I have a very simple question. I will be provided with the first term of Arithmetic Progression and the common difference. I am required to give sum of digits of all numbers in between the range of L and R.Here the sum means the sum <10 that means for a number say 157, the sum of digits is 1+5+7=13 which is further 1+3=4. The range is the index of the elements. L means the Lth number of that series and L starts from 1.Here L and R can be of range 1 to 10^18. How can i find the sum of these digits for such a large range. I know the digit sum of a number n can be calculated as (n-1)%9+1. But i can't iterate over 10^18 numbers.
Example: let say First term of Arithmetic Progression is 14 and common difference 7.Then the sum of digits of all numbers between 2 and 4 will be sum of (2+1)=3 and (2+8)=(1+0)=1 and (3+5)=8 which is equal to 12
for pattern finding
current=first;
ll arr[10]={0};
while(1)// search for the pattern
{
ll dsum=(current-1)%9+1;// calculating digit sum
if(arr[dsum]!=0)
break;
arr[dsum]=ptr;// saving the value in the array by using hashing
ptr++;
current+=c_diff;
}
for sum
for(ll i=1;i<ptr;i++)
{
sum[i]=sum[i-1]+new_arr[i];
}
Since all of your numbers will be (ultimately) reduced to a single digit, you must be having repetition after certain number of terms, and, that is maximum of 9. (because 10 digits, but 0 not possible to repeat).
So, lets start with an example. Say, a=14, d=7, l=2, r=50. I've changed the value of r from your example.
So, trying to find repetition:
Our repetition array is q (say). Then, 1st term of q is 5 (since 14 = 5).
Now, next term is 21 (= 3). Since 3 is not equal to 5, we continue.
We find all terms till we get 5 again. So, q will be like this in this example:
q[] = {5,3,1,8,6,4,2,9,7} ... and then we have 5 again, so stop.
So, our repetitive pattern has 9 members. Now, find a cumulative sum array with this, which will be like:
sum[] = {5,8,9,17,23,27,29,38,45}
Now, since r=50, find sum of r terms, which will be:
(floor)(50/9) * 45 + sum[50%9]
= 5 * 45 + 23
= 248
Now, similarly, find sum of l-1 terms, (since you have to find sum in range l..r inclusive.
Sum of 1st (2-1) = 1 terms will be:
(floor)(1/9) * 45 + sum[1 % 9]
= 0 + 5
= 5
So, the answer is, 248 - 5 = 243.
well you can solve this problem by taking two arrays of elements 9 and finding the lth term element.
from the lth term find the digit sum of lth element and save it in two array respectively .
w1=a+(l-1)*d
for(i=1,j=w1;i<=9;j+=d,i++){ if(j%9==0){array1[i]=9 ;array2[i]=9;}else{array1[i]=j%9;array2[i]=j%9;}
}
now q = r-l+1
w= q/9 and e=q%9
array1[i]=array1[i]*w // in loop from i= 1 to 9
in array2[i]=0 //new loop from i=e+1 to 9
now digitsum += array1[i]+array2[i] // from a a loop i=1 to 9 and digitsum is the sum of all digit of sequence
this digit sum is solution.
Related
I want to find out the number of all permutation of nnumber.Number will be from 1 to n.The given condition is that each ithposition can have number up to Si,where Si is given for each position of number.
1<=n<=10^6
1<=si<=n
For example:
n=5
then its all five element will be
1,2,3,4,5
and given Si for each position is as:
2,3,4,5,5
It shows that at:
1st position can have 1 to 2that is 1,2 but can not be number among 3 to 5.
Similarly,
At 2nd position can have number 1 to 3 only.
At 3rd position can have number 1 to 4 only.
At 4th position can have number 1 to 5 only.
At 5th position can have number 1 to 5 only.
Some of its permutation are:
1,2,3,4,5
2,3,1,4,5
2,3,4,1,5 etc.
But these can not be:
3,1,4,2,5 As 3 is present at 1st position.
1,2,5,3,4 As 5 is present at 3rd position.
I am not getting any idea to count all possible number of permutations with given condition.
Okay, if we have a guarantee that numbers si are given in not descending order then looks like it is possible to calculate the number of permutations in O(n).
