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I've come up with a very efficient algorithm for calculating prime numbers. It uses bit arithmetics ie AND , OR, XOR etc. and its based on the sieve of eratosthenes.
For numbers below 32 it works. For example when n = 31 I get the output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
But when I use a bigger value such as 40 I get a different output:
2,
I am at a loss to explain why this is so, I need guidance.
Below is the code:
https://github.com/rsgilbert/c/blob/master/chp2/sieve-bitset.c
#include <stdio.h>
#include <math.h>
#define MAXLINE 1000
#define A_1 1 // first number in GP
// sieve of eratosthenes
// Using bit arithmetic
long btoi(char s[]);
void itob(long n, char s[]);
long runningPrimes(long max);
void printBits(long num);
long gpSum(long a_1, long r, long n);
long numElementsInGP(long a_1, long a_n, long r);
long lastElementInBitset(long width);
long width(long factor, long n);
long paddedSum(long sum, long n, long i);
long commonRatio(long i);
long flip(long bitset);
void sieve(long n, long primes[]);
long dropRightBits(long bitset, long noToDrop);
void primeBitsetToArray(long bitset, long primes[]);
void printPrimes(long primes[], size_t size);
long greaterFirstBit(long bitset);
long main()
{
long n = 31; // 32 is not supported (??) I dont know why
long runner = runningPrimes(n);
long primes[n];
sieve(n, primes);
printPrimes(primes, n);
}
/**
* sieve of eratosthenes
* Algorithm for finding all prime numbers upto a given limit.
* We go through natural numbers starting with 2 removing out multiples of each.
*/
void sieve(long n, long primes[])
{
// 1. fill in numbers
long runner = runningPrimes(n);
long start_no = 2;
long i = start_no;
while(i <= n)
{
// printf("%d\n", i);
long w = width(i, n);
long lastEl = lastElementInBitset(w);
long r = commonRatio(i);
long numEls = numElementsInGP(A_1, lastEl, r);
long sum = gpSum(A_1, r, numEls);
// printf("non padded sum: w %d , r %d , numEls %d ", w, r, numEls);
// printBits(sum);
sum = paddedSum(sum, n, i);
// We need to flip `sum` bits because currently 1s in sum represent multiples
// If we flip 1010 it becomes 101. But notice we also want to remove 1000.
// If we dont handle this, 4 will show up in the primes
// So we first get a copy of first bit
long grtrBitset = greaterFirstBit(sum);
sum = flip(sum);
sum = grtrBitset | sum;
//
// printf("sum: gr %d ", grtrBitset);
// printBits(grtrBitset);
// printf("sum ");
// printBits(sum);
// Cancel out bits that represent multiples of i
// We are going to drop some bits. The ones that are multiples of i
// We first store some values
long bitsOnRightToDrop = n - (2 * i) + 1;
long notToChangeBits = dropRightBits(runner, bitsOnRightToDrop);
long withoutSum = runner & sum;
// printf("Runner: ");
runner = notToChangeBits | withoutSum;
// printBits(runner);
i++;
}
primeBitsetToArray(runner, primes);
}
long runningPrimes(long max)
{
return pow(2, max) - 1;
}
long paddedSum(long sum, long n, long i) {
return sum * pow(2, (n % i));
}
/** Find sum of a geometric progression
* a_1: first element in GP
* r: common ratio
* n: number of elements in GP
*/
long gpSum(long a_1, long r, long n)
{
return a_1 * (pow(r, n) - 1) / (r - 1);
}
/* Compute common ratio to be used for a given number */
long commonRatio(long i)
{
return pow(2, i);
}
/** Find number of elements in a GP
* a_1: first element in GP
* a_n: last element in GP
* r : common ratio
*/
long numElementsInGP(long a_1, long a_n, long r)
{
return log2(a_n / a_1) / log2(r) + 1;
}
/* Find last element in bitset as decimal integer
* For example if bitset is 10010 , last element is 10000 = 16
* width: Number of characters in bitset.
