I am currently trying to make my chess engine faster, and am looking at implementing magic bitboards for my sliding piece attack generation. I am using a bitboard representation of the chessboard with the a1 square being the furthest right bit, and h8 being the furthest left. I am looking at this site:
https://rhysre.net/fast-chess-move-generation-with-magic-bitboards.html#:~:text=A%20bitboard%20is%20simply%20a,and%20bit%2063%20%3D%20h8)
Specifically the code snippet found towards the bottom of the page that reads:
U64 getBishopAttacksMagic(int square, U64 blockers) {
// Mask blockers to only include bits on diagonals
blockers &= BISHOP_MASKS[square];
// Generate the key using a multiplication and right shift
U64 key = (blockers * BISHOP_MAGICS[square]) >> (64 - BISHOP_INDEX_BITS[square]);
// Return the preinitialized attack set bitboard from the table
return BISHOP_TABLE[square][key];
}
I already have Shallow blues magic numbers(each number corresponding to a square), and I already have pre initialized attack masks for the bishop piece stored in a 64 length array(again each number corresponding to a square). So i know how to get the key. But how do I generate the last array which takes the key, "BISHOP_TABLE" array? I do not understand how to generate that 2d array given an attack mask and magic number for each square. Thank you for your help in advance.
For each square, you need to generate every permutation of blocking pieces inside the bishop mask. For example, using this mask for the square e4 (#28):
8 | 0 0 0 0 0 0 0 0
7 | 0 1 0 0 0 0 0 0
6 | 0 0 1 0 0 0 1 0
5 | 0 0 0 1 0 1 0 0
4 | 0 0 0 0 0 0 0 0
3 | 0 0 0 1 0 1 0 0
2 | 0 0 1 0 0 0 1 0
1 | 0 0 0 0 0 0 0 0
---------------
a b c d e f g h
since there are 9 set bits, there are 2^9 = 512 different patterns of blocking pieces. The permutation number 339 (as binary = 0b101010011) looks like this:
8 | 0 0 0 0 0 0 0 0
7 | 0 1 0 0 0 0 0 0
6 | 0 0 1 0 0 0 0 0
5 | 0 0 0 1 0 0 0 0
4 | 0 0 0 0 0 0 0 0
3 | 0 0 0 0 0 0 0 0
2 | 0 0 1 0 0 0 1 0
1 | 0 0 0 0 0 0 0 0
---------------
a b c d e f g h
Bits are read from right (lsb) to left (msb) in the number and are set in the mask from the a file to the h file, 1st rank to 8th. Permutation 0 is an empty board and 511 (0b111111111) is the full mask.
Here's a method that takes a permutation number along with the bishop mask and returns the corresponding blockers bitboard:
private static long blockersPermutation(int iteration, long mask) {
long blockers = 0;
while (iteration != 0) {
if ((iteration & 1) != 0) {
int shift = Long.numberOfTrailingZeros(mask);
blockers |= (1L << shift);
}
iteration >>>= 1;
mask &= (mask - 1); // used in Kernighan's bit count algorithm
// it pops out the least significant bit in the number
}
return blockers;
}
Using this we can calculate the keys with both the magic numbers and the blockers, and we can create their corresponding values, the attack masks.