The idea of straightforward algorithm is as follows:
At step i multiply the result by current value of si[i];
We chose some number for position i. As long as we need permutation, that number cannot be repeated, so decrement all the rest si[k] where k from i+1 to the end (e.g. n) by 1;
Increase i by 1, go back to (1).
To illustrate on example for si: 2 3 3 4:
result = 1;
current si is "2 3 3 4", result *= si[0] (= 1*2 == 2), decrease 3, 3 and 4 by 1;
current si is "..2 2 3", result *= si[1] (= 2*2 == 4), decrease last 2 and 3 by 1;
current si is "....1 2", result *= si[2] (= 4*1 == 4), decrease last number by 1;
current si is "..... 1", result *= si[3] (= 4*1 == 4), done.
Hovewer this straightforward approach would require O(n^2) due to decreasing steps. To optimize it we can easily observe that at the moment of result *= si[i] our si[i] was already decreased exactly i times (assuming we start from 0 of course).
Thus O(n) way:
unsigned int result = 1;
for (unsigned int i = 0; i < n; ++i)
{
result *= (si[i] - i);
}
for each si count the number of element in your array such that a[i] <= si using binary search, and store the value to an array count[i], now the answer is the product of all count[i], however we have decrease the number of redundancy from the answer ( as same number could be count twice ), for that you can sort si and check how many number is <= s[i], then decrease that number from each count,the complexity is O(nlog(n)), hope at least I give you an idea.
To complete Yuriy Ivaskevych answer, if you don't know if the sis are in increasing order, you can sort the sis and it will also works.
And the result will be null or negative if the permutations are impossible (ex: 1 1 1 1 1)
You can try backtracking, it's a little hardcore approach but will work.
try:
http://www.thegeekstuff.com/2014/12/backtracking-example/
or google backtracking tutorial C++
I want to find the minimum set of prime numbers which would sum to a given value e.g. 9 = 7 + 2 (not 3+3+3).
I have already generated a array of prime numbers using sieve of eratosthens
I am traversing the array in descending order to get the array largest prime number smaller than or equal to given number. This works great if the number is odd.
But fails for even numbers e.g 122 = 113 + 7 + 2 but 122 = 109 +13.
From Golbach's Conjecture we know that any even number can be represented as two sum of two prime numbers. So if a number is even we can directly return 2 as output.
But I am trying to figure out a way other than brute force to find minimum prime numbers.
Although your question didn't say so, I assume you are looking for the set of primes that has the smallest cardinality.
If n is even, then consider the primes p in order, 2, 3, 5, …; eventually n - p will be prime, so n is the sum of two primes. This process typically converges very quickly, with the smaller of the two primes seldom larger than 1000 (and usually much smaller than that).
If n is odd, and n - 2 is prime, then n is the sum of the primes 2 and n - 2.
If n is odd, and n - 2 is not prime, then n - 3 is even and can be written as the sum of two primes, as described above.
Thus you can always find two or three primes that sum to any target n greater than 3.
Try this out!
Not an ideal code but if you want to have a working solution :P
primes = [2,3,5,7]
D = 29
i = -1
sum_ = 0
count = 0
while sum_ != D :
sum_ = sum_ + primes[i]
count += 1
if (sum_ == D) :
break
elif D - sum_ == primes[i-1] :
count += 1
break
elif D - sum_ < ex[i-1] and (D-sum_ not in primes) :
sum_ = sum_ - primes[i]
count = count - 1
i = i - 1
print(count)
This problem was asked to me in Amazon interview -
Given a array of positive integers, you have to find the smallest positive integer that can not be formed from the sum of numbers from array.
Example:
Array:[4 13 2 3 1]
result= 11 { Since 11 was smallest positive number which can not be formed from the given array elements }
What i did was :
sorted the array
calculated the prefix sum
Treverse the sum array and check if next element is less than 1
greater than sum i.e. A[j]<=(sum+1). If not so then answer would
be sum+1
But this was nlog(n) solution.
Interviewer was not satisfied with this and asked a solution in less than O(n log n) time.
There's a beautiful algorithm for solving this problem in time O(n + Sort), where Sort is the amount of time required to sort the input array.
The idea behind the algorithm is to sort the array and then ask the following question: what is the smallest positive integer you cannot make using the first k elements of the array? You then scan forward through the array from left to right, updating your answer to this question, until you find the smallest number you can't make.
