Matrix chain multiplication algorithm - algorithm

I am reading Thoman Cormen's "Introduction to Algorithms" and I have problems understanding the algorithm written below.
Matrix-Chain-Order(p)
1 n ← length[p] − 1
2 for i ← 1 to n
3 do m[i, i] ← 0
4 for l ← 2 to n //l is the chain length.
5 do for i ← 1 to n − l + 1 // what is this?
6 do j ← i + l − 1 // what is this?
7 m[i, j] ← ∞
8 for k ← i to j − 1
9 do q ← m[i, k] + m[k + 1, j] + pi−1pkpj
10 if q < m[i, j]
11 then m[i, j] ← q
12 s[i, j] ← k
13 return m and s
Now, I know how the algorithm works. I know how to proceed in constructing the table and all that. In other words I know what happens up to line 4 and I also know what 9 to 13 is about.
I have problems understanding the subtleties of the "for" loops though. Lines 4 to 8 are difficult to understand. In line 5 why does i go up to n-l+1 and why is j in line 6 set to i+l-1. In line 7 ,m[i, j] is initialized for the comparison in line 10 but then again line 8 is a mystery.

I was just going through the algorithm definition on wikipedia and it's pretty comprehensive there. I'll try to explain you how I understood the solution.
The crux of the problem is we are basically trying to 'parenthesise' i.e. prioritize how we chain our matrices so that they are multiplied most efficiently and it's reflected in this line of code:
q = m[i,k] + m[k+1,j] + p[i-1]*p[k]*p[j];
To understand the above stand, first let's establish that i and j are fixed here i.e. we are trying to compute m[i,j] or the most efficient way to multiply matrices A[i..j] and k is the variable.
So at a very high level if i=1 and j=3 and the matrices are :
(A*B)*C //We are trying to establish where the outer most parenthesis should be
We don't know where it should be, hence we try all possibilities and pick the combination where m[i,j] is minimized. So we try:
i=1 and j=3
A*(B*C) //k=1
(A*B)*C //k=2
So clearly k should vary from i to j-1 which is reflected in the loop as we try all possible combinations and take the most efficient one. So for any k we'll have two partitions: A[i..k] and A[k+1...j]
So the cost of multiplication of A[i..j] for this partition of k is:
m[i,k] //Minimum cost of multiplication of A[i..k]
m[k+1,j] //Minimum cost of multiplication of A[k+1..j]
p[i-1]*p[k]*p[j]; //Final cost of multiplying the two partitions i.e. A[i..k] and A[k+1..j], where p contains the dimensions of the matrices.
A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × 60 matrix. Then,
p[] = [10,30,5,60] i.e. Matrix Ai has dimension p[i-1] x p[i] for i = 1..n
This is what dynamic programming is all about. So we try all combinations of k and calculate m[i,j] but for that we also need to calculate m[i,k] and m[k+1,j] i.e. we break our problem down into smaller sub problems where the concept of chain length comes in.
So for all the matrices A[i..n] we calculate the most efficient way of multiplying a smaller chain of matrices of length l.
The smallest value of l is obviously 2 and the largest is n which is what we would get after we solve the smaller sub problems like I explained.
Let's come to the piece of code you are having trouble understanding:
for l ← 2 to n //l is the chain length.
do for i ← 1 to n − l + 1
do j ← i + l − 1
m[i, j] ← ∞
Now let's again consider a smaller example of 4 matrices H,I,J,K and you are looking at first chain lengths of 2. So when traversing the array of matrices.
A[1..4] = H,I,J,K //where A[1] = H and A[4] = K
For l = 2
Our loop should go from i=1 to i=3, as for every i we are looking at the chain of length 2.
