I am currently trying to solve the above recurrence relation but am having trouble trying to decipher the pattern an rewrite it as a sum. Could anyone help me out?
k >= 0. T(n<=2) = 1.
This recurrence relation was obtained from an algorithm I wrote to obtain a single sorted array from an array that is k sorted (meaning that every k'th element is in sorted order). This algorithm runs at most k times. Each time k is reduced by one and every k'th element is added to another array. Finally each array is merged using the merge from merge sort (n time). This algorithm is called recursively until k = 0, meaning we have found each sorted sub array.
I have a feeling that this is O(k*n), but I am not sure.
It might help to note that
n - n/k = ((k - 1) / k)n,
so your recurrence relation represents n decaying geometrically by a factor of (k-1)/k at each step. To see how much work is done, let a = (k-1)/k. Then the work done is upper-bounded by
n + an + a2n + a3n + ...
= n / (1 - a)
= n / (1 / k)
= nk.
So your total work is O(nk).
As a note, I haven’t checked whether the recurrence relation you have matches your code - I’m just showing the math here. :-)
Related
int recursiveFunc(int n) {
if (n == 1) return 0;
for (int i = 2; i < n; i++)
if (n % i == 0) return i + recursiveFunc(n / i);
return n;
}
I know Complexity = length of tree from root node to leaf node * number of leaf nodes, but having hard time to come to an equation.
This one is tricky, because the runtime is highly dependent on what number you provide in as input in a way that most recursive functions are not.
For starters, notice that the way that this recursion works, it takes in a number and then either
returns without making any further calls if the number is prime, or
recursively calls itself on number divided by that proper factor.
This means that in one case, the function, called on a number n, will do Θ(n) work and make no calls (which happens if the number is prime), and in the other case will do Θ(d) work and then make a recursive call on the number n / d, which happens if n is composite and is the largest divisor of n.
One useful fact we'll use to analyze this function is that given a composite number n, the smallest factor d of n is never any greater than √n. If it were, then we would have that n = df for some other factor f, and since d is the smallest proper divisor, we'd have that f ≥ d, so df > √n √ n = n, which would be impossible.
With that in mind, we can argue that the worst-case runtime of this function is O(n), and in fact that happens when n is prime. Here's how to see this. Imagine the worst-case amount of time this function can take if it ends up making a recursive call. In that case, the function will do at most Θ(√n) work (let's assume our smallest divisor is as large as possible), then recursively makes a call on a number whose size is at most n / 2 (which is the absolute largest number we could get as part of the recursive call. In that case, we'd get this recurrence relation under the pessimistic assumption that we do the maximum work possible
T(n) = T(n / 2) + √n
This solves, by the Master Theorem, to Θ(√n), which is less work than what we'd do if we had a prime number as an input.
But what happens if, instead, we do the maximum amount of work possible for some number of iterations, and then end up with a prime number and stop? In that case, using the iteration method, we'd see that the work done would be
n1/2 + n1/4 + ... + n / 2k,
which would happen if we stopped after k iterations. In this case, notice that this expression is maximized when we pick k to be as small as possible - which would correspond to stopping as soon as possible, which happens if we pick a prime number for n.
So in this sense, the worst-case runtime of this function is Θ(n), which happens for n being a prime number, with composite numbers terminating much faster than this.
So how fast can this function be? Well, imagine, for example, that we have a number of the form pk, where p is some prime number. In that case, this function will do Θ(p) work to discover p as a prime factor, then recursively call itself on the number pk-1. If you think about what this will look like, this function will end up doing Θ(p) work Θ(k) times for a total runtime of Θ(pk). And since n = pk, we'd have k = logp n, so the runtime would be Θ(p logp n). That's minimized at either p = 2 or p = 3, and in either case gives us a runtime of Θ(log n) in this case.
I strongly suspect that's the best case here, though I'm not entirely sure. But what this does mean is that
the worst-case runtime is definitely Θ(n), occurring at prime numbers, and
the best-case runtime is O(log n), which I'm fairly certain is a tight bound but I'm not 100% sure how to prove.