Here's how it works. Initially, the smallest number you can't make is 1. Then, going from left to right, do the following:
If the current number is bigger than the smallest number you can't make so far, then you know the smallest number you can't make - it's the one you've got recorded, and you're done.
Otherwise, the current number is less than or equal to the smallest number you can't make. The claim is that you can indeed make this number. Right now, you know the smallest number you can't make with the first k elements of the array (call it candidate) and are now looking at value A[k]. The number candidate - A[k] therefore must be some number that you can indeed make with the first k elements of the array, since otherwise candidate - A[k] would be a smaller number than the smallest number you allegedly can't make with the first k numbers in the array. Moreover, you can make any number in the range candidate to candidate + A[k], inclusive, because you can start with any number in the range from 1 to A[k], inclusive, and then add candidate - 1 to it. Therefore, set candidate to candidate + A[k] and increment k.
In pseudocode:
Sort(A)
candidate = 1
for i from 1 to length(A):
if A[i] > candidate: return candidate
else: candidate = candidate + A[i]
return candidate
Here's a test run on [4, 13, 2, 1, 3]. Sort the array to get [1, 2, 3, 4, 13]. Then, set candidate to 1. We then do the following:
A[1] = 1, candidate = 1:
A[1] ≤ candidate, so set candidate = candidate + A[1] = 2
A[2] = 2, candidate = 2:
A[2] ≤ candidate, so set candidate = candidate + A[2] = 4
A[3] = 3, candidate = 4:
A[3] ≤ candidate, so set candidate = candidate + A[3] = 7
A[4] = 4, candidate = 7:
A[4] ≤ candidate, so set candidate = candidate + A[4] = 11
A[5] = 13, candidate = 11:
A[5] > candidate, so return candidate (11).
So the answer is 11.
The runtime here is O(n + Sort) because outside of sorting, the runtime is O(n). You can clearly sort in O(n log n) time using heapsort, and if you know some upper bound on the numbers you can sort in time O(n log U) (where U is the maximum possible number) by using radix sort. If U is a fixed constant, (say, 109), then radix sort runs in time O(n) and this entire algorithm then runs in time O(n) as well.
Hope this helps!
Use bitvectors to accomplish this in linear time.
Start with an empty bitvector b. Then for each element k in your array, do this:
b = b | b << k | 2^(k-1)
To be clear, the i'th element is set to 1 to represent the number i, and | k is setting the k-th element to 1.
After you finish processing the array, the index of the first zero in b is your answer (counting from the right, starting at 1).
b=0
process 4: b = b | b<<4 | 1000 = 1000
process 13: b = b | b<<13 | 1000000000000 = 10001000000001000
process 2: b = b | b<<2 | 10 = 1010101000000101010
process 3: b = b | b<<3 | 100 = 1011111101000101111110
process 1: b = b | b<<1 | 1 = 11111111111001111111111
First zero: position 11.
Consider all integers in interval [2i .. 2i+1 - 1]. And suppose all integers below 2i can be formed from sum of numbers from given array. Also suppose that we already know C, which is sum of all numbers below 2i. If C >= 2i+1 - 1, every number in this interval may be represented as sum of given numbers. Otherwise we could check if interval [2i .. C + 1] contains any number from given array. And if there is no such number, C + 1 is what we searched for.
Here is a sketch of an algorithm:
For each input number, determine to which interval it belongs, and update corresponding sum: S[int_log(x)] += x.
Compute prefix sum for array S: foreach i: C[i] = C[i-1] + S[i].
Filter array C to keep only entries with values lower than next power of 2.
Scan input array once more and notice which of the intervals [2i .. C + 1] contain at least one input number: i = int_log(x) - 1; B[i] |= (x <= C[i] + 1).
Find first interval that is not filtered out on step #3 and corresponding element of B[] not set on step #4.
If it is not obvious why we can apply step 3, here is the proof. Choose any number between 2i and C, then sequentially subtract from it all the numbers below 2i in decreasing order. Eventually we get either some number less than the last subtracted number or zero. If the result is zero, just add together all the subtracted numbers and we have the representation of chosen number. If the result is non-zero and less than the last subtracted number, this result is also less than 2i, so it is "representable" and none of the subtracted numbers are used for its representation. When we add these subtracted numbers back, we have the representation of chosen number. This also suggests that instead of filtering intervals one by one we could skip several intervals at once by jumping directly to int_log of C.