For each permutation of blocking pieces, create the corresponding attack mask and store it in the table. Include the blocking pieces and the squares on the side of the board in the mask. The attack mask for the blockers #339 on square #28 is:
8 | 0 0 0 0 0 0 0 0
7 | 0 0 0 0 0 0 0 1
6 | 0 0 0 0 0 0 1 0
5 | 0 0 0 1 0 1 0 0
4 | 0 0 0 0 0 0 0 0
3 | 0 0 0 1 0 1 0 0
2 | 0 0 1 0 0 0 1 0
1 | 0 0 0 0 0 0 0 0
---------------
a b c d e f g h
Here's a Java method to initialize the 64 bishop lookup tables:
private final long[][] BISHOP_LOOKUP = new long[64][512];
private static int getFile(int square) {
return square % 8;
}
private static int getRank(int square) {
return square / 8;
}
private static int getSquare(int rank, int file) {
return rank * 8 + file;
}
// just like the code snippet, generates the key
private static int transform (long blockers, long magic, int shift) {
return (int) ((blockers * magic) >>> (64 - shift));
}
private void initBishopLookup() {
for (int square = 0; square < 64; square++) {
long mask = BISHOP_MASKS[square];
int permutationCount = (1 << Long.bitCount(mask));
for (int i = 0; i < permutationCount; i++) {
long blockers = blockersPermutation(i, mask);
long attacks = 0L;
int rank = getRank(square), r;
int file = getFile(square), f;
for (r = rank + 1, f = file + 1; r <= 7 && f <= 7; r++, f++) {
attacks |= (1L << getSquare(r, f));
if ((blockers & (1L << getSquare(r, f))) != 0) {
break;
}
}
for (r = rank - 1, f = file + 1; r >= 0 && f <= 7; r--, f++) {
attacks |= (1L << getSquare(r, f));
if ((blockers & (1L << getSquare(r, f))) != 0) {
break;
}
}
for (r = rank - 1, f = file - 1; r >= 0 && f >= 0; r--, f--) {
attacks |= (1L << getSquare(r, f));
if ((blockers & (1L << getSquare(r, f))) != 0) {
break;
}
}
for (r = rank + 1, f = file - 1; r <= 7 && f >= 0; r++, f--) {
attacks |= (1L << getSquare(r, f));
if ((blockers & (1L << getSquare(r, f))) != 0) {
break;
}
}
int key = transform(blockers, BISHOP_MAGICS[square], Long.bitCount(mask));
BISHOP_LOOKUP[square][key] = attacks;
}
}
}
This uses plain magic bitboards with fixed size lookup tables, all of length 512 when masks with less set bits could fit in less space. Like on square b1 the mask uses 5 bits on the diagonal and the table could fit in an array of length 2^5 = 32. We're wasting (512 - 32) * (8 bytes per 64 bits number) / 1024 bytes per Kio = 3.75Kio for this square. Plain magic bitboards take 2Mib of memory for rooks and 256Kib for bishops, using fancy magic bitboards can reduce the total to ~800Kib. It's not really needed though, 2.25Mib of memory is small.
I'm writing a method to store a number's prime factor.
I'm asked to use 2-d array to store its prime factor and the factor's number.
public static int[][] getMatrix (long x){
int[][] matrix =new int[10][2];
int count;
for (int i = 2, j = 0; i <=x / 2; i++) {
count=0;
while (x % i == 0) {
x = x/i;
count++;
}
matrix[j][0] = i;
matrix[j][1] = count;
j++;
}
return matrix;
}
But this code only store data into the first row of the array.
Could someone help me correct it or privide other ideas?
if I use the following code to output the result.
for(int row=0;row<b_matrix.length;row++)
{
for(int column=0;column<2;column++)
{
System.out.print(b_matrix[row][column]+" ");
}
}
And x=9 I got this:
2 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
x=6 I got this:
2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
such as :6
matrix[0][0]=2 matrix[0][1]=1
matrix[1][0]=3 matrix[1][1]=1 //can't store
such as :9
matrix[0][0]=2 matrix[0][1]=0//only output the next row when this equals to 0
matrix[1][0]=3 matrix[1][1]=2
Your logic is right except, in the for loop i should go till x instead of x/2 as follows,
for (int i = 2, j = 0; i <= x; i++)
Output for getMatrix(60):
2 2
3 1
4 0
5 1
0 0
0 0
0 0
0 0
0 0
0 0
I need to find the number of heavy integers between two integers A and B, where A <= B at all times.
An integer is considered heavy whenever the average of it's digit is larger than 7.
For example: 9878 is considered heavy, because (9 + 8 + 7 + 8)/4 = 8
, while 1111 is not, since (1 + 1 + 1 + 1)/4 = 1.
