Given two numbers a and b where b is of form 2k where k is unknown.What would be the efficient way of computing a%b using bitwise operator.
a AND (b-1) == a%b (when b is 2^k)
ex. a = 11 (1011b), b = 4 (0100b)
11 / 4 = 2 R3
11 % 4 == 11 AND (4-1)
11 (1011b) AND 3 (0011b) == 3 (0011b)
Related
Today, while practicing some Algorithm questions I found an interesting question.
The question is
You have to divide 1 to n (with one missing value x ) into two equal
halfs such that sum of the two halfs are equal.
Example:
If n = 7 and x = 4
The solution will be {7, 5} and {1, 2, 3, 6}
I can answer it with brute force method but i want an efficient solution
Can any one help me out?
If the sum of the elements 1→N without x is odd then there is no solution.
Otherwise you can find your solution in O(N) with balanced selection.
4 in a row
First let us consider that any sequence of four contiguous numbers can be split in two sets with equal sum given that:
[x, x+1, x+2, x+3] → [x+3, x];[x+2, x+1]
Thus selecting them and placing them in sets A B B A balances sets A and B.
4 across
Moreover, when we have two couples across an omitted value, it can hold a similar property:
[x-2, x-1, x+1, x+2] → [x+2, x-2]; [x+1, x-1]
so still A B B A
At this point we can fix the following cases:
we have a quadruplet: we split it as in case 1
we have 2 numbers, x and other 2 numbers: we split as in case 2
Alright, but it can happen we have 3 numbers, x and other 3 numbers, or other conditions. How can we select in balanced manner anyway?
+2 Gap
If we look again at the gap across x:
[x-1, x+1]
we can notice that somehow if we split the two neighbors in two separate sets we must balance a +2 on the set with bigger sum.
Balancing Tail
We can do this by using the last four numbers of the sequence:
[4 3 2 1] → [4, 2] ; [3, 1] → 6 ; 4
Finally we have to consider that we might not have one of them, so let's build the other case:
[3 2 1] → [2] ; [3, 1] → 2 ; 4
and let us also realize we can do the very same at the other end of the sequence with an A B A B (or B A B A) pattern - if our +2 stands on B (or A);
4 across +
It is amazing that 4 across still holds if we jump h (odd!) numbers:
[x+3, x+2, x-2, x-3] → [x+3, x-3]; [x+2, x-2]
So, exploring the array we can draw the solution step by step
An example:
11 10 9 8 7 6 5 4 3 2 1
the sum it's even, so x can be only an even number:
x = 10
11 - 9 | 8 7 6 5 | 4 3 2 1 → (+2 gap - on A) (4 in a row) (balancing tail)
A B A B B A B A B A
x = 8
11 10 | 9 - 7 | 6 5 | 4 3 2 1 → (4 across +) (+2 gap - on A) (balancing tail)
a b A B | b a | B A B A
x = 6
11 10 9 8 | 7 - 5 | 4 3 2 1 → (4 in a row) (+2 gap - on A) (balancing tail)
A B B A A B A B B B
x = 4 we have no balancing tail - we have to do that with head
11 10 9 8 | 7 6 | 5 - 3 | 2 1 → (balancing head) (4 across +) (+2 gap)
A B A B A B | b a | B A
x = 2
11 10 9 8 | 7 6 5 4 | 3 - 1 → (balancing head) (4 in a row) (+2 gap)
A B A B A B B A B A
It is interesting to notice the symmetry of the solutions. Another example.
10 9 8 7 6 5 4 3 2 1
the sum it's odd, so x can be only an odd number, and the number of elements now is odd.
x = 9
10 - 8 | 7 6 5 4 | 3 2 1 → (+2 gap - on A) (4 in a row) (balancing tail)
A B A B B A B A B
x = 7
10 9 | 8 - 6 | 5 4 | 3 2 1 → (4 across +) (+2 gap - on A) (balancing tail)
a b | A B | b a B A B
x = 5
10 9 8 7 | 6 - 4 | 3 2 1 → (4 in a row) (+2 gap - on A) (balancing tail)
A B B A A B B A B
x = 3
10 9 8 7 | 6 5 | 4 - 2 | 1 → (balancing head) (4 across + virtual 0) (+2 gap)
A B A B B A | a b | A
x = 1
10 9 8 7 | 6 5 4 3 | 2 → (balancing head) (4 in a row) (+2 gap virtual 0)
A B A B A B B A B
Finally it is worth to notice we can switch from A to B whenever we have a full balanced segment (i.e. 4 in a row or 4 across)
Funny said - but the property requesting the sum([1 ... N]-x) to be even makes the cases quite redundant if you try yourself.
