How to count number of (weakly) decreasing sequences of numbers? - algorithm

I want to solve a problem without loops. Assume we have N place, and for each place you can select a number between [p,q] but the number in place i+1 should be less or equal than the number in place i. Now how to count all possible strings without brute force.
For example assume we have 2 places and you can select a number between [2,3] then the possible sequences can be:
3 3
3 2
2 2
As the number of places is not limited also the p an q then it is impossible to solve it with simple loops.

It's choose(q-p+N, N) where choose is the binomial coefficient.
Weakly decreasing sequences that are between p and q are in bijection with sequences of 0's and 1's of length q-p+N, where the sequences have exactly N ones. It's obvious that the number of such sequences is choose(q-p+N, N) because that's the number of ways of choosing N things from q-p+N things.
The proof that the two sets are in bijection
Given a sequence xs of 0's and 1's of length q-p+N with N 1's, this pseudocode generates a weakly decreasing sequence of numbers between p and q:
c = q
for x in xs
if x = 1 then output c
if x = 0 then c <- c - 1
Conversely, given an weakly decreasing sequence xs of length N between p and q, this pseudocode generates a sequence of 0's and 1's of length q-p+N with exactly N 1's.
c = q
while xs is not empty
if c = head(hs) then
output 1
xs <- tail(xs)
else
output 0
c <- c - 1
while c > p
output 0
c <- c - 1
Here head(xs) denotes the first thing in the xs list, and tail(xs) the remainder of the list.

Related

XOR of numbers = X

I found this problem in a hiring contest(which is over now). Here it is:
You are given two natural numbers N and X. You are required to create an array of N natural numbers such that the bitwise XOR of these numbers is equal to X. The sum of all the natural numbers that are available in the array is as minimum as possible.
If there exist multiple arrays, print the smallest one
Array A< Array B if
A[i] < B[i] for any index i, and A[i]=B[i] for all indices less than i
Sample Input: N=3, X=2
Sample output : 1 1 2
Explanation: We have to print 3 natural numbers having the minimum sum Thus the N-spaced numbers are [1 1 2]
My approach:
If N is odd, I put N-1 ones in the array (so that their xor is zero) and then put X
If N is even, I put N-1 ones again and then put X-1(if X is odd) and X+1(if X is even)
But this algorithm failed for most of the test cases. For example, when N=4 and X=6 my output is
1 1 1 7 but it should be 1 1 2 4
Anyone knows how to make the array sum minimum?
In order to have the minimum sum, you need to make sure that when your target is X, you are not cancelling the bits of X and recreating them again. Because this will increase the sum. For this, you have create the bits of X one by one (ideally) from the end of the array. So, as in your example of N=4 and X=6 we have: (I use ^ to show xor)
X= 7 = 110 (binary) = 2 + 4. Note that 2^4 = 6 as well because these numbers don't share any common bits. So, the output is 1 1 2 4.
So, we start by creating the most significant bits of X from the end of the output array. Then, we also have to handle the corner cases for different values of N. I'm going with a number of different examples to make the idea clear:
``
A) X=14, N=5:
X=1110=8+4+2. So, the array is 1 1 2 4 8.
B) X=14, N=6:
X=8+4+2. The array should be 1 1 1 1 2 12.
C) X=15, N=6:
X=8+4+2+1. The array should be 1 1 1 2 4 8.
D) X=15, N=5:
The array should be 1 1 1 2 12.
E) X=14, N=2:
The array should be 2 12. Because 12 = 4^8
``
So, we go as follows. We compute the number of powers of 2 in X. Let this number be k.
Case 1 - If k <= n (example E): we start by picking the smallest powers from left to right and merge the remaining on the last position in the array.
Case 2 - If k > n (example A, B, C, D): we compute h = n - k. If h is odd we put h = n-k+1. Now, we start by putting h 1's in the beginning of the array. Then, the number of places left is less than k. So, we can follow the idea of Case 1 for the remaining positions. Note that in case 2, instead of having odd number of added 1's we put and even number of 1's and then do some merging at the end. This guarantees that the array is the smallest it can be.
We have to consider that we have to minimize the sum of the array for solution and that is the key point.
First calculate set bits in N suppose if count of setbits are less than or equal to X then divide N in X integers based on set bits like
N = 15, X = 2
setbits in 15 are 4 solution is 1 14
if X = 3 solution is 1 2 12
this minimizes array sum too.
other case if setbits are greater than X
calculate difference = setbits(N) - X
If difference is even then add ones as needed and apply above algorithm all ones will cancel out.
If difference is odd then add ones but now you have take care of that 1 extra one in the answer array.
Check for the corner cases too.