So when i = 1, we would compute
m[1,2] i.e. minimum cost to multiply chain (H,I)
and when i = 3, we would compute
m[3,4] i.e. minimum cost to multiply chain (J,K)
When chain length is 3, we would have:
For i=1, j=3
m[i,j] -> m[1,3] i.e. minimum cost to multiply chain (H,I,J)
For i=2, j=4
m[i,j] -> m[2,4] i.e. minimum cost to multiply chain (I,J,K)
Hence when we define i to not exceed n-l+1 and j=i+l-1, we are making sure we are covering all the elements of the array and not exceeding the boundary condition i.e. the size of the array which is n and j defines the end of the chain starting from i with length l.
So the problem comes down to calculating m[i,j] for some i and j which as I explained earlier is solved by taking a partition k and trying out all possible values of k and then re-defining m[i,j] as the minimum value which is why it is initialized as ∞.
I hope my answer wasn't too long and it gives you clarity as to how the algorithm flows and helps you appreciate the sheer vastness of dynamic programming.

Related

Get highest score in this game: choosing and removing elements in an array

Given an array arr of n integers, what is the highest score that a player can reach, playing the following game?
Choose an index 0 < i < n-1 in the array
Add arr[i-1] * arr[i+1] points to the score (initially the score is 0)
Shrink the array by removing element i (forall j >= i: arr[j] = arr[j+1]; then n = n - 1
Repeat steps 1-3 until n == 2.
Do the above until there are only 2 elements (which are the first and the last element because you can't remove them).
What is the highest score you can get ?
Example
arr = [1 2 3 4]
Choose i=2, get: 2*4 = 8 points, remove 3
Remaining: arr = [1 2 4]
Choose i=1, get 1*4 = 4 points, remove 2
Remaining: arr = [1 4].
The sum of points is 8 + 4 = 12, which is the highest possible score on this example.
I think it is related to Dynamic programming but I'm not sure how to solve it.
This problem has a dynamic programming approach similar to Matrix-chain multiplication problem. You can find further explanation in the book "Introduction to Algorithms", 3rd Edition (Cormen, page 370).
Let's find the optimal substructure property and then use it to construct an optimal solution to the problem from optimal solutions to subproblems.
Notation: Ci..j, where i ≤ j, stands for elements Ci,Ci+1,...,Cj.
Definition: A removal sequence for Ci..j is a permutation of i+1,i+2,...,j-1.
A removal sequence for Ci..j is optimal if the score achieved by removing the elements of Ci..j in that order is maximum among all possible removal sequences for Ci..j.
1. Characterize the structure of an optimal solution
If the problem is nontrivial, i.e. i + 1 < j, then any solution has a last removed element which corresponding index is k in the range
i < k < j. Such k split the problem into Ci..k and Ck..j. That is, for some value k, we first remove non extremal elements of Ci..k and Ck..j and then we remove element k. As removing non extremal elements of Ci..k doesn't affect score obtained by removing non extremal elements of Ck..j and an analogous reasoning for removing non extremal elements of Ck..j is also true we state that both subproblems are independent. Then, for a given removal sequence where kth-element is last, the score of Ci..j is equal to the sum of scores of Ci..k and Ck..j, plus the score of removing kth-element (C[i] * C[j]).
The optimal substructure of this problem is as follows. Suppose there is an optimal removal sequence O for Ci..j that ends at kth-element, then the ordering of removed elements from Ci..k must be optimal too. We can prove it by contradiction: If there was a removal sequence for Ci..k that scored higher than removal subsequence extracted from O for Ci..k then we can produce another removal sequence for Ci..j with higher score than optimal removal sequence (contradiction). A similar observation holds for the ordering of removed elements from Ck..j in the optimal removal sequence for Ci..j: it must be optimal too.
We can build an optimal solution for nontrivial instances of the problem by splitting the problem into two subproblems, finding optimal solutions to subproblem instances, and them combining these optimal subproblem solutions.
2. Recursively define the value of an optimal solution.
For this problem our subproblems are the maximum score obtained in Ci..j for 1 ≤ i ≤ j ≤ N. Let S[i, j] be the maximum score obtained in Ci..j; for the full problem, the highest score when evaluating the given rules is S[1, N].
