Can I say that:
log n + log (n-1) + log (n-2) + .... + log (n - k) = theta(k * log n)?
Formal way to write the above:
Sigma (i runs from 0 to k) log (n-i) = theta (k* log n)?
If the above statement is right, how can I prove it?
If it is wrong, how can I express it (the left side of the equation, of course) as an asymptotic run time function of n and k?
Thanks.
Denote:
LHS = log(n) + log(n-1) + ... + log(n-k)
RHS = k * log n
Note that:
LHS = log(n*(n-1)*...*(n-k)) = log(polynomial of (k+1)th order)
It follows that this is equal to:
(k+1)*log(n(1 + terms that are 0 in limit))
If we consider a division:
(k+1)*log(n(1 + terms that are 0 in limit)) / RHS
we get in limit:
(k+1)/k = 1 + 1/k
So if k is a constant, both terms grow equally fast. So LHS = theta(RHS).
Wolfram Alpha seems to agree.
When n is constant, terms that previously were 0 in limit don't disappear but instead you get:
(k+1) * some constant number / k * (some other constant number)
So it's:
(1 + 1/k)*(another constant number). So also LHS = theta(RHS).
When proving Θ, you want to prove O and Ω.
Upper bound is proven easily:
log(n(n-1)...(n-k)) ≤ log(n^k) = k log n = O(k log n)
For the lower bound, if k ≥ n/2,
then in the product there is n/2 terms greater than n/2:
log(n(n-1)...(n-k)) ≥ (n/2)log(n/2) = Ω(n log n) ≥ Ω(k log n)
and if k ≤ n/2, all terms are greater than n/2:
log(n(n-1)...(n-k)) ≥ log((n/2)^k) = k log(n/2) = Ω(k log n)
Related
For n/2 + 5 log n, I would of thought the lower order terms of 5 and 2 would be dropped, thus leaving n log n
Where am I going wrong?
Edit:
Thank you, I believe I can now correct my mistake:
O(n/2 + 5 log n) = O(n/2 + log n) = O(n + log n) = O(n)
n/2 + 5 log n <= 2n, for all n >= 1 (c = 2, n0=1)
Let us define the function f as follows for n >= 1:
f(n) = n/2 + 5*log(n)
This function is not O(log n); it grows more quickly than that. To show that, we can show that for any constant c > 0, there is a choice of n0 such that for n > n0, f(n) > c * log(n). For 0 < c <= 5, this is trivial, since f(n) > [5log(n)] by definition. For c > 5, we get
n/2 + 5*log(n) > c*log(n)
<=> n/2 > (c - 5)*log(n)
<=> (1/(2(c - 5))*n/log(n) > 1
We can now note that the expression on the LHS is monotonically increasing for n > 1 and find the limit as n grows without bound using l'Hopital:
lim(n->infinity) (1/(2(c - 5))*n/log(n)
= (1/(2(c - 5))* lim(n->infinity) n/log(n)
= (1/(2(c - 5))* lim(n->infinity) 1/(1/n)
= (1/(2(c - 5))* lim(n->infinity) n
-> infinity
Using l'Hopital we find there is no limit as n grows without bound; the value of the LHS grows without bound as well. Because the LHS is monotonically increasing and grows without bound, there must be an n0 after which the value of the LHS exceeds the value 1, as required.
This all proves that f is not O(log n).
It is true that f is O(n log n). This is not hard to show at all: choose c = (5+1/2), and it is obvious that
f(n) = n/2 + 5log(n) <= nlog(n)/2 + 5nlog(n) = (5+1/2)nlog(n) for all n.
However, this is not the best bound we can get for your function. Your function is actually O(n) as well. Choosing the same value for c as before, we need only notice that n > log(n) for all n >= 1, so
f(n) = n/2 + 5log(n) <= n/2 + 5n = (5+1/2)n
So, f is also O(n). We can show that f(n) is Omega(n) which proves it is also Theta(n). That is left as an exercise but is not difficult to do either. Hint: what if you choose c = 1/2?
It's neither O(log n) nor O(n*log n). It'll be O(n) because for large value of n log n is much smaller than n hence it'll be dropped.
It's neither O(log n) nor O(n*log n). It'll be O(n) because for larger values of n log(n) is much smaller than n hence it'll be dropped.
Consider n=10000, now 5log(n) i.e 5*log(10000)=46(apprx) which is less than n/2(= 5000).