Time complexity is determined by function int_log(), which is integer logarithm or index of the highest set bit in the number. If our instruction set contains integer logarithm or any its equivalent (count leading zeros, or tricks with floating point numbers), then complexity is O(n). Otherwise we could use some bit hacking to implement int_log() in O(log log U) and obtain O(n * log log U) time complexity. (Here U is largest number in the array).
If step 1 (in addition to updating the sum) will also update minimum value in given range, step 4 is not needed anymore. We could just compare C[i] to Min[i+1]. This means we need only single pass over input array. Or we could apply this algorithm not to array but to a stream of numbers.
Several examples:
Input: [ 4 13 2 3 1] [ 1 2 3 9] [ 1 1 2 9]
int_log: 2 3 1 1 0 0 1 1 3 0 0 1 3
int_log: 0 1 2 3 0 1 2 3 0 1 2 3
S: 1 5 4 13 1 5 0 9 2 2 0 9
C: 1 6 10 23 1 6 6 15 2 4 4 13
filtered(C): n n n n n n n n n n n n
number in
[2^i..C+1]: 2 4 - 2 - - 2 - -
C+1: 11 7 5
For multi-precision input numbers this approach needs O(n * log M) time and O(log M) space. Where M is largest number in the array. The same time is needed just to read all the numbers (and in the worst case we need every bit of them).
Still this result may be improved to O(n * log R) where R is the value found by this algorithm (actually, the output-sensitive variant of it). The only modification needed for this optimization is instead of processing whole numbers at once, process them digit-by-digit: first pass processes the low order bits of each number (like bits 0..63), second pass - next bits (like 64..127), etc. We could ignore all higher-order bits after result is found. Also this decreases space requirements to O(K) numbers, where K is number of bits in machine word.
If you sort the array, it will work for you. Counting sort could've done it in O(n), but if you think in a practically large scenario, range can be pretty high.
Quicksort O(n*logn) will do the work for you:
def smallestPositiveInteger(self, array):
candidate = 1
n = len(array)
array = sorted(array)
for i in range(0, n):
if array[i] <= candidate:
candidate += array[i]
else:
break
return candidate
I am having a Algorithm question, in which numbers are been given from 1 to N and a number of operations are to be performed and then min/max has to be found among them.
Two operations - Addition and subtraction
and operations are in the form a b c d , where a is the operation to be performed,b is the starting number and c is the ending number and d is the number to be added/subtracted
for example
suppose numbers are 1 to N
and
N =5
1 2 3 4 5
We perform operations as
1 2 4 5
2 1 3 4
1 4 5 6
By these operations we will have numbers from 1 to N as
1 7 8 9 5
-3 3 4 9 5
-3 3 4 15 11
So the maximum is 15 and min is -3
My Approach:
I have taken the lower limit and upper limit of the numbers in this case it is 1 and 5 only stored in an array and applied the operations, and then had found the minimum and maximum.
Could there be any better approach?
I will assume that all update (addition/subtraction) operations happen before finding max/min. I don't have a good solution for update and min/max operations mixing together.
You can use a plain array, where the value at index i of the array is the difference between the index i and index (i - 1) of the original array. This makes the sum from index 0 to index i of our array to be the value at index i of the original array.
Subtraction is addition with the negated number, so they can be treated similarly. When we need to add k to the original array from index i to index j, we will add k to index i of our array, and subtract k to index (j + 1) of our array. This takes O(1) time per update.
You can find the min/max of the original array by accumulating summing the values and record the max/min values. This takes O(n) time per operation. I assume this is done once for the whole array.
Pseudocode:
a[N] // Original array
d[N] // Difference array
// Initialization
d[0] = a[0]
for (i = 1 to N-1)
d[i] = a[i] - a[i - 1]
// Addition (subtraction is similar)
add(from_idx, to_idx, amount) {
d[from_idx] += amount
d[to_idx + 1] -= amount
}
// Find max/min for the WHOLE array after add/subtract
current = max = min = d[0];
for (i = 1 to N - 1) {
current += d[i]; // Sum from d[0] to d[i] is a[i]
max = MAX(max, current);
min = MIN(min, current);
}
Generally there is no "best way" to find the min/max in the performance point of view because it depends on how this application will be used.