Original problem:
Let N be a positive integer (actually, N <= 2000) and P - set of all possible partitions of the N, where with and . Let A be the number of partitions . Find the A.
Input: N. Output: A - the number of partitions .
What have I tried:
I think that this problem can be solved by dynamic-based algorithm. Let p(n,a,b) be the function, which returns the number of partitons of n using only numbers a. . .b. Then we can compute the A with the code like:
int Ans = 2; // the 1+1+...+1=N & N=N partitions
for(int a = 2; a <= N/2; a += 1){ //a - from 2 to N/2
int b = a*2-1;
Ans += p[N][a][b]; // add all partitions using a..b to Answer
if(a < (a-1)*2-1){ // if a < previous b [ (a-1)*2-1 ]
Ans -= p[N][a][(a-1)*2-1]; // then we counted number of partitions
} // using numbers a..prev_b twice.
}
Next I tried to find the dynamic algorithm computing p(n,a,b) for any integer a <= b <= n. This paper (.pdf) provides the folowing algorithm:
, were I(n<=b) = 1 if n<=b and =0 otherwise.
Question(s):
How should I realize the algorithm from the paper? I'm new at d-p problems and as I can see, this problem has 3 dimensions (n,a & b), which is quite tricky for me.
How actually that algorithm works? I know how work the algorithms for computing p(n,0,b) or p(n,a,n), but a little explanation for p(n,a,b) will be very helpful.
Does original problem have simpler solution? I'm quite sure that there's another clean solution, but I didn't found it.
I calculated all A(1)-A(600) in 23 seconds with memoization approach (top-down dynamic programming). 3D table requires 1.7 GB of memory.
For reference: A[50] = 278, A(200)=465202, A(600)=38860513616
N=2000 requires too large table for 32-bit environment, and map approach worked too slow.
I can make 2D table with reasonable size, but this approach requires table zeroing at every iteration of external loop - slow again.
A(1000) = 107292471486730 in 131 sec. And I think that long arithmetic might be needed for larger values to avoid Int64 overflow.
Given an array of size n (1<=n<=200000) with each entry a[i] ( 1<=a[i]<=100), find all the number of subsequences that form Arithmetic progression.
Subsequences are the sequences where you can leave any number of elements in the original sequence.
For example, the sequence A,B,D is a subsequence of A,B,C,D,E,F obtained after removal of elements C, E and F. The relation of one sequence being the subsequence of another is a preorder.
I have written O(n^2) solution using DP. But n^2 = 10^10. So, It'll not get accepted.
Here is what I did.
Pseudocode:
for every element A[i]:
for every element A[k] such that k<i:
diff = A[i] - A[k] + 100: (adding 100, -ve differences A.P.)
dp[i][diff] += dp[i-1][diff] + 1;
for every element A[i]:
for every diff, d:
ans = ans + dp[i][d];
return ans;
This is giving correct output but TLE for 3 big cases.
P.S. Please suggest better solution..!!
Is divide and conquer optimization DP required here?? If yes, tell me how to build the solution.
The josephus problem can be solved by the below recursion:
josephus(n, k) = (josephus(n - 1, k) + k-1) % n + 1
josephus(1, k) = 1
How this recurrence relation has been derived?
josephus(n, k) = (josephus(n - 1, k) + k-1) % n + 1 ...... (1)
To put it in simple words -
starting with the "+1" in the formula. It implies that 1 iteration of the recurrence has already been done. Now, we would be left with n-1 persons/elements. We need to process n-1 elements recursively at intervals of k. But, now, since the last element to be removed was at kth location, we would continue from thereof. Thus, k-1 added. Further, this addition might upset the indexing of the array. Thus %n done to keep the array index within the bounds.
Hope it is lucid and elaborate enough :) .
This paragraph is sufficient from wikipedia..
When the index starts from one, then the person at s shifts from the
first person is in position ((s-1)\bmod n)+1, where n is the total
number of persons. Let f(n,k) denote the position of the survivor.