*/
long lastElementInBitset(long width)
{
return pow(2, (width - 1));
}
/* Produces the number of bits from first multiple greater than factor to last multiple less than n. */
long width(long factor, long n)
{
long firstMultGR = 2 * factor;
long lastMultLess = n - (n % factor);
return lastMultLess - firstMultGR + 1;
}
// -- Bit functions --
/** Convert binary to decimal integer */
long btoi(char s[])
{
long result = 0;
long i = 0;
while (s[i] != 0)
{
if (s[i] != '0' && s[i] != '1')
return -1;
result *= 2;
result += s[i] - '0';
i++;
}
return result;
}
/** convert decimal integer to binary */
void itob(long n, char s[])
{
if (n == 0)
{
s[0] = '0';
s[1] = 0;
return;
};
if (n < 0)
{
s[0] = '-';
s[1] = '1';
s[2] = 0;
return;
}
long pos = log2(n);
s[pos + 1] = 0;
while (pos != 0)
{
s[pos] = '0' + n % 2;
n = n / 2;
pos--;
}
// pos will be 0
s[pos] = '1';
}
/**
* Prints binary representation of set
*/
void printBits(long num)
{
char result[MAXLINE];
itob(num, result);
printf("%s\n", result);
}
// Flip bits. For example 10110 becomes 1001
long flip(long bitset)
{
long mask = pow(2, (long) log2(bitset) + 1) - 1;
return bitset ^ mask;
}
// Drop some bits from the right side of a bitset. For example dropRighBits(btoi("1001100110"), 6) produces 1001000000
long dropRightBits(long bitset, long noToDrop) {
long mask = pow(2, noToDrop) - 1;
long rightFlippedBitset = bitset ^ mask;
return bitset & rightFlippedBitset;
}
/* Copy bitset representing prime positions into an array of prime numbers */
void primeBitsetToArray(long bitset, long primes[])
{
char temp[MAXLINE];
itob(bitset, temp);
// printf("bitset %s\n", temp);
// in the bitset, the first position represents number 1 then 2 ... etc So our primes will start at index 1
long i = 1;
long j = 0;
while(temp[i] != 0)
{
if(temp[i] == '1')
{
// in the bitset, the first position is 1 then 2 ... etc
primes[j] = i + 1;
// printf("%d\n", i + 1);
j++;
}
i++;
}
primes[j] = -1;
}
/* Return bitset for the first bit in a given bitset.
* For example bitsetForFirstBit(10) = 8
*/
long bitsetForFirstBit(long bitset) {
return pow(2, log2(bitset));
}
/* Return bitset that is greater than given bitset but also a multiple of 2.
* For example bitsetForFirstBit(10) = 16
*/
long greaterFirstBit(long bitset) {
return pow(2, (long) log2(bitset) + 1);
}
void printPrimes(long primes[], size_t size)
{
long i = 0;
while(i < size && primes[i] != -1)
{
printf("%d, ", primes[i]);
i++;
}
printf("\n");
}
I am trying to construct a simple program which adds together the digits of a long number. I attempted to do this by using a loop employing the modulo operator and some basic arithmetic. I want to increment the modulo operator by multiplying it by ten on each iteration of the loop in order to reach the next digit. I want to check if my code is correct, however, I receive errors pertaining to the lines involving the modulo operations and I'm not quite sure why.
This was my attempted construction:
{
long i = 0;
long b;
int m = 1;
do
{
long number = get_long("Number?\n");
long a = number % m;
b = number - a;
long c = b % m x 10;
long d = c / m;
{
i = i + d;
}
{
m = m x 10
}
}
while (b > 0);
printf("%ld\n", i);
}
Edit:
I made the basic error of writing "x" instead of "*". However, having fixed this, I no longer receive errors, but the program simply returns "0". Any diagnosis would be appreciated.
int main(void)
{
long i = 0;
long b;
int m = 10;
long number = get_long("Number?\n");
do
{
long a = number % m;
b = number - a;
long c = b % m * 10;
long d = c / m;
{
i = i + d;
}
{
m = m * 10;
}
}
while (b > 0);
printf("%ld\n", i);
}
For your revised code:
long c = b % m * 10;
this line will evaluate (b % m) and then multiply it by 10 because of the order of operations.
I presume what you actually want is:
long c = b % (m * 10);
Secondly, the following line determines which digit you start at:
int m = 10;
and this line determines how many digits between the ones you include in your total:
m = m * 10;
So for this configuration, it will start at the 2nd digit from the right and add every digit.
So for the number 1234, you'd get 3 + 2 + 1 = 6.
If you want to add every digit, you could set:
int m = 10;
and you'd get 4 + 3 + 2 + 1 = 10.
Alternatively, if you had used:
m = m * 10;
you'd have 3 + 1 = 4.
First, you're likely getting errors due to these lines:
long c = b % m x 10;
m = m x 10
This is because x is not a valid operator.
The multiplication operator is *:
long c = b % m * 10;
m = m * 10;
As for your approach, I would suggest, instead of changing the modulo operand, you simply divide the original number by 10 to shift it one digit each operation.