Find the highest minimum difference of a number from a given range

Say I have an array of N numbers {A1, A2, ... , An} and 2 numbers P, Q
I have to find an integer M between P and Q , such that, min {|Ai-M|, 1 ≤ i ≤ N} is maximised.
1 < N < P ≤ Q ≤ 10^6
in simpler words:
for each number, find the minimum absolute difference between this number and the array.
then out of all those minimum differences, find the number who has the highest minimum difference.
I have to do this in O(NlogN) or less.
I have tried the following:
sort the array A (NlogN)
iterate over all the numbers between P and Q and for each number find the minimum difference using modified binary search and keep track who has the highest difference - O((Q-P)logN)
I'm guessing there is some kind of math "trick" like using average I'm missing..
edit (add example):
for example if you have the array
{5 8 14}
and P=4 Q=9
the answer is 4,6,7, or 9.
lets look at the numbers 4-9
|4-5| = 1
|4-8| = 4
|4-14| = 10
so minimum diff for 4 is 1
|5-5| = 0
|5-8| = 3
|5-14| = 9
so minimum diff for 4 is 0
we keep going and find minimum diff for all the numbers and then we need to say which number (4/5/6/7/8/9) had the highest minimum diff (in this example 4,6,7 and 9 have 1 minimum difference which is max among all minimum differences)
First you have to sort an array. Then you have to notice that your solution is either P or Q or some point x[i] = (A[i] + A[i+1]) // 2. Basically x[i] is in the middle between consecutive elements in the array (if this x[i] is between P, Q).
Because N is really small, this will run basically in O(1).

Dividing N items in p groups

You are given N total number of item, P group in which you have to divide the N items.
Condition is the product of number of item held by each group should be max.
example N=10 and P=3 you can divide the 10 item in {3,4,3} since 3x3x4=36 max possible product.
You will want to form P groups of roughly N / P elements. However, this will not always be possible, as N might not be divisible by P, as is the case for your example.
So form groups of floor(N / P) elements initially. For your example, you'd form:
floor(10 / 3) = 3
=> groups = {3, 3, 3}
Now, take the remainder of the division of N by P:
10 mod 3 = 1
This means you have to distribute 1 more item to your groups (you can have up to P - 1 items left to distribute in general):
for i = 0 up to (N mod P) - 1:
groups[i]++
=> groups = {4, 3, 3} for your example
Which is also a valid solution.
For fun I worked out a proof of the fact that it in an optimal solution either all numbers = N/P or the numbers are some combination of floor(N/P) and ceiling(N/P). The proof is somewhat long, but proving optimality in a discrete context is seldom trivial. I would be interested if anybody can shorten the proof.
Lemma: For P = 2 the optimal way to divide N is into {N/2, N/2} if N is even and {floor(N/2), ceiling(N/2)} if N is odd.
This follows since the constraint that the two numbers sum to N means that the two numbers are of the form x, N-x.
The resulting product is (N-x)x = Nx - x^2. This is a parabola that opens down. Its max is at its vertex at x = N/2. If N is even this max is an integer. If N is odd, then x = N/2 is a fraction, but such parabolas are strictly unimodal, so the closer x gets to N/2 the larger the product. x = floor(N/2) (or ceiling, it doesn't matter by symmetry) is the closest an integer can get to N/2, hence {floor(N/2),ceiling(N/2)} is optimal for integers.
General case: First of all, a global max exists since there are only finitely many integer partitions and a finite list of numbers always has a max. Suppose that {x_1, x_2, ..., x_P} is globally optimal. Claim: given and i,j we have
|x_i - x_ j| <= 1
In other words: any two numbers in an optimal solution differ by at most 1. This follows immediately from the P = 2 lemma (applied to N = x_i + x_ j).
From this claim it follows that there are at most two distinct numbers among the x_i. If there is only 1 number, that number is clearly N/P. If there are two numbers, they are of the form a and a+1. Let k = the number of x_i which equal a+1, hence P-k of the x_i = a. Hence
(P-k)a + k(a+1) = N, where k is an integer with 1 <= k < P
But simple algebra yields that a = (N-k)/P = N/P - k/P.
Hence -- a is an integer < N/P which differs from N/P by less than 1 (k/P < 1)
Thus a = floor(N/P) and a+1 = ceiling(N/P).
QED