I have the solution below, but it's absolutely terrible and it times out when run with large inputs. What can I do to make it more efficient?
int countHeavy(int A, int B) {
int countHeavy = 0;
while(A <= B){
if(averageOfDigits(A) > 7){
countHeavy++;
}
A++;
}
return countHeavy;
}
float averageOfDigits(int a) {
float result = 0;
int count = 0;
while (a > 0) {
result += (a % 10);
count++;
a = a / 10;
}
return result / count;
}
Counting the numbers with a look-up table
You can generate a table that stores how many integers with d digits have a sum of their digits that is greater than a number x. Then, you can quickly look up how many heavy numbers there are in any range of 10, 100, 1000 ... integers. These tables hold only 9×d values, so they take up very little space and can be quickly generated.
Then, to check a range A-B where B has d digits, you build the tables for 1 to d-1 digits, and then you split the range A-B into chunks of 10, 100, 1000 ... and look up the values in the tables, e.g. for the range A = 782, B = 4321:
RANGE DIGITS TARGET LOOKUP VALUE
782 - 789 78x > 6 table[1][ 6] 3 <- incomplete range: 2-9
790 - 799 79x > 5 table[1][ 5] 4
800 - 899 8xx >13 table[2][13] 15
900 - 999 9xx >12 table[2][12] 21
1000 - 1999 1xxx >27 table[3][27] 0
2000 - 2999 2xxx >26 table[3][26] 1
3000 - 3999 3xxx >25 table[3][25] 4
4000 - 4099 40xx >24 impossible 0
4100 - 4199 41xx >23 impossible 0
4200 - 4299 42xx >22 impossible 0
4300 - 4309 430x >21 impossible 0
4310 - 4319 431x >20 impossible 0
4320 - 4321 432x >19 impossible 0 <- incomplete range: 0-1
--
48
If the first and last range are incomplete (not *0 - *9), check the starting value or the end value against the target. (In the example, 2 is not greater than 6, so all 3 heavy numbers are included in the range.)
Generating the look-up table
For 1-digit decimal integers, the number of integers n that is greater than value x is:
x: 0 1 2 3 4 5 6 7 8 9
n: 9 8 7 6 5 4 3 2 1 0
As you can see, this is easily calculated by taking n = 9-x.
For 2-digit decimal integers, the number of integers n whose sum of digits is greater than value x is:
x: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
n: 99 97 94 90 85 79 72 64 55 45 36 28 21 15 10 6 3 1 0
For 3-digit decimal integers, the number of integers n whose sum of digits is greater than value x is:
x: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
n: 999 996 990 980 965 944 916 880 835 780 717 648 575 500 425 352 283 220 165 120 84 56 35 20 10 4 1 0
Each of these sequences can be generated from the previous one: start with value 10d and then subtract from this value the previous sequence in reverse (skipping the first zero). E.g. to generate the sequence for 3 digits from the sequence for 2 digits, start with 103 = 1000, and then:
0. 1000 - 1 = 999
1. 999 - 3 = 996
2. 996 - 6 = 990
3. 990 - 10 = 980
4. 980 - 15 = 965
5. 965 - 21 = 944
6. 944 - 28 = 916
7. 916 - 36 = 880
8. 880 - 45 = 835
9. 835 - 55 = 780
10. 780 - 64 + 1 = 717 <- after 10 steps, start adding the previous sequence again
11. 717 - 72 + 3 = 648
12. 648 - 79 + 6 = 575
13. 575 - 85 + 10 = 500
14. 500 - 90 + 15 = 425
15. 425 - 94 + 21 = 352
16. 352 - 97 + 28 = 283
17. 283 - 99 + 36 = 220
18. 220 - 100 + 45 = 165 <- at the end of the sequence, keep subtracting 10^(d-1)
19. 165 - 100 + 55 = 120
20. 120 - 100 + 64 = 84
21. 84 - 100 + 72 = 56
22. 56 - 100 + 79 = 35
23. 35 - 100 + 85 = 20
24. 20 - 100 + 90 = 10
25. 10 - 100 + 94 = 4
26. 4 - 100 + 97 = 1
27. 1 - 100 + 99 = 0
By the way, you can use the same tables if "heavy" numbers are defined with a value other than 7.