For example:
#include <stdio.h>
int main()
{
int sumofdigits = 0;
int num = 12345;
while(num > 0) {
sumofdigits += num % 10;
num /= 10;
}
printf("%d", sumofdigits);
return 0;
}
The reduced-sum of the digits of a number is the same as that number modulo 9.
Example:
#include <stdio.h>
int main(void) {
int number = 57283;
printf("%d \n", number%9);
// 5 + 7 + 2 + 8 + 3 == 25 ==> 2 + 5 == 7
// 57283 % 9 == 7
return 0;
}
If you want to use loops to get the reduced sum:
int sum_of_digits(int num)
{
int sum;
do
{
sum = 0;
while(num)
{
sum += num%10;
num /= 10;
}
num = sum;
} while (sum >9);
return sum;
}
But if you only want the simple sum of digits (one pass only):
int sum_of_digits(int num)
{
int sum = 0;
while(num)
{
sum += num%10;
num /= 10;
}
return sum;
}
You have to find the sum of the digits of a variable of type long by the two operators modulo (%) and division (/), you start with the operator modulo to find the remainder of the division (the digits) then, you add this degit to the sum, then you do the division / 10 to overwrite (the summed digit) until the number is equal to 0 like this:
int main()
{
long number=0,m=0;
printf("Give a number :");
scanf("%ld",&number);
long s=0,temp=number;
while(number != 0)
{
m=number%10;
s+=m;
number/=10;
}
printf("\n%The sum of the digits of the Number %ld is : %ld\n",temp,s);
}
Problem statement :
Given a 32-bit signed integer, reverse digits of an integer.
Note: Assume we are dealing with an environment that could only store
integers within the 32-bit signed integer range: [ −2^31, 2^31 − 1]. For
the purpose of this problem, assume that your function returns 0 when
the reversed integer overflows.
I'm trying to implement the recursive function reverseRec(), It's working for smaller values but it's a mess for the edge cases.
int reverseRec(int x)
{
if(abs(x)<=9)
{
return x;
}
else
{
return reverseRec(x/10) + ((x%10)*(pow(10, (floor(log10(abs(x)))))));
}
}
I've implemented non recursive function which is working just fine :
int reverse(int x)
{
long long val = 0;
do{
val = val*10 + (x%10);
x /= 10;
}while(x);
return (val < INT_MIN || val > INT_MAX) ? 0 : val;
}
Here I use variable val of long long type to check the result with MAX and MIN of signed int type but the description of the problem specifically mentioned that we need to deal within the range of 32-bit integer, although somehow it got accepted but I'm just curious If there is a way to implement a recursive function using only int datatype ?
One more thing even if I consider using long long I'm failing to implement it in the recursive function reverseRec().
If there is a way to implement a recursive function using only int datatype ?
(and) returns 0 when the reversed integer overflows
Yes.
For such +/- problems, I like to fold the int values to one side and negate as needed. The folding to one side (- or +) simplifies overflow detection as only a single side needs testing
I prefer folding to the negative side as there are more negatives, than positives. (With 32-bit int, really didn't make any difference for this problem.)
As code forms the reversed value, test if the following r * 10 + least_digit may overflow before doing it.
An int only recursive solution to reverse an int. Overflow returns 0.
#include <limits.h>
#include <stdio.h>
static int reverse_recurse(int i, int r) {
if (i) {
int least_digit = i % 10;
if (r <= INT_MIN / 10 && (r < INT_MIN / 10 || least_digit < INT_MIN % 10)) {
return 1; /// Overflow indication
}
r = reverse_recurse(i / 10, r * 10 + least_digit);
}
return r;
}
// Reverse an int, overflow returns 0
int reverse_int(int i) {
// Proceed with negative values, they have more range than + side
int r = reverse_recurse(i > 0 ? -i : i, 0);
if (r > 0) {
return 0;
}
if (i > 0) {
if (r < -INT_MAX) {
return 0;
}
r = -r;
}
return r;
}
Test
int main(void) {
int t[] = {0, 1, 42, 1234567890, 1234567892, INT_MAX, INT_MIN};
for (unsigned i = 0; i < sizeof t / sizeof t[0]; i++) {
printf("%11d %11d\n", t[i], reverse_int(t[i]));
if (t[i] != INT_MIN) {
printf("%11d %11d\n", -t[i], reverse_int(-t[i]));
}
}
}
Output
0 0
0 0
1 1
-1 -1
42 24
-42 -24
1234567890 987654321
-1234567890 -987654321
1234567892 0
-1234567892 0
2147483647 0
-2147483647 0
-2147483648 0
You could add a second parameter:
int reverseRec(int x, int reversed)
{
if(x == 0)
{
return reversed;
}
else
{
return reverseRec(x/10, reversed * 10 + x%10);
}
}
And call the function passing the 0 for the second parameter. If you want negative numbers you can check the sign before and pass the absolute value to this function.