Find cardinality of set

I have faced the following problem recently:
We have a sequence A of M consecutive integers, beginning at A[1] = 1:
1,2,...M (example: M = 8 , A = 1,2,3,4,5,6,7,8 )
We have the set T consisting of all possible subsequences made from L_T consecutive terms of A.
(example L_T = 3 , subsequences are {1,2,3},{2,3,4},{3,4,5},...). Let's call the elements of T "tiles".
We have the set S consisting of all possible subsequences of A that have length L_S. ( example L_S = 4, subsequences like {1,2,3,4} , {1,3,7,8} ,...{4,5,7,8} ).
We say that an element s of S can be "covered" by K "tiles" of T if there exist K tiles in T such that the union of their sets of terms contains the terms of s as a subset. For example, subsequence {1,2,3} is possible to cover with 2 tiles of length 2 ({1,2} and {3,4}), while subsequnce {1,3,5} is not possible to "cover" with 2 "tiles" of length 2, but is possible to cover with 2 "tiles" of length 3 ({1,2,3} and {4,5,6}).
Let C be the subset of elements of S that can be covered by K tiles of T.
Find the cardinality of C given M, L_T, L_S, K.
Any ideas would be appreciated how to tackle this problem.
Assume M is divisible by T, so that we have an integer number of tiles covering all elements of the initial set (otherwise the statement is currently unclear).
First, let us count F (P): it will be almost the number of subsequences of length L_S which can be covered by no more than P tiles, but not exactly that.
Formally, F (P) = choose (M/T, P) * choose (P*T, L_S).
We start by choosing exactly P covering tiles: the number of ways is choose (M/T, P).
When the tiles are fixed, we have exactly P * T distinct elements available, and there are choose (P*T, L_S) ways to choose a subsequence.
Well, this approach has a flaw.
Note that, when we chose a tile but did not use its elements at all, we in fact counted some subsequences more than once.
For example, if we fixed three tiles numbered 2, 6 and 7, but used only 2 and 7, we counted the same subsequences again and again when we fixed three tiles numbered 2, 7 and whatever.
The problem described above can be countered by a variation of the inclusion-exclusion principle.
Indeed, for a subsequence which uses only Q tiles out of P selected tiles, it is counted choose (M-Q, P-Q) times instead of only once: Q of P choices are fixed, but the other ones are arbitrary.
Define G (P) as the number of subsequences of length L_S which can be covered by exactly P tiles.
Then, F (P) is sum for Q from 0 to P of the products G (Q) * choose (M-Q, P-Q).
Working from P = 0 upwards, we can calculate all the values of G by calculating the values of F.
For example, we get G (2) from knowing F (2), G (0) and G (1), and also the equation connecting F (2) with G (0), G (1) and G (2).
After that, the answer is simply sum for P from 0 to K of the values G (P).

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

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