I am pretty sure this algorithm can be generalized - I'll probably provide a revised version soon.
This problem can be solved by wrapping the standard subset sum problem of dynamic programming with preprocessing steps. These steps are of O(1) com
Algorithm (n, x):
sum = n * (n+1) / 2
neededSum = sum - x
If (neededSum % 2 != 0): return 0
create array [1..n] and remove x from it
call standard subsetsum(arr, 0, neededSum/2, [])
Working python implementation of subsetsum algorithm - printing all subsets is given below.
def subsetsum(arr, i, sum, ss):
if i >= len(arr):
if sum == 0:
print ss
return 1
else:
return 0
ss1 = ss[:]
count = subsetsum(arr, i + 1, sum, ss1)
ss1.append(arr[i])
count += subsetsum(arr, i + 1, sum - arr[i], ss1)
return count
arr = [1, 2, 3, 10, 5, 7]
sum = 14
a = []
print subsetsum(arr, 0, sum, a)
Hope it helps!
Suppose I have an array X of size n by p by q. I would like to reshape it as a matrix with p rows, and in each row put the concatenation of the n rows of size q, resulting in a matrix of size p by nq.
I managed to do it with a loop but it takes a while say if n=1000, p=300, q=300.
F0=[];
for k=1:size(F,1)
F0=[F0,squeeze(X(k,:,:))];
end
Is there a faster way?
I think this is what you want:
Y = reshape(permute(X, [2 1 3]), size(X,2), []);
Example with n=2, p=3, q=4:
>> X
X(:,:,1) =
0 6 9
8 3 0
X(:,:,2) =
4 7 1
3 7 4
X(:,:,3) =
4 7 2
6 7 6
X(:,:,4) =
6 1 9
1 4 3
>> Y = reshape(permute(X, [2 1 3]), size(X,2), [])
Y =
0 8 4 3 4 6 6 1
6 3 7 7 7 7 1 4
9 0 1 4 2 6 9 3
Try this -
reshape(permute(X,[2 3 1]),p,[])
Thus, for code verification, one can look into a sample case run -
n = 2;
p = 3;
q = 4;
X = rand(n,p,q)
F0=[];
for k=1:n
F0=[F0,squeeze(X(k,:,:))];
end
F0
F0_noloop = reshape(permute(X,[2 3 1]),p,[])
Output is -
F0 =
0.4134 0.6938 0.3782 0.4775 0.2177 0.0098 0.7043 0.6237
0.1257 0.8432 0.7295 0.2364 0.3089 0.9223 0.2243 0.1771
0.7261 0.7710 0.2691 0.8296 0.7829 0.0427 0.6730 0.7669
F0_noloop =
0.4134 0.6938 0.3782 0.4775 0.2177 0.0098 0.7043 0.6237
0.1257 0.8432 0.7295 0.2364 0.3089 0.9223 0.2243 0.1771
0.7261 0.7710 0.2691 0.8296 0.7829 0.0427 0.6730 0.7669
Rather than using vectorization to solve the problem, you could look at the code to try and figure out what may improve performance. In this case, since you know the size of your output matrix F0 should be px(n*q), you could pre-allocate memory to F0 and avoid the constant resizing of the matrix at each iteration of the for loop
n=1000;
p=300;
q=300;
F0=zeros(p,n*q);
for k=1:size(F,1)
F0(:,(k-1)*q+1:k*q) = squeeze(F(k,:,:));
end
While probably not as efficient as the other two solutions, it is an alternative. Try the above and see what happens!
I have N integers numbers: 1,2,3...N
The task is to use +,-,*,/ to make expression 0.
For example -1*2+3+4-5=0
How can I do it?
May be some code on C/C++ ?
If N % 4 == 0, for every four consecutive integers a, b, c, d, take a - b - c + d
If N % 4 == 1, use 1 * 2 to start, then proceed as before. (i.e., 1*2 - 3 - 4 + 5 + 6 - 8 - 8 + 9 ...)
If N % 4 == 2, start with 1 - 2 + 3 * 4 - 5 - 6, then proceed as in the N % 4 == 0 example.
If N % 4 == 3, start with 1 + 2 - 3, then proceed as in the N%4 == 0 example.
All of these find a way to get zero out of the first few integers, leaving a multiple of four integers to work on, then take advantage of the fact that the pattern a - b - c + d = 0 for any four consecutive integers.
This is essentially SAT, or do you know that the numbers are a sequence (e.g. 2 1 8 is forbidden). What about negative numbers?