In trying to learn C programming I programed this question and get some correct results and some incorrect. I don't see the reason for the difference.
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <math.h> // requires adding link to math -lm as in: gcc b.c -lm -o q11
int ReverseInt(int startValue, int decimalPlace)
{
if(decimalPlace == 0) // if done returns value
{
return startValue;
}
int temp = startValue % 10; // gets units digit
int newStart = (startValue -temp)/10; // computes new starting value after removing one digit
int newDecimal = decimalPlace -1;
int value = temp*pow(10,decimalPlace);
return value + ReverseInt(newStart,newDecimal); // calls itself recursively until done
}
int main()
{
int x, decimalP, startValue;
printf("Input number to be reversed \n Please note number must be less than 214748364 :");
scanf("%d", &x);
if (x > 214748364)
{
printf("Input number to be reversed \n Please note number must be less than 214748364 :");
scanf("%d", &x);
}
decimalP = round(log10(x)); // computes the number of powers of 10 - 0 being units etc.
startValue = ReverseInt(x, decimalP); // calls function with number to be reversed and powers of 10
printf("\n reverse of %d is %d \n", x, startValue);
}
Output is: reverse of 1234 is 4321 but then reverse of 4321 is 12340
It's late and nothing better does not come into my mind. No float calculations. Of course, integer has to be big enough to accommodate the result. Otherwise it is an UB.
int rev(int x, int partial, int *max)
{
int result;
if(x / partial < 10 && (int)(x / partial) > -10)
{
*max = partial;
return abs(x % 10) * partial;
}
result = rev(x, partial * 10, max) + abs(((x / (int)(*max / partial)) % 10) * partial);
return result;
}
int reverse(int x)
{
int max;
return rev(x, 1, &max) * ((x < 0) ? -1 : 1);
}
int main(void){
printf("%d", reverse(-456789));
}
https://godbolt.org/z/M1eezf
unsigned rev(unsigned x, unsigned partial, unsigned *max)
{
unsigned result;
if(x / partial < 10)
{
*max = partial;
return (x % 10) * partial;
}
result = rev(x, partial * 10, max) + (x / (*max / partial) % 10) * partial;
return result;
}
unsigned reverse(unsigned x)
{
unsigned max;
return rev(x, 1, &max);
}
int main(void){
printf("%u", reverse(123456));
}
when using long long to store the result all possible integers can be reversed
long long rev(int x, long long partial, long long *max)
{
long long result;
if(x / partial < 10 && (int)(x / partial) > -10)
{
*max = partial;
return abs(x % 10) * partial;
}
result = rev(x, partial * 10, max) + abs(((x / (int)(*max / partial)) % 10) * partial);
return result;
}
long long reverse(int x)
{
long long max;
return rev(x, 1, &max) * ((x < 0) ? -1 : 1);
}
int main(void){
printf("%d reversed %lld\n", INT_MIN, reverse(INT_MIN));
printf("%d reversed %lld\n", INT_MAX, reverse(INT_MAX));
}
https://godbolt.org/z/KMfbxz
I am assuming by reversing an integer you mean turning 129 to 921 or 120 to 21.
You need an initial method to initialize your recursive function.
Your recursive function must figure out how many decimal places your integer uses. This can be found by using log base 10 with the value and then converting the result to a integer.
log10 (103) approx. 2.04 => 2
Modulus the initial value by 10 to get the ones place and store it in a variable called temp
Subtract the ones place from the initial value and store that in a variable called newStart.
divide this value by 10
Subtract one from the decimal place and store in another variable called newDecimal.
Return the ones place times 10 to the power of the decimal place and add it to the function where the initial value is newStart and the decimalPlace is newDecimal.