Can we say O(K + (N-K)logK) is equivalent to O(K + N logK) for 1 < = K <= N?
The short answer is they are not equivalent and it depends on the value of k. If k is equal to N, then the first complexity is O(N), and the second complexity is O(N + Nlog N) which is equivalent to O(NlogN). However, O(N) is not equivalent to O(N log N).
Moreover, if a function is in O(K + (N-K) log K) is in O(K + N log K) (definitely for every positive K), and the proof of this is straightforward.
Yes because in the worst case (N-K) logK is at most N logK given your constraints since 1 <= K <= N.
Not exactly.
If they are equivalent, then every function in O(k + (n-k)log k) is also in O(k + n log k) and vice-versa.
Let f(n,k) = n log k
This function is certainly in O(k + n log k), but not in O(k + (n-k)log k).
Let g(n,k) = k + (n-k)log k
Then as x approaches infinity, f(x,x)/g(x,x) grows without bound, since:
f(x,x) / g(x,x)
= (x log x) / x
= log x
See the definition of big-O notation for multiple variables: http://mathwiki.cs.ut.ee/asymptotics/04_multiple_variables
Wikipedia provides the same information, but in less accessible notation:
https://en.wikipedia.org/wiki/Big_O_notation#Multiple_variables
I have an algorithm that have the following complexity in big O:
log(N) + log(N+1) + log(N+2) + ... + log(N+M)
Is it the same as log(N+M) since it is the largest element?
OR is it M*log(N+M) because it is a sum of M elements?
The important rules to know in order to solve it are:
Log a + Log b = Log ab, and
Log a - Log b = Log a/b
Add and subtract Log 2, Log 3, ... Log N-1 to the given value.
This will give you Log 2 + Log 3 + ... + Log (N+M) - (Log 2 + Log 3 + ... + Log (N-1))
The first part will compute to Log ((N+M)!) and the part after the subtraction sign will compute to Log ((N-1)!)
Hence, this complexity comes to Log ( (N+M)! / (N-1)! ).
UPDATE after OP asked another good question in the comment:
If we have N + N^2 + N^3, it will reduce to just N^3 (the largest element), right? Why we can't apply the same logic here - log(N+M) - largest element?
If we have just two terms that look like Log(N) + Log(M+N), then we can combine them both and say that they will definitely be less than 2 * Log(M+N) and will therefore be O(Log (M+N)).
However, if there is a relation between the number of items being summed and highest value of the item, then the presence of such a relation makes the calculation slightly not so straightforward.
For example, the big O of addition of 2 (Log N)'s, is O(Log N), while the big O of summation of N Log N's is not O(Log N) but is O(N * Log N).
In the given summation, the value and the number of total values is dependent on M and N, therefore we cannot put this complexity as Log(M+N), however, we can definitely write it as M * (Log (M+N)).
How? Each of the values in the given summation is less than or equal to Log(M + N), and there are total M such values. Hence the summation of these values will be less than M * (Log (M+N)) and therefore will be O(M * (Log (M+N))).
Thus, both answers are correct but O(Log ( (N+M)! / (N-1)! )) is a tighter bound.
If M does not depend on N and does not vary then the complexity is O(log(N))
For k such as 0 <= k <= M and N>=M and N>=2,
log(N+k)=log(N(1+k/N)) = log(N) + log(1+k/N) <= log(N) + log(2)
<= log(N) + log(N) <= 2 log(N)
So
log(N) + log(N+1) + log(N+2) + ... + log(N+M) <= (M+1)2 log(N)
So the complexity in big O is: log(N)
To answer your questions:
1) yes because there is a fixed number of elements all less or equal than log(N+M)
2) In fact there are M + 1 elements (from 0 to M)
I specify that O((M+1)log(N+M)) is a O(log(N))
The question comes from Introduction to Algorithms 3rd Edition, P63, Problem 3-6, where it's introduced as Iterated functions. I rewrite it as follows:
int T(int n){
for(int count = 0; n > 2 ; ++count)
{
n = n/log₂(n);
}
return count;
}
Then give as tight a bound as possible on T(n).
I can make it O(log n) and Ω(log n / log log n), but can it be tighter?
PS: Using Mathematica, I've learned that when n=1*10^3281039, T(n)=500000
and the same time, T(n)=1.072435*log n/ log log n
and the coefficient declines with n from 1.22943 (n = 2.07126*10^235) to 1.072435 (n = 1*10^3281039).