After the k-th person is killed, we're left with a circle of n-1, and
we start the next count with the person whose number in the original
problem was (k\bmod n)+1. The position of the survivor in the
remaining circle would be f(n-1,k) if we start counting at 1;
shifting this to account for the fact that we're starting at (k\bmod
n)+1 yields the recurrence
f(n,k)=((f(n-1,k)+k-1) \bmod n)+1,\text{ with }f(1,k)=1\,,
Recently I have been studying recursion; how to write it, analyze it, etc. I have thought for a while that recurrence and recursion were the same thing, but some problems on recent homework assignments and quizzes have me thinking there are slight differences, that 'recurrence' is the way to describe a recursive program or function.
This has all been very Greek to me until recently, when I realized that there is something called the 'master theorem' used to write the 'recurrence' for problems or programs. I've been reading through the wikipedia page, but, as usual, things are worded in such a way that I don't really understand what it's talking about. I learn much better with examples.
So, a few questions:
Lets say you are given this recurrence:
r(n) = 2*r(n-2) + r(n-1);
r(1) = r(2)
= 1
Is this, in fact, in the form of the master theorem? If so, in words, what is it saying? If you were to be trying to write a small program or a tree of recursion based on this recurrence, what would that look like? Should I just try substituting numbers in, seeing a pattern, then writing pseudocode that could recursively create that pattern, or, since this may be in the form of the master theorem, is there a more straightforward, mathematical approach?
Now, lets say you were asked to find the recurrence, T(n), for the number of additions performed by the program created from the previous recurrence. I can see that the base case would probably be T(1) = T(2) = 0, but I'm not sure where to go from there.
Basically, I am asking how to go from a given recurrence to code, and the opposite. Since this looks like the master theorem, I'm wondering if there is a straightforward and mathematical way of going about it.
EDIT: Okay, I've looked through some of my past assignments to find another example of where I'm asked, 'to find the recurrence', which is the part of this question I'm having the post trouble with.
Recurrence that describes in the best
way the number of addition operations
in the following program fragment
(when called with l == 1 and r == n)
int example(A, int l, int r) {
if (l == r)
return 2;
return (A[l] + example(A, l+1, r);
}
A few years ago, Mohamad Akra and Louay Bazzi proved a result that generalizes the Master method -- it's almost always better. You really shouldn't be using the Master Theorem anymore...
See, for example, this writeup: http://courses.csail.mit.edu/6.046/spring04/handouts/akrabazzi.pdf
Basically, get your recurrence to look like equation 1 in the paper, pick off the coefficients, and integrate the expression in Theorem 1.
Zachary:
Lets say you are given this
recurrence:
r(n) = 2*r(n-2) + r(n-1); r(1) = r(2)
= 1
Is this, in fact, in the form of the
master theorem? If so, in words, what
is it saying?
I think that what your recurrence relation is saying is that for function of "r" with "n" as its parameter (representing the total number of data sets you're inputting), whatever you get at the nth position of the data-set is the output of the n-1 th position plus twice whatever is the result of the n-2 th position, with no non-recursive work being done. When you try to solve a recurrence relation, you're trying to go about expressing it in a way that doesn't involve recursion.
However, I don't think that that is in the correct form for the Master Theorem Method. Your statement is a "second order linear recurrence relation with constant coefficients". Apparently, according to my old Discrete Math textbook, that's the form you need to have in order to solve the recurrence relation.
Here's the form that they give:
r(n) = a*r(n-1) + b*r(n-2) + f(n)
For 'a' and 'b' are some constants and f(n) is some function of n. In your statement, a = 1, b = 2, and f(n) = 0. Whenever, f(n) is equal to zero the recurrence relation is known as "homogenous". So, your expression is homogenous.
I don't think that you can solve a homogenous recurrence relation using the Master Method Theoerm because f(n) = 0. None of the cases for Master Method Theorem allow for that because n-to-the-power-of-anything can't equal zero. I could be wrong, because I'm not really an expert at this but I don't that it's possible to solve a homogenous recurrence relation using the Master Method.