#include <stdio.h>
#include <math.h>
int ReverseInt(int startValue, int decimalPlace);
int main()
{
int i = -54;
int positive = i < 0? i*-1 : i;
double d = log10(positive);
int output = ReverseInt(positive,(int)d);
int correctedOutput = i < 0? output*-1 : output;
printf("%d \n",correctedOutput);
return 0;
}
int ReverseInt(int startValue, int decimalPlace)
{
if(decimalPlace == 0)
{
return startValue;
}
int temp = startValue % 10;
int newStart = (startValue -temp)/10;
int newDecimal = decimalPlace -1;
int value = temp*pow(10,decimalPlace);
return value + ReverseInt(newStart,newDecimal);
}
Let's say I've been given two integers a, b where a is a positive integer and is smaller than b. I have to find an efficient algorithm that's going to give me the sum of number of base2 digits (number of bits) over the interval [a, b]. For example, in the interval [0, 4] the sum of digits is equal to 9 because 0 = 1 digit, 1 = 1 digit, 2 = 2 digits, 3 = 2 digits and 4 = 3 digits.
My program is capable of calculating this number by using a loop but I'm looking for something more efficient for large numbers. Here are the snippets of my code just to give you an idea:
int numberOfBits(int i) {
if(i == 0) {
return 1;
}
else {
return (int) log2(i) + 1;
}
}
The function above is for calculating the number of digits of one number in the interval.
The code below shows you how I use it in my main function.
for(i = a; i <= b; i++) {
l = l + numberOfBits(i);
}
printf("Digits: %d\n", l);
Ideally I should be able to get the number of digits by using the two values of my interval and using some special algorithm to do that.
Try this code, i think it gives you what you are needing to calculate the binaries:
int bit(int x)
{
if(!x) return 1;
else
{
int i;
for(i = 0; x; i++, x >>= 1);
return i;
}
}
The main thing to understand here is that the number of digits used to represent a number in binary increases by one with each power of two:
+--------------+---------------+
| number range | binary digits |
+==============+===============+
| 0 - 1 | 1 |
+--------------+---------------+
| 2 - 3 | 2 |
+--------------+---------------+
| 4 - 7 | 3 |
+--------------+---------------+
| 8 - 15 | 4 |
+--------------+---------------+
| 16 - 31 | 5 |
+--------------+---------------+
| 32 - 63 | 6 |
+--------------+---------------+
| ... | ... |
A trivial improvement over your brute force algorithm would then be to figure out how many times this number of digits has increased between the two numbers passed in (given by the base two logarithm) and add up the digits by multiplying the count of numbers that can be represented by the given number of digits (given by the power of two) with the number of digits.
A naive implementation of this algorithm is:
int digits_sum_seq(int a, int b)
{
int sum = 0;
int i = 0;
int log2b = b <= 0 ? 1 : floor(log2(b));
int log2a = a <= 0 ? 1 : floor(log2(a)) + 1;
sum += (pow(2, log2a) - a) * (log2a);
for (i = log2b; i > log2a; i--)
sum += pow(2, i - 1) * i;
sum += (b - pow(2, log2b) + 1) * (log2b + 1);
return sum;
}
It can then be improved by the more efficient versions of the log and pow functions seen in the other answers.
First, we can improve the speed of log2, but that only gives us a fixed factor speed-up and doesn't change the scaling.
Faster log2 adapted from: https://graphics.stanford.edu/~seander/bithacks.html#IntegerLogLookup
The lookup table method takes only about 7 operations to find the log
of a 32-bit value. If extended for 64-bit quantities, it would take
roughly 9 operations. Another operation can be trimmed off by using
four tables, with the possible additions incorporated into each. Using
int table elements may be faster, depending on your architecture.
Second, we must re-think the algorithm. If you know that numbers between N and M have the same number of digits, would you add them up one by one or would you rather do (M-N+1)*numDigits?
But if we have a range where multiple numbers appear what do we do? Let's just find the intervals of same digits, and add sums of those intervals. Implemented below. I think that my findEndLimit could be further optimized with a lookup table.