We can define S[i, j] recursively as follows:
If j ≤ i + 1 then S[i, j] = 0
If i + 1 < j then S[i, j] = maxi < k < j{S[i, k] + S[k, j] + C[i] * C[j]}
We ensure that we search for the correct place to split because we consider all possible places, so that we are sure of having examined the optimal one.
3. Compute the value of an optimal solution
You can use your favorite method to compute S:
top-down approach (recursive)
bottom-up approach (iterative)\
I would use bottom-up for computing the solution since it would be < 5 lines long in almost any programming language.
Example in C++11:
for(int l = 2; l <= N; ++l) \\ increasing length intervals
for(int i = 1, j = i + l; j <= N; ++i, ++j)
for(int k = i + 1; k < j; ++k)
S[i, j] = max(S[i, j], S[i, k] + S[k, j] + C[i] * C[j])
4. Time Complexity and Space Complexity
There are nC2 + n = Θ(n2) subproblems and every subproblem do an operation which running time is Θ(l) where l is length of the subproblem so the math yield a running time of Θ(n3) for the algorithm (it's easy to spot the O(n3) part :-)). Also, the algorithm requires Θ(n2) space to store the S table.

arrangement with constraint on the sum

I'm looking to construct an algorithm which gives the arrangements with repetition of n sequences of a given step S (which can be a positive real number), under the constraint that the sum of all combinations is k, with k a positive integer.
My problem is thus to find the solutions to the equation:
x 1 + x 2 + ⋯ + x n = k
where
0 ≤ x i ≤ b i
and S (the step) a real number with finite decimal.
For instance, if 0≤xi≤50, and S=2.5 then xi = {0, 2.5 , 5,..., 47.5, 50}.
The point here is to look only through the combinations having a sum=k because if n is big it is not possible to generate all the arrangements, so I would like to bypass this to generate only the combinations that match the constraint.
I was thinking to start with n=2 for instance, and find all linear combinations that match the constraint.
ex: if xi = {0, 2.5 , 5,..., 47.5, 50} and k=100, then we only have one combination={50,50}
For n=3, we have the combination for n=2 times 3, i.e. {50,50,0},{50,0,50} and {0,50,50} plus the combinations {50,47.5,2.5} * 3! etc...
If xi = {0, 2.5 , 5,..., 37.5, 40} and k=100, then we have 0 combinations for n=2 because 2*40<100, and we have {40,40,20} times 3 for n=3... (if I'm not mistaken)
I'm a bit lost as I can't seem to find a proper way to start the algorithm, knowing that I should have the step S and b as inputs.
Do you have any suggestions?
Thanks
You can transform your problem into an integer problem by dividing everything by S: We want to find all integer sequences y1, ..., yn with:
(1) 0 ≤ yi ≤ ⌊b / S⌋
(2) y1 + ... + yn = k / S
We can see that there is no solution if k is not a multiple of S. Once we have reduced the problem, I would suggest using a pseudopolynomial dynamic programming algorithm to solve the subset sum problem and then reconstruct the solution from it. Let f(i, j) be the number of ways to make sum j with i elements. We have the following recurrence:
f(0,0) = 1
f(0,j) = 0 forall j > 0
f(i,j) = sum_{m = 0}^{min(floor(b / S), j)} f(i - 1, j - m)
We can solve f in O(n * k / S) time by filling it row by row. Now we want to reconstruct the solution. I'm using Python-style pseudocode to illustrate the concept:
def reconstruct(i, j):
if f(i,j) == 0:
return
if i == 0:
yield []
return
for m := 0 to min(floor(b / S), j):
for rest in reconstruct(i - 1, j - m):
yield [m] + rest
result = reconstruct(n, k / S)
result will be a list of all possible combinations.
What you are describing sounds like a special case of the subset sum problem. Once you put it in those terms, you'll find that Pisinger apparently has a linear time algorithm for solving a more general version of your problem, since your weights are bounded. If you're interested in designing your own algorithm, you might start by reading Pisinger's thesis to get some ideas.