Code example
Below is a Javascript code snippet (I don't speak Java) that demonstrates the method. It is very much unoptimised, but it does the 0→100,000,000 example in less than 0.07ms. It also works for weights other than 7. Translated to Java, it should easily beat any algorithm that actually runs through the numbers and checks their weight.
function countHeavy(A, B, weight) {
var a = decimalDigits(A), b = decimalDigits(B); // create arrays
while (a.length < b.length) a.push(0); // add leading zeros
var digits = b.length, table = weightTable(); // create table
var count = 0, diff = B - A + 1, d = 0; // calculate range
for (var i = digits - 1; i >= 0; i--) if (a[i]) d = i; // lowest non-0 digit
while (diff) { // increment a until a=b
while (a[d] == 10) { // move to higher digit
a[d++] = 0;
++a[d]; // carry 1
}
var step = Math.pow(10, d); // value of digit d
if (step <= diff) {
diff -= step;
count += increment(d); // increment digit d
}
else --d; // move to lower digit
}
return count;
function weightTable() { // see above for details
var t = [[],[9,8,7,6,5,4,3,2,1,0]];
for (var i = 2; i < digits; i++) {
var total = Math.pow(10, i), final = total / 10;
t[i] = [];
for (var j = 9 * i; total > 0; --j) {
if (j > 9) total -= t[i - 1][j - 10]; else total -= final;
if (j < 9 * (i - 1)) total += t[i - 1][j];
t[i].push(total);
}
}
return t;
}
function increment(d) {
var sum = 0, size = digits;
for (var i = digits - 1; i >= d; i--) {
if (a[i] == 0 && i == size - 1) size = i; // count used digits
sum += a[i]; // sum of digits
}
++a[d];
var target = weight * size - sum;
if (d == 0) return (target < 0) ? 1 : 0; // if d is lowest digit
if (target < 0) return table[d][0] + 1; // whole range is heavy
return (target > 9 * d) ? 0 : table[d][target]; // use look-up table
}
function decimalDigits(n) {
var array = [];
do {array.push(n % 10);
n = Math.floor(n / 10);
} while (n);
return array;
}
}
document.write("0 → 100,000,000 = " + countHeavy(0, 100000000, 7) + "<br>");
document.write("782 → 4321 = " + countHeavy(782, 4321, 7) + "<br>");
document.write("782 → 4321 = " + countHeavy(782, 4321, 5) + " (weight: 5)");
I really liked the post of #m69 so I wrote implementation inspired by it. The table creation is not that elegant, but works. For n+1 digits long integer I sum (at most) 10 values from n digits long integer, one for every digit 0-9.
I use this simplification to avoid arbitrary range calculation:
countHeavy(A, B) = countHeavy(0, B) - countHeavy(0, A-1)
The result is calculated in two loops. One for numbers shorter than the given number and one for the rest. I was not able to merge them easily. getResultis just lookup into the tablewith range checking, the rest of the code should be quite obvious.