May this information be helpful.
It looks like the lower bound is pretty good, so I tried to proof that the upper bound is O(log n / log log n).
But let me first explain the other bounds (just for a better understanding).
TL;DR
T(n) is in Θ(log n / log log n).
T(n) is in O(log n)
This can be seen by modifying n := n/log₂n to n := n/2.
It needs O(log₂ n) steps until n ≤ 2 holds.
T(n) is in Ω(log n / log log n)
This can be seen by modifying n := n/log₂(n) to n := n/m, where m is the initial value of log n.
Solving the equation
n / (log n)x < 2 for x leads us to
log n - x log log n < log 2
⇔ log n - log 2 < x log log n
⇔ (log n - log 2) / log log n < x
⇒ x ∈ Ω(log n / log log n)
Improving the upper bound: O(log n) → O(log n / log log n)
Now let us try to improve the upper bound. Instead of dividing n by a fixed constant (namely 2 in the above proof) we divide n as long by the initial value of log(n)/2 as the current value of log(n) is bigger. To be more clearer have a look at the modified code:
int T₂(int n){
n_old = n;
for(int count=0; n>2 ;++count)
{
n = n / (log₂(n_old)/2);
if(log₂(n)) <= log₂(n_old)/2)
{
n_old = n;
}
}
return count;
}
The complexity of the function T₂ is clearly an upper bound for the function T, since log₂(n_old)/2 < log₂(n) holds for the whole time.
Now we need to know how many times we divide by each 1/2⋅log(n_old):
n / (log(sqrt(n)))x ≤ sqrt(n)
⇔ n / sqrt(n) ≤ log(sqrt(n))x
⇔ log(sqrt(n)) ≤ x log(log(sqrt(n)))
⇔ log(sqrt(n)) / log(log(sqrt(n))) ≤ x
So we get the recurrence formula T₂(n) = T(sqrt(n)) + O(log(sqrt(n)) / log(log(sqrt(n)))).
Now we need to know how often this formula has to be expanded until n < 2 holds.
n2-x < 2
⇔ 2-x⋅log n < log 2
⇔ -x log 2 + log log n < log 2
⇔ log log n < log 2 + x log 2
⇔ log log n < (x + 1) log 2
So we need to expand the formula about log log n times.
Now it gets a little bit harder. (Have also a look at the Mike_Dog's answer)
T₂(n) = T(sqrt(n)) + log(sqrt(n)) / log(log(sqrt(n)))
= Σk=1,...,log log n - 1 2-k⋅log(n) / log(2-k⋅log n))
= log(n) ⋅ Σk=1,...,log log n - 1 2-k / (-k + log log n))
(1) = log(n) ⋅ Σk=1,...,log log n - 1 2k - log log n / k
= log(n) ⋅ Σk=1,...,log log n - 1 2k ⋅ 2- log log n / k
= log(n) ⋅ Σk=1,...,log log n - 1 2k / (k ⋅ log n)
= Σk=1,...,log log n - 1 2k / k
In the line marked with (1) I reordered the sum.
So, at the end we "only" have to calculate Σk=1,...,t 2k / k for t = log log n - 1. At this point Maple solves this to
Σk=1,...,t 2k / k = -I⋅π - 2t⋅LerchPhi(2, 1, t) +2t/t
where I is the imaginary unit and LerchPhi is the Lerch transcendent. Since the result for the sum above is a real number for all relevant cases, we can just ignore all imaginary parts. The Lerch transcendent LerchPhi(2,1,t) seems to be in O(-1/t), but I'm not 100% sure about it. Maybe someone will prove this.
Finally this results in
T₂(n) = -2t⋅O(-1/t) + 2t/t = O(2t/t) = O(log n / log log n)
All together we have T(n) ∈ Ω(log n / log log n) and T(n) ∈ O(log n/ log log n),
so T(n) ∈ Θ(log n/ log log n) holds. This result is also supported by your example data.
I hope this is understandable and it helps a little.