Since you are looking for all possible solutions and not just a single solution, the dynamic programming approach is probably your best bet.

Levenstein distance from particular group of numbers

My input are three numbers - a number s and the beginning b and end e of a range with 0 <= s,b,e <= 10^1000. The task is to find the minimal Levenstein distance between s and all numbers in range [b, e]. It is not necessary to find the number minimizing the distance, the minimal distance is sufficient.
Obviously I have to read the numbers as string, because standard C++ type will not handle such large numbers. Calculating the Levenstein distance for every number in the possibly huge range is not feasible.
Any ideas?
[EDIT 10/8/2013: Some cases considered in the DP algorithm actually don't need to be considered after all, though considering them does not lead to incorrectness :)]
In the following I describe an algorithm that takes O(N^2) time, where N is the largest number of digits in any of b, e, or s. Since all these numbers are limited to 1000 digits, this means at most a few million basic operations, which will take milliseconds on any modern CPU.
Suppose s has n digits. In the following, "between" means "inclusive"; I will say "strictly between" if I mean "excluding its endpoints". Indices are 1-based. x[i] means the ith digit of x, so e.g. x[1] is its first digit.
Splitting up the problem
The first thing to do is to break up the problem into a series of subproblems in which each b and e have the same number of digits. Suppose e has k >= 0 more digits than s: break up the problem into k+1 subproblems. E.g. if b = 5 and e = 14032, create the following subproblems:
b = 5, e = 9
b = 10, e = 99
b = 100, e = 999
b = 1000, e = 9999
b = 10000, e = 14032
We can solve each of these subproblems, and take the minimum solution.
The easy cases: the middle
The easy cases are the ones in the middle. Whenever e has k >= 1 more digits than b, there will be k-1 subproblems (e.g. 3 above) in which b is a power of 10 and e is the next power of 10, minus 1. Suppose b is 10^m. Notice that choosing any digit between 1 and 9, followed by any m digits between 0 and 9, produces a number x that is in the range b <= x <= e. Furthermore there are no numbers in this range that cannot be produced this way. The minimum Levenshtein distance between s (or in fact any given length-n digit string that doesn't start with a 0) and any number x in the range 10^m <= x <= 10^(m+1)-1 is necessarily abs(m+1-n), since if m+1 >= n it's possible to simply choose the first n digits of x to be the same as those in s, and delete the remainder, and if m+1 < n then choose the first m+1 to be the same as those in s and insert the remainder.
In fact we can deal with all these subproblems in a single constant-time operation: if the smallest "easy" subproblem has b = 10^m and the largest "easy" subproblem has b = 10^u, then the minimum Levenshtein distance between s and any number in any of these ranges is m-n if n < m, n-u if n > u, and 0 otherwise.
The hard cases: the end(s)
The hard cases are when b and e are not restricted to have the form b = 10^m and e = 10^(m+1)-1 respectively. Any master problem can generate at most two subproblems like this: either two "ends" (resulting from a master problem in which b and e have different numbers of digits, such as the example at the top) or a single subproblem (i.e. the master problem itself, which didn't need to be subdivided at all because b and e already have the same number of digits). Note that due to the previous splitting of the problem, we can assume that the subproblem's b and e have the same number of digits, which we will call m.
Super-Levenshtein!
What we will do is design a variation of the Levenshtein DP matrix that calculates the minimum Levenshtein distance between a given digit string (s) and any number x in the range b <= x <= e. Despite this added "power", the algorithm will still run in O(n^2) time :)
First, observe that if b and e have the same number of digits and b != e, then it must be the case that they consist of some number q >= 0 of identical digits at the left, followed by a digit that is larger in e than in b. Now consider the following procedure for generating a random digit string x:
Set x to the first q digits of b.
Append a randomly-chosen digit d between b[i] and e[i] to x.