public class HeavyNumbers {
private static int maxDigits = String.valueOf(Long.MAX_VALUE).length();
private int[][] table = null;
public HeavyNumbers(){
table = new int[maxDigits + 1][];
table[0] = new int[]{1};
for (int s = 1; s < maxDigits + 1; ++s) {
table[s] = new int[s * 9 + 1];
for (int k = 0; k < table[s].length; ++k) {
for (int d = 0; d < 10; ++d) {
if (table[s - 1].length > k - d) {
table[s][k] += table[s - 1][Math.max(0, k - d)];
}
}
}
}
}
private int[] getNumberAsArray(long number) {
int[] tmp = new int[maxDigits];
int cnt = 0;
while (number != 0) {
int remainder = (int) (number % 10);
tmp[cnt++] = remainder;
number = number / 10;
}
int[] ret = new int[cnt];
for (int i = 0; i < cnt; ++i) {
ret[i] = tmp[i];
}
return ret;
}
private int getResult(int[] sum, int digits, int fixDigitSum, int heavyThreshold) {
int target = heavyThreshold * digits - fixDigitSum + 1;
if (target < sum.length) {
return sum[Math.max(0, target)];
}
return 0;
}
public int getHeavyNumbersCount(long toNumberIncl, int heavyThreshold) {
if (toNumberIncl <= 0) return 0;
int[] numberAsArray = getNumberAsArray(toNumberIncl);
int res = 0;
for (int i = 0; i < numberAsArray.length - 1; ++i) {
for (int d = 1; d < 10; ++d) {
res += getResult(table[i], i + 1, d, heavyThreshold);
}
}
int fixDigitSum = 0;
int fromDigit = 1;
for (int i = numberAsArray.length - 1; i >= 0; --i) {
int toDigit = numberAsArray[i];
if (i == 0) {
toDigit++;
}
for (int d = fromDigit; d < toDigit; ++d) {
res += getResult(table[i], numberAsArray.length, fixDigitSum + d, heavyThreshold);
}
fixDigitSum += numberAsArray[i];
fromDigit = 0;
}
return res;
}
public int getHeavyNumbersCount(long fromIncl, long toIncl, int heavyThreshold) {
return getHeavyNumbersCount(toIncl, heavyThreshold) -
getHeavyNumbersCount(fromIncl - 1, heavyThreshold);
}
}
It is used like this:
HeavyNumbers h = new HeavyNumbers();
System.out.println( h.getHeavyNumbersCount(100000000,7));
prints out 569484, the repeated calculation time without initialization of the table is under 1us
I looked at the problem differently than you did. My perception is that the problem is based on the base-10 representation of a number, so the first thing you should do is to put the number into a base-10 representation. There may be a nicer way of doing it, but Java Strings represent Integers in base-10, so I used those. It's actually pretty fast to turn a single character into an integer, so this doesn't really cost much time.
Most importantly, your calculations in this matter never need to use division or floats. The problem is, at its core, about integers only. Do all the digits (integers) in the number (integer) add up to a value greater than or equal to seven (integer) times the number of digits (integer)?
Caveat - I don't claim that this is the fastest possible way of doing it, but this is probably faster than your original approach.
Here is my code:
package heavyNum;
public class HeavyNum
{
public static void main(String[] args)
{
HeavyNum hn = new HeavyNum();
long startTime = System.currentTimeMillis();
hn.countHeavy(100000000, 1);
long endTime = System.currentTimeMillis();
System.out.println("Time elapsed: "+(endTime- startTime));
}
private void countHeavy(int A, int B)
{
int heavyFound = 0;
for(int i = B+1; i < A; i++)
{
if(isHeavy(i))
heavyFound++;
}
System.out.println("Found "+heavyFound+" heavy numbers");
}
private boolean isHeavy(int i)
{
String asString = Integer.valueOf(i).toString();
int length = asString.length();
int dividingLine = length * 7, currTotal = 0, counter = 0;
while(counter < length)
{
currTotal += Character.getNumericValue(asString.charAt(counter++));
}
return currTotal > dividingLine;
}
}
Credit goes to this SO Question for how I get the number of digits in an integer and this SO Question for how to quickly convert characters to integers in java
Running on a powerful computer with no debugger for numbers between one and 100,000,000 resulted in this output:
Found 569484 heavy numbers
Time elapsed: 6985
EDIT: I initially was looking for numbers whose digits were greater than or equal to 7x the number of digits. I previously had results of 843,453 numbers in 7025 milliseconds.
Here's a pretty barebones recursion with memoization that enumerates the digit possibilities one by one for a fixed-digit number. You may be able to set A and B by controlling the range of i when calculating the corresponding number of digits.
Seems pretty fast (see the result for 20 digits).