-Finding the max and min in a list needs O(n) Time, so if you want to run many (many in the context of the input) operations, your approach to find the min/max after all the operations took place is fine.
-But if the list will hold many elements and you don’t want to run that many operations, you better check each result of the op if its a new max/min and update if necessary.
This question already has answers here:
Closed 11 years ago.
Possible Duplicate:
Generating the partitions of a number
Prime number sum
The number 7 can be expressed in 5 ways as a sum of primes:
2 + 2 + 3
2 + 3 + 2
2 + 5
3 + 2 + 2
5 + 2
Make a program that calculates, in how many ways number n can be
expressed as a sum of primes. You can assume that n is a number
between 0-100. Your program should print the answer in less than a
second
Example 1:
Give number: 7 Result: 5
Example 2:
Give number: 20 Result: 732
Example 3:
Give number: 80 Result: 10343662267187
I've been at this problem for hours. I can't figure out how to get n from (n-1).
Here are the sums from the first 30 numbers by a tree search
0 0 0 1 2 2 5 6 10 16 19 35 45 72 105 152 231 332 500 732 1081 1604 2351 3493 5136 7595 11212 16534 24441
I thought I had something with finding the biggest chain 7 = 5+2 and somehow using the knowledge that five can be written as 5, 3+2, 2+3, but somehow I need to account for the duplicate 2+3+2 replacement.
Look up dynamic programming, specifically Wikipedia's page and the examples there for the fibonacci sequence, and think about how you might be able to adapt that to your problem here.
Okay so this is a complicated problem. you are asking how to write code for the Partition Function; I suggest that you read up on the partition function itself first. Next you should look at algorithms to calculate partitions. It is a complex subject here is a starting point ... Partition problem is [NP complete] --- This question has already been asked and answered here and that may also help you start with algorithms.
There're several options. Since you know the number is between 0-100, there is the obvious: cheat, simply make an array and fill in the numbers.
The other way would be a loop. You'd need all the primes under 100, because a number which is smaller than 100 can't be expressed using the sum of a prime which is larger than 100. Eg. 99 can't be expressed as the sum of 2 and any prime larger than 100.
What you also know is: the maximum length of the sum for even numbers is the number divided by 2. Since 2 is the smallest prime. For odd numbers the maximum length is (number - 1) / 2.
Eg.
8 = 2 + 2 + 2 + 2, thus length of the sum is 4
9 = 2 + 2 + 2 + 3, thus length of the sum is 4
If you want performance you could cheat in another way by using GPGPU, which would significantly increase performance.
Then they're is the shuffling method. If you know 7 = 2 + 2 + 3, you know 7 = 2 + 3 + 2. To do this you'd need a method of calculating the different possibilities of shuffling. You could store the combinations of possibilities or keep them in mind while writing your loop.
Here is a relative brute force method (in Java):
int[] primes = new int[]{/* fill with primes < 100 */};
int number = 7; //Normally determined by user
int maxLength = (number % 2 == 0) ? number / 2 : (number - 1) / 2; //If even number maxLength = number / 2, if odd, maxLength = (number - 1) / 2
int possibilities = 0;
for (int i = 1; i <= maxLength; i++){
int[][] numbers = new int[i][Math.pow(primes.length, i)]; //Create an array which will hold all combinations for this length
for (int j = 0; j < Math.pow(primes.length, i); j++){ //Loop through all the possibilities
int value = 0; //Value for calculating the numbers making up the sum
for (int k = 0; k < i; k++){
numbers[k][j] = primes[(j - value) % (Math.pow(primes.length, k))]; //Setting the numbers making up the sum
value += numbers[k][j]; //Increasing the value
}
}
for (int x = 0; x < primes.length; x++){
int sum = 0;
for (int y = 0; y < i; y++){
sum += numbers[y];
if (sum > number) break; //The sum is greater than what we're trying to reach, break we've gone too far
}
if (sum == number) possibilities++;
}
}
I understand this is complicated. I will try to use an analogy. Think of it as a combination lock. You know the maximum number of wheels, which you have to try, hence the "i" loop. Next you go through each possibility ("j" loop) then you set the individual numbers ("k" loop). The code in the "k" loop is used to go from the current possibility (value of j) to the actual numbers. After you entered all combinations for this amount of wheels, you calculate if any were correct and if so, you increase the number of possibilities.
I apologize in advance if I made any errors in the code.