If d == b[i], we "hug" the lower bound:
For i from q+1 to m:
If b[i] == 9 then append b[i]. [EDIT 10/8/2013: Actually this can't happen, because we chose q so that e[i] will be larger then b[i], and there is no digit larger than 9!]
Otherwise, flip a coin:
Heads: Append b[i].
Tails: Append a randomly-chosen digit d > b[i], then goto 6.
Stop.
Else if d == e[i], we "hug" the upper bound:
For i from q+1 to m:
If e[i] == 0 then append e[i]. [EDIT 10/8/2013: Actually this can't happen, because we chose q so that b[i] will be smaller then e[i], and there is no digit smaller than 0!]
Otherwise, flip a coin:
Heads: Append e[i].
Tails: Append a randomly-chosen digit d < e[i], then goto 6.
Stop.
Otherwise (if d is strictly between b[i] and e[i]), drop through to step 6.
Keep appending randomly-chosen digits to x until it has m digits.
The basic idea is that after including all the digits that you must include, you can either "hug" the lower bound's digits for as long as you want, or "hug" the upper bound's digits for as long as you want, and as soon as you decide to stop "hugging", you can thereafter choose any digits you want. For suitable random choices, this procedure will generate all and only the numbers x such that b <= x <= e.
In the "usual" Levenshtein distance computation between two strings s and x, of lengths n and m respectively, we have a rectangular grid from (0, 0) to (n, m), and at each grid point (i, j) we record the Levenshtein distance between the prefix s[1..i] and the prefix x[1..j]. The score at (i, j) is calculated from the scores at (i-1, j), (i, j-1) and (i-1, j-1) using bottom-up dynamic programming. To adapt this to treat x as one of a set of possible strings (specifically, a digit string corresponding to a number between b and e) instead of a particular given string, what we need to do is record not one but two scores for each grid point: one for the case where we assume that the digit at position j was chosen to hug the lower bound, and one where we assume it was chosen to hug the upper bound. The 3rd possibility (step 5 above) doesn't actually require space in the DP matrix because we can work out the minimal Levenshtein distance for the entire rest of the input string immediately, very similar to the way we work it out for the "easy" subproblems in the first section.
Super-Levenshtein DP recursion
Call the overall minimal score at grid point (i, j) v(i, j). Let diff(a, b) = 1 if characters a and b are different, and 0 otherwise. Let inrange(a, b..c) be 1 if the character a is in the range b..c, and 0 otherwise. The calculations are:
# The best Lev distance overall between s[1..i] and x[1..j]
v(i, j) = min(hb(i, j), he(i, j))
# The best Lev distance between s[1..i] and x[1..j] obtainable by
# continuing to hug the lower bound
hb(i, j) = min(hb(i-1, j)+1, hb(i, j-1)+1, hb(i-1, j-1)+diff(s[i], b[j]))
# The best Lev distance between s[1..i] and x[1..j] obtainable by
# continuing to hug the upper bound
he(i, j) = min(he(i-1, j)+1, he(i, j-1)+1, he(i-1, j-1)+diff(s[i], e[j]))
At the point in time when v(i, j) is being calculated, we will also calculate the Levenshtein distance resulting from choosing to "stop hugging", i.e. by choosing a digit that is strictly in between b[j] and e[j] (if j == q) or (if j != q) is either above b[j] or below e[j], and thereafter freely choosing digits to make the suffix of x match the suffix of s as closely as possible:
# The best Lev distance possible between the ENTIRE STRINGS s and x, given that
# we choose to stop hugging at the jth digit of x, and have optimally aligned
# the first i digits of s to these j digits
sh(i, j) = if j >= q then shc(i, j)+abs(n-i-m+j)
else infinity
shc(i, j) = if j == q then
min(hb(i, j-1)+1, hb(i-1, j-1)+inrange(s[i], (b[j]+1)..(e[j]-1)))
else
min(hb(i, j-1)+1, hb(i-1, j-1)+inrange(s[i], (b[j]+1)..9),
he(i, j-1)+1, he(i-1, j-1)+inrange(s[i], (0..(e[j]-1)))
The formula for shc(i, j) doesn't need to consider "downward" moves, since such moves don't involve any digit choice for x.