The guts of the problem of verifying the conjectured estimate is to get a good estimate of plugging the value
n / log(n)
into the function
n --> log(n) / log(log(n))
Theorem:
log( n/log(n) ) / log(log( n/log(n) )) = log(n)/log(log(n)) - 1 + o(1)
(in case of font readibility issues, that's little-oh, not big-oh)
Proof:
To save on notation, write
A = n
B = log(n)
C = log(log(n))
The work is based on the first-order approximation to the (natural) logarithm: when 0 < y < x,
log(x) - y/x < log(x - y) < log(x)
The value we're trying to estimate is
log(A/B) / log(log(A/B)) = (B - C) / log(B - C)
Applying the bounds for the logarithm of a difference gives
(B-C) / log(B) < (B-C) / log(B-C) < (B-C) / (log(B) - C/B)
that is,
(B-C) / C < (B-C) / log(B-C) < (B-C)B / (C (B-1))
Both the recursion we're trying to satisfy and the lower bound suggest we should estimate this with B/C - 1. Pulling that off of both sides gives
B/C - 1 < (B-C) / log(B-C) < B/C - 1 + (B-C)/(C(B-1))
and thus we conclude
(B-C) / log(B-C) = B/C - 1 + o(1)
If you take away one idea from this analysis to use on your own, let it be the point of using differential approximations (or even higher order Taylor series) to replace complicated functions with simpler ones. e.g. once you have the idea to use
log(x-y) = log(x) + Θ(y/x) when y = o(x)
then all of the algebraic calculations you need for your problem simply follow directly.
Thanks for the answer of #AbcAeffchen
I'm the owner of the question, using the knowledge of "the master method" I learned yesterday, the "a little bit harder" part of proof can be done as follows simply.
I will start here:
T(n) = T(sqrt(n)) + O(log(sqrt(n)) / log(log(sqrt(n))))
⇔ T(n)=T(sqrt(n)) + O(log n / log log n)
Let
n=2k , S(k)=T(2k)
then we have
T(2k) =T(2k/2) + O(log 2k / log log
2k) ⇔ S(k) =S(k/2) + O( k/log k)
with the master method
S(k)=a*S(k/b)+f(k), where a=1, b=2, f(k)=k/log k
= Ω(klog21 +ε) = Ω(kε)，
as long as ε∈(0,1)
so we can apply case 3. Then
S(k) = O(k/log k)
T(n) = S(k) = O(k/log k) = O(log n/ log log n)
Consider I get f(n)=log(n*log n). Should I say that its O(log(n*log n)?
Or should I do log(n*log n)=log n + log(log n) and then say that the function f(n) is O(log n)?
First of all, as you have observed:
log(n*log n) = log(n) + log(log(n))
but think about log(log N) as N->large (as Floris suggests).
For example, let N = 1000, then log N = 3 (i.e. a small number) and log(3) is even smaller,
this holds as N gets huge, i.e. way more than the number of instructions your code could ever generate.
Thus, O(log(n * log n)) = O(log n + k) = O(log(n)) + k = O(log n)
Another way to look at this is that: n * log n << n^2, so in the worse case:
O(log(n^2)) > O(log(n * log n))
So, 2*O(log(n)) is an upper bound, and O(log(n * log n)) = O(log n)
Use the definition. If f(n) = O(log(n*log(n))), then there must exist a positive constant M and real n0 such that:
|f(n)| ≤ M |log(n*log(n))|
for all n > n0.
Now let's assume (without loss of generality) that n0 > 0. Then
log(n) ≥ log(log(n))
for all n > n0.
From this, we have:
log(n(log(n)) = log(n) + log(log(n)) ≤ 2 * log(n)
Substituting, we find that
|f(n)| ≤ 2*M|log(n))| for all n > n0
Since 2*M is also a positive constant, it immediately follows that f(n) = O(log(n)).
Of course in this case simple transformations show both functions differ by a constant factor asymptotically, as shown.
However, I feel like it is worthwhile remind a classic test for analyzing how two functions relate to each other asymptotically. So here's a little more formal proof.
You can check how does f(x) relates to g(x) by analyzing lim f(x)/g(x) when x->infinity.
There are 3 cases:
lim = infinty <=> O(f(x)) > O(g(x))
inf > lim > 0 <=> O(f(x)) = O(g(x))
lim = 0 <=> O(f(x)) < O(g(x))
So
lim ( log( n * log(n) ) / log n ) =
lim ( log n + log log (n) ) / log n =
lim 1 + log log (n) / log n =
1 + 0 = 1
Note: I assumed log log n / log n to be trivial but you can do it by de l'Hospital Rule.