I that that the way to solve a homogeneous recurrence relation is to go by 5 steps:
1) Form the characteristic equation, which is something of the form of:
x^k - c[1]*x^k-1 - c[2]*x^k-2 - ... - c[k-1]*x - c[k] = 0
If you've only got 2 recursive instances in your homogeneous recurrence relation then you only need to change your equation into the Quadratic Equation where
x^2 - a*x - b = 0
This is because a recurrence relation of the form of
r(n) = a*r(n-1) + b*r(n-2)
Can be re-written as
r(n) - a*r(n-1) - b*r(n-2) = 0
2) After your recurrence relation is rewritten as a characteristic equation, next find the roots (x[1] and x[2]) of the characteristic equation.
3) With your roots, your solution will now be one of the two forms:
if x[1]!=x[2]
c[1]*x[1]^n + c[2]*x[2]^n
else
c[1]*x[1]^n + n*c[2]*x[2]^n
for when n>2.
4) With the new form of your recursive solution, you use the initial conditions (r(1) and r(2)) to find c[1] and c[2]
Going with your example here's what we get:
1)
r(n) = 1*r(n-1) + 2*r(n-2)
=> x^2 - x - 2 = 0
2) Solving for x
x = (-1 +- sqrt(-1^2 - 4(1)(-2)))/2(1)
x[1] = ((-1 + 3)/2) = 1
x[2] = ((-1 - 3)/2) = -2
3) Since x[1] != x[2], your solution has the form:
c[1](x[1])^n + c[2](x[2])^n
4) Now, use your initial conditions to find the two constants c[1] and c[2]:
c[1](1)^1 + c[2](-2)^1 = 1
c[1](1)^2 + c[2](-2)^2 = 1
Honestly, I'm not sure what your constants are in this situation, I stopped at this point. I guess you'd have to plug in numbers until you'd somehow got a value for both c[1] and c[2] which would both satisfy those two expressions. Either that or perform row reduction on a matrix C where C equals:
[ 1 1 | 1 ]
[ 1 2 | 1 ]
Zachary:
Recurrence that describes in the best
way the number of addition operations
in the following program fragment
(when called with l == 1 and r == n)
int example(A, int l, int r) {
if (l == r)
return 2;
return (A[l] + example(A, l+1, r);
}
Here's the time complexity values for your given code for when r>l:
int example(A, int l, int r) { => T(r) = 0
if (l == r) => T(r) = 1
return 2; => T(r) = 1
return (A[l] + example(A, l+1, r); => T(r) = 1 + T(r-(l+1))
}
Total: T(r) = 3 + T(r-(l+1))
Else, when r==l then T(r) = 2, because the if-statement and the return both require 1 step per execution.
Your method, written in code using a recursive function, would look like this:
function r(int n)
{
if (n == 2) return 1;
if (n == 1) return 1;
return 2 * r(n-2) + r(n-1); // I guess we're assuming n > 2
}
I'm not sure what "recurrence" is, but a recursive function is simply one that calls itself.
Recursive functions need an escape clause (some non-recursive case - for example, "if n==1 return 1") to prevent a Stack Overflow error (i.e., the function gets called so much that the interpreter runs out of memory or other resources)
A simple program that would implement that would look like:
public int r(int input) {
if (input == 1 || input == 2) {
return 1;
} else {
return 2 * r(input - 2) + r(input -1)
}
}
You would also need to make sure that the input is not going to cause an infinite recursion, for example, if the input at the beginning was less than 1. If this is not a valid case, then return an error, if it is valid, then return the appropriate value.
"I'm not exactly sure what 'recurrence' is either"
The definition of a "recurrence relation" is a sequence of numbers "whose domain is some infinite set of integers and whose range is a set of real numbers." With the additional condition that that the function describing this sequence "defines one member of the sequence in terms of a previous one."
And, the objective behind solving them, I think, is to go from a recursive definition to one that isn't. Say if you had T(0) = 2 and T(n) = 2 + T(n-1) for all n>0, you'd have to go from the expression "T(n) = 2 + T(n-1)" to one like "2n+2".
sources:
1) "Discrete Mathematics with Graph Theory - Second Edition", by Edgar G. Goodair and Michael M. Parmenter
2) "Computer Algorithms C++," by Ellis Horowitz, Sartaj Sahni, and Sanguthevar Rajasekaran.