Code
#include <stdio.h>
#include <limits.h>
#include <time.h>
unsigned int fastLog2(unsigned int v)
{
static const char LogTable256[256] =
{
#define LT(n) n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n
-1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3,
LT(4), LT(5), LT(5), LT(6), LT(6), LT(6), LT(6),
LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7)
};
register unsigned int t, tt; // temporaries
if (tt = v >> 16)
{
return (t = tt >> 8) ? 24 + LogTable256[t] : 16 + LogTable256[tt];
}
else
{
return (t = v >> 8) ? 8 + LogTable256[t] : LogTable256[v];
}
}
unsigned int numberOfBits(unsigned int i)
{
if (i == 0) {
return 1;
}
else {
return fastLog2(i) + 1;
}
}
unsigned int findEndLimit(unsigned int sx, unsigned int ex)
{
unsigned int sy = numberOfBits(sx);
unsigned int ey = numberOfBits(ex);
unsigned int mx;
unsigned int my;
if (sy == ey) // this also means sx == ex
return ex;
// assumes sy < ey
mx = (ex - sx) / 2 + sx; // will eq. sx for sx + 1 == ex
my = numberOfBits(mx);
while (ex - sx != 1) {
mx = (ex - sx) / 2 + sx; // will eq. sx for sx + 1 == ex
my = numberOfBits(mx);
if (my == ey) {
ex = mx;
ey = numberOfBits(ex);
}
else {
sx = mx;
sy = numberOfBits(sx);
}
}
return sx+1;
}
int main(void)
{
unsigned int a, b, m;
unsigned long l;
clock_t start, end;
l = 0;
a = 0;
b = UINT_MAX;
start = clock();
unsigned int i;
for (i = a; i < b; ++i) {
l += numberOfBits(i);
}
if (i == b) {
l += numberOfBits(i);
}
end = clock();
printf("Naive\n");
printf("Digits: %ld; Time: %fs\n",l, ((double)(end-start))/CLOCKS_PER_SEC);
l=0;
start = clock();
do {
m = findEndLimit(a, b);
l += (b-m + 1) * (unsigned long)numberOfBits(b);
b = m-1;
} while (b > a);
l += (b-a+1) * (unsigned long)numberOfBits(b);
end = clock();
printf("Binary search\n");
printf("Digits: %ld; Time: %fs\n",l, ((double)(end-start))/CLOCKS_PER_SEC);
}
Output
From 0 to UINT_MAX
$ ./main
Naive
Digits: 133143986178; Time: 25.722492s
Binary search
Digits: 133143986178; Time: 0.000025s
My findEndLimit can take long time in some edge cases:
From UINT_MAX/16+1 to UINT_MAX/8
$ ./main
Naive
Digits: 7784628224; Time: 1.651067s
Binary search
Digits: 7784628224; Time: 4.921520s
Conceptually, you would need to split the task to two subproblems -
1) find the sum of digits from 0..M, and from 0..N, then subtract.
2) find the floor(log2(x)), because eg for the number 77 the numbers 64,65,...77 all have 6 digits, the next 32 have 5 digits, the next 16 have 4 digits and so on, which makes a geometric progression.
Thus:
int digits(int a) {
if (a == 0) return 1; // should digits(0) be 0 or 1 ?
int b=(int)floor(log2(a)); // use any all-integer calculation hack
int sum = 1 + (b+1) * (a- (1<<b) +1); // added 1, due to digits(0)==1
while (--b)
sum += (b + 1) << b; // shortcut for (b + 1) * (1 << b);
return sum;
}
int digits_range(int a, int b) {
if (a <= 0 || b <= 0) return -1; // formulas work for strictly positive numbers
return digits(b)-digits(a-1);
}
As efficiency depends on the tools available, one approach would be doing it "analog":
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
unsigned long long pow2sum_min(unsigned long long n, long long unsigned m)
{
if (m >= n)
{
return 1;
}
--n;
return (2ULL << n) + pow2sum_min(n, m);
}
#define LN(x) (log2(x)/log2(M_E))
int main(int argc, char** argv)
{
if (2 >= argc)
{
fprintf(stderr, "%s a b\n", argv[0]);
exit(EXIT_FAILURE);
}
long a = atol(argv[1]), b = atol(argv[2]);
if (0L >= a || 0L >= b || b < a)
{
puts("Na ...!");
exit(EXIT_FAILURE);
}
/* Expand intevall to cover full dimensions: */
unsigned long long a_c = pow(2, floor(log2(a)));
unsigned long long b_c = pow(2, floor(log2(b+1)) + 1);
double log2_a_c = log2(a_c);
double log2_b_c = log2(b_c);
unsigned long p2s = pow2sum_min(log2_b_c, log2_a_c) - 1;
/* Integral log2(x) between a_c and b_c: */
double A = ((b_c * (LN(b_c) - 1))
- (a_c * (LN(a_c) - 1)))/LN(2)
+ (b+1 - a);
/* "Integer"-integral - integral of log2(x)'s inverse function (2**x) between log(a_c) and log(b_c): */
double D = p2s - (b_c - a_c)/LN(2);
/* Corrective from a_c/b_c to a/b : */
double C = (log2_b_c - 1)*(b_c - (b+1)) + log2_a_c*(a - a_c);
printf("Total used digits: %lld\n", (long long) ((A - D - C) +.5));
}
:-)
The main thing here is the number and kind of iterations done.