The overall minimal Levenshtein distance is the minimum of v(n, m) and sh(i, j), for all 0 <= i <= n and 0 <= j <= m.
Complexity
Take N to be the largest number of digits in any of s, b or e. The original problem can be split in linear time into at most 1 set of easy problems that collectively takes O(1) time to solve and 2 hard subproblems that each take O(N^2) time to solve using the super-Levenshtein algorithm, so overall the problem can be solved in O(N^2) time, i.e. time proportional to the square of the number of digits.
A first idea to speed up the computation (works if |e-b| is not too large):
Question: how much can the Levestein distance change when we compare s with n and then with n+1?
Answer: not too much!
Let's see the dynamic-programming tables for s = 12007 and two consecutive n
n = 12296
0 1 2 3 4 5
1 0 1 2 3 4
2 1 0 1 2 3
3 2 1 1 2 3
4 3 2 2 2 3
5 4 3 3 3 3
and
n = 12297
0 1 2 3 4 5
1 0 1 2 3 4
2 1 0 1 2 3
3 2 1 1 2 3
4 3 2 2 2 3
5 4 3 3 3 2
As you can see, only the last column changes, since n and n+1 have the same digits, except for the last one.
If you have the dynamic-programming table for the edit-distance of s = 12001 and n = 12296, you already have the table for n = 12297, you just need to update the last column!
Obviously if n = 12299 then n+1 = 12300 and you need to update the last 3 columns of the previous table.. but this happens just once every 100 iteration.
In general, you have to
update the last column on every iterations (so, length(s) cells)
update the second-to-last too, once every 10 iterations
update the third-to-last, too, once every 100 iterations
so let L = length(s) and D = e-b. First you compute the edit-distance between s and b. Then you can find the minimum Levenstein distance over [b,e] looping over every integer in the interval. There are D of them, so the execution time is about:
Now since
we have an algorithm wich is

Time complexity for this relation - matrix chain multiplication

I think an (inefficient) recursive procedure for Matrix chain multiplication problem can be this (based on recurrence relation given in Cormen):
MATRIX-CHAIN(i,j)
if i == j
return 0
if i < j
q = INF
for k = i to j-1
q = min (q, MATRIX-CHAIN(i,k) + MATRIX-CHAIN(k+1, j) + c)
//c = cost of multiplying two sub-matrices.
return q
Time complexity for this will be:
T(n) = summation over k varying from i to j [T(k) + T(n-k)]
Here, n = number of matrices to be multiplied.
What will be the value of T(n) and how?
This is http://en.wikipedia.org/wiki/Catalan_number
You can view the recurrence relation as doing parenthesis. The wiki page describes in depth how to arrive to the formula.
This might help:
you only have to work out each matrix-chain once (and store its value).
start = anywhere between i and j
end = anywhere between start and j
k = anywhere between start and end
if we think of a number with all 0's apart from three 1's (which represent start, k, end)
this special number has j-i+1 digits.
e.g. if i = 3 and j = 6 we need 4 digits giving us the following options:
1101 (i=3, k=4, j=6)
1011 (i=3, k=5, j=6)
0111 (i=4, k=5, j=6)
1110 (i=3, k=4, j=5)
number of choices for i,j,k = Combinations(3, j-i+1)
this is n!/(k! * (n-k)!) = (j-i+1)! / (3! * (j-i+1-3)!)

Number of Positive Solutions to a1 x1+a2 x2+......+an xn=k (k<=10^18)

The question is Number of solutions to a1 x1+a2 x2+....+an xn=k with constraints: 1)ai>0 and ai<=15 2)n>0 and n<=15 3)xi>=0 I was able to formulate a Dynamic programming solution but it is running too long for n>10^10. Please guide me to get a more efficient soution.