JavaScript code:
var hash = {}
function f(k,soFar,count){
if (k == 0){
return 1;
}
var key = [k,soFar].join(",");
if (hash[key]){
return hash[key];
}
var res = 0;
for (var i=Math.max(count==0?1:0,7*(k+count)+1-soFar-9*(k-1)); i<=9; i++){
res += f(k-1,soFar+i,count+1);
}
return hash[key] = res;
}
// Output:
console.log(f(3,0,0)); // 56
hash = {};
console.log(f(6,0,0)); // 12313
hash = {};
console.log(f(20,0,0)); // 2224550892070475
You can indeed use strings to get the number of digits and then add the values of the individual digits to see if their sum > 7 * length, as Jeutnarg seems to do. I took his code and added my own, simple isHeavyRV(int):
private boolean isHeavyRV(int i)
{
int sum = 0, count = 0;
while (i > 0)
{
sum += i % 10;
count++;
i = i / 10;
}
return sum >= count * 7;
}
Now, instead of
if(isHeavy(i))
I tried
if(isHeavyRV(i))
I actually first tested his implementation of isHeavy(), using strings, and that ran in 12388 milliseconds on my machine (an older iMac), and it found 843453 heavy numbers.
Using my implementation, I found exactly the same number of heavy numbers, but in a time of a mere 5416 milliseconds.
Strings may be fast, but they can't beat a simple loop doing basically what Integer.toString(i, 10) does as well, but without the string detour.
When you add 1 to a number, you are incrementing one digit, and changing all the smaller digits to zero. If incrementing changes from a heavy to a non-heavy number, its because too many low-order digits were zeroed. In this case, it's pretty easy to find the next heavy number without checking all the numbers in between:
public class CountHeavy
{
public static void main(String[] args)
{
long startTime = System.currentTimeMillis();
int numHeavy = countHeavy(1, 100000000);
long endTime = System.currentTimeMillis();
System.out.printf("Found %d heavy numbers between 1 and 100000000\n", numHeavy);
System.out.println("Time elapsed: "+(endTime- startTime)+" ms");
}
static int countHeavy(int from, int to)
{
int numdigits=1;
int maxatdigits=9;
int numFound = 0;
if (from<1)
{
from=1;
}
for(int i = from; i < to;)
{
//keep track of number of digits in i
while (i > maxatdigits)
{
long newmax = 10L*maxatdigits+9;
maxatdigits = (int)Math.min(Integer.MAX_VALUE, newmax);
++numdigits;
}
//get sum of digits
int digitsum=0;
for(int digits=i;digits>0;digits/=10)
{
digitsum+=(digits%10);
}
//calculate a step size that increments the first non-zero digit
int step=1;
int stepzeros=0;
while(step <= (Integer.MAX_VALUE/10) && to-i >= step*10 && i%(step*10) == 0)
{
step*=10;
stepzeros+=1;
}
//step is a 1 followed stepzeros zeros
//how much is our sum too small by?
int need = numdigits*7+1 - digitsum;
if (need <= 0)
{
//already have enough. All the numbers between i and i+step are heavy
numFound+=step;
}
else if (need <= stepzeros*9)
{
//increment to the smallest possible heavy number. This puts all the
//needed sum in the lowest-order digits
step = need%9;
for(;need >= 9;need-=9)
{
step = step*10+9;
}
}
//else there are no heavy numbers between i and i+step
i+=step;
}
return numFound;
}
}
Found 569484 heavy numbers between 1 and 100000000
Time elapsed: 31 ms
Note that the answer is different from #JeutNarg's, because you asked for average > 7, not average >= 7.
I was given a Codility test lately and I was wondering how can I negate -2 base numbers?
For example the array [1,0,0,1,1] represents 9 in base -2:
-2 bases:
1,-2,4,-8,16
1 + (-8) + 16 = 9
[1,0,0,1,1]
Negative 9 in base -2 is:
-2 bases:
1,-2,4,-8
1 + (-2) + -8 = -9
[1,1,0,1]
I'm in the dark regarding the question. There must be some intuitive solution for this. Do you have any hints?