Number is
log(floor(b_c)) - log(floor(a_c))
times
doing one
n - 1 /* Integer decrement */
2**n + s /* One bit-shift and one integer addition */
for each iteration.
Here's an entirely look-up based approach. You don't even need the log2 :)
Algorithm
First we precompute interval limits where the number of bits would change and create a lookup table. In other words we create an array limits[2^n], where limits[i] gives us the biggest integer that can be represented with (i+1) bits. Our array is then {1, 3, 7, ..., 2^n-1}.
Then, when we want to determine the sum of bits for our range, we must first match our range limits a and b with the smallest index for which a <= limits[i] and b <= limits[j] holds, which will then tell us that we need (i+1) bits to represent a, and (j+1) bits to represent b.
If the indexes are the same, then the result is simply (b-a+1)*(i+1), otherwise we must separately get the number of bits from our value to the edge of same number of bits interval, and add up total number of bits for each interval between as well. In any case, simple arithmetic.
Code
#include <stdio.h>
#include <limits.h>
#include <time.h>
unsigned long bitsnumsum(unsigned int a, unsigned int b)
{
// generate lookup table
// limits[i] is the max. number we can represent with (i+1) bits
static const unsigned int limits[32] =
{
#define LTN(n) n*2u-1, n*4u-1, n*8u-1, n*16u-1, n*32u-1, n*64u-1, n*128u-1, n*256u-1
LTN(1),
LTN(256),
LTN(256*256),
LTN(256*256*256)
};
// make it work for any order of arguments
if (b < a) {
unsigned int c = a;
a = b;
b = c;
}
// find interval of a
unsigned int i = 0;
while (a > limits[i]) {
++i;
}
// find interval of b
unsigned int j = i;
while (b > limits[j]) {
++j;
}
// add it all up
unsigned long sum = 0;
if (i == j) {
// a and b in the same range
// conveniently, this also deals with j == 0
// so no danger to do [j-1] below
return (i+1) * (unsigned long)(b - a + 1);
}
else {
// add sum of digits in range [a, limits[i]]
sum += (i+1) * (unsigned long)(limits[i] - a + 1);
// add sum of digits in range [limits[j], b]
sum += (j+1) * (unsigned long)(b - limits[j-1]);
// add sum of digits in range [limits[i], limits[j]]
for (++i; i<j; ++i) {
sum += (i+1) * (unsigned long)(limits[i] - limits[i-1]);
}
return sum;
}
}
int main(void)
{
clock_t start, end;
unsigned int a=0, b=UINT_MAX;
start = clock();
printf("Sum of binary digits for numbers in range "
"[%u, %u]: %lu\n", a, b, bitsnumsum(a, b));
end = clock();
printf("Time: %fs\n", ((double)(end-start))/CLOCKS_PER_SEC);
}
Output
$ ./lookup
Sum of binary digits for numbers in range [0, 4294967295]: 133143986178
Time: 0.000282s
Algorithm
The main idea is to find the n2 = log2(x) rounded down. That is the number of digits in x. Let pow2 = 1 << n2. n2 * (pow2 - x + 1) is the number of digits in the values [x...pow2]. Now find the sun of digits in the powers of 2 from 1 to n2-1
Code
I am certain various simplifications can be made.
Untested code. Will review later.
// Let us use unsigned for everything.
unsigned ulog2(unsigned value) {
unsigned result = 0;
if (0xFFFF0000u & value) {
value >>= 16; result += 16;
}
if (0xFF00u & value) {
value >>= 8; result += 8;
}
if (0xF0u & value) {
value >>= 4; result += 4;
}
if (0xCu & value) {
value >>= 2; result += 2;
}
if (0x2 & value) {
value >>= 1; result += 1;
}
return result;
}
unsigned bit_count_helper(unsigned x) {
if (x == 0) {
return 1;
}
unsigned n2 = ulog2(x);
unsigned pow2 = 1u << n;
unsigned sum = n2 * (pow2 - x + 1u); // value from pow2 to x
while (n2 > 0) {
// ... + 5*16 + 4*8 + 3*4 + 2*2 + 1*1
pow2 /= 2;
sum += n2 * pow2;
}
return sum;
}
unsigned bit_count(unsigned a, unsigned b) {
assert(a < b);
return bit_count_helper(b - 1) - bit_count_helper(a);
}
For this problem your solution is the simplest, the one called "naive" where you look for every element in the sequence or in your case interval for check something or execute operations.
Naive Algorithm
Assuming that a and b are positive integers with b greater than a let's call the dimension/size of the interval [a,b], n = (b-a).