The code
int dp[]=new int[16];
dp[0]=1;
BigInteger seen=new BigInteger("0");
while(true)
{
for(int i=0;i<arr[0];i++)
{
if(dp[0]==0)
break;
dp[arr[i+1]]=(dp[arr[i+1]]+dp[0])%1000000007;
}
for(int i=1;i<15;i++)
dp[i-1]=dp[i];
seen=seen.add(new BigInteger("1"));
if(seen.compareTo(n)==0)
break;
}
System.out.println(dp[0]);
arr is the array containing coefficients and answer should be mod 1000000007 as the number of ways donot fit into an int.
Update for real problem:
The actual problem is much simpler. However, it's hard to be helpful without spoiling it entirely.
Stripping it down to the bare essentials, the problem is
Given k distinct positive integers L1, ... , Lk and a nonnegative integer n, how many different finite sequences (a1, ..., ar) are there such that 1. for all i (1 <= i <= r), ai is one of the Lj, and 2. a1 + ... + ar = n. (In other words, the number of compositions of n using only the given Lj.)
For convenience, you are also told that all the Lj are <= 15 (and hence k <= 15), and n <= 10^18. And, so that the entire computation can be carried out using 64-bit integers (the number of sequences grows exponentially with n, you wouldn't have enough memory to store the exact number for large n), you should only calculate the remainder of the sequence count modulo 1000000007.
To solve such a problem, start by looking at the simplest cases first. The very simplest cases are when only one L is given, then evidently there is one admissible sequence if n is a multiple of L and no admissible sequence if n mod L != 0. That doesn't help yet. So consider the next simplest cases, two L values given. Suppose those are 1 and 2.
0 has one composition, the empty sequence: N(0) = 1
1 has one composition, (1): N(1) = 1
2 has two compositions, (1,1); (2): N(2) = 2
3 has three compositions, (1,1,1);(1,2);(2,1): N(3) = 3
4 has five compositions, (1,1,1,1);(1,1,2);(1,2,1);(2,1,1);(2,2): N(4) = 5
5 has eight compositions, (1,1,1,1,1);(1,1,1,2);(1,1,2,1);(1,2,1,1);(2,1,1,1);(1,2,2);(2,1,2);(2,2,1): N(5) = 8
You may see it now, or need a few more terms, but you'll notice that you get the Fibonacci sequence (shifted by one), N(n) = F(n+1), thus the sequence N(n) satisfies the recurrence relation
N(n) = N(n-1) + N(n-2) (for n >= 2; we have not yet proved that, so far it's a hypothesis based on pattern-spotting). Now, can we see that without calculating many values? Of course, there are two types of admissible sequences, those ending with 1 and those ending with 2. Since that partitioning of the admissible sequences restricts only the last element, the number of ad. seq. summing to n and ending with 1 is N(n-1) and the number of ad. seq. summing to n and ending with 2 is N(n-2).
That reasoning immediately generalises, given L1 < L2 < ... < Lk, for all n >= Lk, we have
N(n) = N(n-L1) + N(n-L2) + ... + N(n-Lk)
with the obvious interpretation if we're only interested in N(n) % m.
Umm, that linear recurrence still leaves calculating N(n) as an O(n) task?
Yes, but researching a few of the mentioned keywords quickly leads to an algorithm needing only O(log n) steps ;)
Algorithm for misinterpreted problem, no longer relevant, but may still be interesting:
The question looks a little SPOJish, so I won't give a complete algorithm (at least, not before I've googled around a bit to check if it's a contest question). I hope no restriction has been omitted in the description, such as that permutations of such representations should only contribute one to the count, that would considerably complicate the matter. So I count 1*3 + 2*4 = 11 and 2*4 + 1*3 = 11 as two different solutions.
Some notations first. For m-tuples of numbers, let < | > denote the canonical bilinear pairing, i.e.