If the sequence is not too large, i would recommend to simply bootforce it. A greedy solution would be to reduce the problem by finding subsets which can be evaluated to zero.
Exercise 2.2 in Warren's Abstract Machine: A Tutorial Reconstruction
asks for representations for the terms f(X, g(X, a)) and f(b, Y) and then to perform unification on the address of these terms (denoted a1 and a2 respectively).
I've constructed the heap representations for the terms, and they are as follows:
f(X, g(X, a)):
0 STR 1
1 a/0
2 STR 3
3 g/2
4 REF 4
5 STR 1
6 STR 7
7 f/2
8 REF 4
9 STR 3
f(b, Y):
10 STR 11
11 b/0
12 STR 7
13 STR 11
14 REF 14
and I am now asked to trace unify(a1, a2), but following the algorithm on page 20 in 1 I get:
d1 = deref(a1) = deref(10) = 10
d2 = deref(a2) = deref(0) = 0
0 != 10 so we continue
<t1, v1> = STORE(d1) = STORE(10) = <STR, 11>
<t2, v2> = STORE(d2) = STORE(0) = <STR, 1>
t1 != REF and t2 != REF so we continue
f1 / n1 = STORE(v1) = STORE(11) = b / 0
f2 / n2 = STORE(v2) = STORE(1) = a / 0
and now b != a so the algorithm terminated with fail = true,
and thus unification failed, but obviously there exists
a solution with X = b and Y = g(b, a).
Where is my mistake?
I found the solution myself. Here's my corrections:
Each term should have their own definitions of the functors (ie. the f-functor in the second term should not just link to the first f-functor in the first term, but should have its own) and pointers to the terms (a1 and a2) should point to the outermost term functor.
This means that a1 = 6 and a2 = 12 in the following layout
f(X, g(X, a)):
0 STR 1
1 a/0
2 STR 3
3 g/2
4 REF 4
5 STR 1
6 STR 7
7 f/2
8 REF 4
9 STR 3
f(b, Y):
10 STR 11
11 b/0
12 STR 13
13 f/2
14 REF 11
15 REF 15
We are given an array A of integers. I want to find 2 contiguous subarrays of the largest length(both subarrays must be equal in length) that have the same weighted average. The weights are the positions in the subarray. For example
A=41111921111119
Subarrays:: (11119) and (11119)
Ive tried to find the weighted average of all subarrays by DP and then sorting them columnwise to find 2 with same length.But I cant proceed further and my approach seems too vague/bruteforce.I would appreciate any help. Thanks in advance.
The first step should be to sort the array. Any pairs of equal values can then be identified and factored out. The remaining numbers will all be different, like this:
2, 3, 5, 9, 14, 19 ... etc
The next step would be to compare pairs to their center:
2 + 5 == 2 * 3 ?
3 + 9 == 2 * 5 ?
5 + 14 == 2 * 9 ?
9 + 19 == 2 * 14 ?
The next step is to compare nested pairs, meaning if you have A B C D, you compare A+D to B+C. So for the above example it would be:
2+9 == 3+5 ?
3+15 == 5+9 ?
5+19 == 9+14 ?
Next you would compare triples to the two inside values:
2 + 3 + 9 == 3 * 5 ?
2 + 5 + 9 == 3 * 3 ?
3 + 5 + 14 == 3 * 9 ?
3 + 9 + 14 == 3 * 5 ?
5 + 9 + 19 == 3 * 14 ?
5 + 14 + 19 == 3 * 9 ?
Then you would compare pairs of triples:
2 + 3 + 19 == 5 + 9 + 14 ?
2 + 5 + 19 == 3 + 9 + 14 ?
2 + 9 + 19 == 3 + 5 + 14 ?
and so on. There are different ways to do the ordering. One way is to create an initial bracket, for example, given A B C D E F G H, the initial bracket is ABGH versus CDEF, ie the outside compared to the center. Then switch values according to the comparison. For example, if ABGH > CDEF, then you can try all switches where the left value is greater than the right value. In this case G and H are greater than E and F, so the possible switches are:
G <-> E
G <-> F
H <-> E
H <-> F
GH <-> EF
First, as the length of the two subarray must be equal, you can consider the length from 1 to n step by step.
For length i, you can calculate the weighted sum of every subarray in a total complexity of O(n). Then sort the sums to determine if there's an equal pair.
Because you sort n times the time would be O(n^2 log n) while the space is O(n).
Maybe I just repeated your solution mentioned in the question? But I don't think it can be optimized any more...