Having our number of elements n and using some notations of algorithms (like big-O notation link), the worst case cost is O(n*(numberOfBits_cost)).
From this we can see that we can speed up our algorithm by using a faster algorithm for computing numberOfBits() or we need to find a way to not look at every element of the interval that costs us n operations.
Intuition
Now looking at a possible interval [6,14] you can see that for 6 and 7 we need 3 digits, with 4 need for 8,9,10,11,12,13,14. This results in calling numberOfBits() for every number that use the same number of digits to be represented, while the following multiplication operation would be faster:
(number_in_subinterval)*digitsForThisInterval
((14-8)+1)*4 = 28
((7-6)+1)*3 = 6
So we reduced the looping on 9 elements with 9 operations to only 2.
So writing a function that use this intuition will give us a more efficient in time, not necessarily in memory, algorithm. Using your numberOfBits() function I have created this solution:
int intuitionSol(int a, int b){
int digitsForA = numberOfBits(a);
int digitsForB = numberOfBits(b);
if(digitsForA != digitsForB){
//because a or b can be that isn't the first or last element of the
// interval that a specific number of digit can rappresent there is a need
// to execute some correction operation before on a and b
int tmp = pow(2,digitsForA) - a;
int result = tmp*digitsForA; //will containt the final result that will be returned
int i;
for(i = digitsForA + 1; i < digitsForB; i++){
int interval_elements = pow(2,i) - pow(2,i-1);
result = result + ((interval_elements) * i);
//printf("NumOfElem: %i for %i digits; sum:= %i\n", interval_elements, i, result);
}
int tmp1 = ((b + 1) - pow(2,digitsForB-1));
result = result + tmp1*digitsForB;
return result;
}
else {
int elements = (b - a) + 1;
return elements * digitsForA; // or digitsForB
}
}
Let's look at the cost, this algorithm costs is the cost of doing correction operation on a and b plus the most expensive one that of the for-loop. In my solution however I'm not looping over all elements but only on numberOfBits(b)-numberOfBits(a) that in the worst case, when [0,n], become log(n)-1 thats equivalent to O(log n).
To resume we passed from a linear operations cost O(n) to a logartmic one O(log n) in the worst case. Look on this diagram the diferinces between the two.
Note
When I talk about interval or sub-interval I refer to the interval of elements that use the same number of digits to represent the number in binary.
Following there are some output of my tests with the last one that shows the difference:
Considered interval is [0,4]
YourSol: 9 in time: 0.000015s
IntuitionSol: 9 in time: 0.000007s
Considered interval is [0,0]
YourSol: 1 in time: 0.000005s
IntuitionSol: 1 in time: 0.000005s
Considered interval is [4,7]
YourSol: 12 in time: 0.000016s
IntuitionSol: 12 in time: 0.000005s
Considered interval is [2,123456]
YourSol: 1967697 in time: 0.005010s
IntuitionSol: 1967697 in time: 0.000015s
I have read a lot of good algos to calculate n! mod m but they were usually valid when m was prime . I wanted to know whether some good algo exists when m is not prime .I would be helpful if someone could write the basic function of the algo too.I have been using
long long factMOD(long long n,long long mod)
{
long long res = 1;
while (n > 0)
{
for (long long i=2, m=n%mod; i<=m; i++)
res = (res * i) % mod;
if ((n/=mod)%2 > 0)
res = mod - res;
}
return res;
}
but getting wrong answer when I try to print factMOD(4,3) even. source of this algo is :
http://comeoncodeon.wordpress.com/category/algorithm/
The basic algorithm is valid for any value of m:
product := 1
for i := 2 to n
product := (product * i) mod m
return product
and an easy optimization is that you can bail out early and return 0 whenever product becomes 0. You can also return 0 at the beginning if n > m, since that guarantees that n! is a multiple of m.
This is what I've come up with:
#include <stdio.h>
#include <stdlib.h>
unsigned long long nfactmod(unsigned long long n, unsigned long long m)
{
unsigned long long i, f;
for (i = 1, f = 1; i <= n; i++) {
f *= i;
if (f > m) {
f %= m;
}
}
return f;
}
int main(int argc, char *argv[])
{
unsigned long long n = strtoull(argv[1], NULL, 10);
unsigned long long m = strtoull(argv[2], NULL, 10);
printf("%llu\n", nfactmod(n, m));
return 0;
}
and this:
h2co3-macbook:~ h2co3$ ./mod 1000000 1001001779
744950559
h2co3-macbook:~ h2co3$
runs in a fraction of a second.