<a|x> = a_1*x_1 + ... + a_m*x_m. For a positive integer B, let A_B = {1, 2, ..., B} be the set of positive integers not exceeding B. Let N denote the set of natural numbers, i.e. of nonnegative integers.
For 0 <= m, k and B > 0, let C(B,m,k) = card { (a,x) \in A_B^m × N^m : <a|x> = k }.
Your problem is then to find \sum_{m = 1}^15 C(15,m,k) (modulo 1000000007).
For completeness, let us mention that C(B,0,k) = if k == 0 then 1 else 0, which can be helpful in theoretical considerations. For the case of a positive number of summands, we easily find the recursion formula
C(B,m+1,k) = \sum_{j = 0}^k C(B,1,j) * C(B,m,k-j)
By induction, C(B,m,_) is the convolution¹ of m factors C(B,1,_). Calculating the convolution of two known functions up to k is O(k^2), so if C(B,1,_) is known, that gives an O(n*k^2) algorithm to compute C(B,m,k), 1 <= m <= n. Okay for small k, but our galaxy won't live to see you calculating C(15,15,10^18) that way. So, can we do better? Well, if you're familiar with the Laplace-transformation, you'll know that an analogous transformation will convert the convolution product to a pointwise product, which is much easier to calculate. However, although the transformation is in this case easy to compute, the inverse is not. Any other idea? Why, yes, let's take a closer look at C(B,1,_).
C(B,1,k) = card { a \in A_B : (k/a) is an integer }
In other words, C(B,1,k) is the number of divisors of k not exceeding B. Let us denote that by d_B(k). It is immediately clear that 1 <= d_B(k) <= B. For B = 2, evidently d_2(k) = 1 if k is odd, 2 if k is even. d_3(k) = 3 if and only if k is divisible by 2 and by 3, hence iff k is a multiple of 6, d_3(k) = 2 if and only if one of 2, 3 divides k but not the other, that is, iff k % 6 \in {2,3,4} and finally, d_3(k) = 1 iff neither 2 nor 3 divides k, i.e. iff gcd(k,6) = 1, iff k % 6 \in {1,5}. So we've seen that d_2 is periodic with period 2, d_3 is periodic with period 6. Generally, like reasoning shows that d_B is periodic for all B, and the minimal positive period divides B!.
Given any positive period P of C(B,1,_) = d_B, we can split the sum in the convolution (k = q*P+r, 0 <= r < P):
C(B,m+1, q*P+r) = \sum_{c = 0}^{q-1} (\sum_{j = 0}^{P-1} d_B(j)*C(B,m,(q-c)*P + (r-j)))
+ \sum_{j = 0}^r d_B(j)*C(B,m,r-j)
The functions C(B,m,_) are no longer periodic for m >= 2, but there are simple formulae to obtain C(B,m,q*P+r) from C(B,m,r). Thus, with C(B,1,_) = d_B and C(B,m,_) known up to P, calculating C(B,m+1,_) up to P is an O(P^2) task², getting the data necessary for calculating C(B,m+1,k) for arbitrarily large k, needs m such convolutions, hence that's O(m*P^2).
Then finding C(B,m,k) for 1 <= m <= n and arbitrarily large k is O(n^2*P^2), in time and O(n^2*P) in space.
For B = 15, we have 15! = 1.307674368 * 10^12, so using that for P isn't feasible. Fortunately, the smallest positive period of d_15 is much smaller, so you get something workable. From a rough estimate, I would still expect the calculation of C(15,15,k) to take time more appropriately measured in hours than seconds, but it's an improvement over O(k) which would take years (for k in the region of 10^18).
¹ The convolution used here is (f \ast g)(k) = \sum_{j = 0}^k f(j)*g(k-j).
² Assuming all arithmetic operations are O(1); if, as in the OP, only the residue modulo some M > 0 is desired, that holds if all intermediate calculations are done modulo M.

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