to run a program which can run in O(sqrt(n)) in O(n).. We always look for the best one.
Go through the book of Cormen Introduction to Algorithms.
I will also request you to read following stackoverflow question and answers they will clear all the doubts for sure :)
Are there any O(1/n) algorithms?
Plain english explanation Big-O
Which one is better?
How do you calculte big-O complexity?
This link provides a very basic beginner understanding of O() i.e., O(sqrt n) time complexity. It is the last example in the video, but I would suggest that you watch the whole video.
https://www.youtube.com/watch?v=9TlHvipP5yA&list=PLDN4rrl48XKpZkf03iYFl-O29szjTrs_O&index=6
The simplest example of an O() i.e., O(sqrt n) time complexity algorithm in the video is:
p = 0;
for(i = 1; p <= n; i++) {
p = p + i;
}
Mr. Abdul Bari is reknowned for his simple explanations of data structures and algorithms.
Primality test
Solution in JavaScript
const isPrime = n => {
for(let i = 2; i <= Math.sqrt(n); i++) {
if(n % i === 0) return false;
}
return true;
};
Complexity
O(N^1/2) Because, for a given value of n, you only need to find if its divisible by numbers from 2 to its root.
JS Primality Test
O(sqrt(n))
A slightly more performant version, thanks to Samme Bae, for enlightening me with this. 😉
function isPrime(n) {
if (n <= 1)
return false;
if (n <= 3)
return true;
// Skip 4, 6, 8, 9, and 10
if (n % 2 === 0 || n % 3 === 0)
return false;
for (let i = 5; i * i <= n; i += 6) {
if (n % i === 0 || n % (i + 2) === 0)
return false;
}
return true;
}
isPrime(677);