In base −2, a 1 at position i means (−2)i.
So, a [1,1] in positions [i,i+1] means (−2)i + (−2)i+1 = (−2)i + (−2)(−2)i = (1 + −2)(−2)i = −(−2)i.
So you can negate any occurrence of a [1,0] by changing it to a [1,1], and vice versa.
Any other occurrences of 0, of course, can be left intact: −0 = 0.
So in your example, we split [1,0,0,1,1] into [{1,0}, {0}, {1,1}], negate each part to get [{1,1}, {0}, {1,0}], i.e., [1,1,0,1,0], and remove the unnecessary high 0, producing [1,1,0,1].
Let's try a few examples:
(16 -8 4 -2 1)
1 = 0 0 0 0 1
-1 = 0 0 0 1 1
2 = 0 0 1 1 0
-2 = 0 0 0 1 0
3 = 0 0 1 1 1
-3 = 0 1 1 0 1
4 = 0 0 1 0 0
-4 = 0 1 1 0 0
5 = 0 0 1 0 1
-5 = 0 1 1 1 1
We can try to define this mathematically:
Given input I(b) (where B is the bit number),
I = ∑(-2)bI(b) -- definition of base -2)
O = -I -- what we're trying to solve for
O = -∑(-2)bI(b) -- substitution
O = ∑-(-2)bI(b) -- distribution
-(-2)b = (-2)b + (-2)b+1
O = ∑((-2)b + (-2)b+1)I(b) -- substitution
O = ∑((-2)bI(b) + (-2)b+1I(b)) -- substitution
O = ∑(-2)bI(b) + ∑(-2)b+1I(b)
O(b) = I(b) + I(b-1)
Now, this leaves the possibility that O(b) is 0, 1, or 2, since I(b) is always 0 or 1.
If O(b) is a 2, that is a "carry", Let's look at a few examples of carries:
(16 -8 4 -2 1) (16 -8 4 -2 1)
1+1 = 0 0 0 0 2 = 0 0 1 1 0
-2-2 = 0 0 0 2 0 = 0 1 1 0 0
4+4 = 0 0 2 0 0 = 1 1 0 0 0
for each b, starting at 0, if O(b) >= 2, subtract 2 from O(b) and increment O(b+1) and O(b+2). Do this until you reach your maximum B.
Hopefully this explains it in enough detail.
Imagine you have a number A. Then -A = A - 2*A
Or -A = A + (-2)*A. Luckily you have base -2. And (-2)*A is equivalent to left shift by one digit. All you need now is just to implement A << 1 + A. Array shifting is easy. And then you need to implement binary addition with one small difference: each time you carry over a bit you need to multiply it by -1.
public int[] solution(int[] input)
{
var A = new int[input.Length + 1];
var B = new int[input.Length + 1];
input.CopyTo(B, 1);
input.CopyTo(A, 0);
return GetResult(A, B).ToArray();
}
public IEnumerable<int> GetResult(int[] A, int[] B)
{
var r = 0;
for (int i = 0; i < A.Length; i++)
{
var currentSum = A[i] + B[i] + r;
r = -currentSum / 2;
yield return currentSum % 2;
}
}
Sorry, but the example is in C#
I'm trying to understand the bitwise and the shift operators. I wrote a simple code to show me the bits in a short type.
class Shift {
public static void main (String args[]) {
short b = 16384;
for (int t = 32768; t > 0; t = t / 2) {
if ((b&t) != 0) System.out.print("1 ");
else System.out.print ("0 ");
}
System.out.println();
b = (short)(b + 2);
for (long t = 2147483648L; t > 0; t = t / 2) {
if ((b&t) != 0) System.out.print ("1 ");
else System.out.print ("0 ");
}
System.out.println();
}
}
And the output is:
C:\>java Shift
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
I used with the second for an AND with a short (16 bits) and a long (64 bits) and the output is 32-bits.
I don't understand why the output of the second for is 32-bits.
Thank you.
You start your loop with long t = 2147483648L, which is 2^31. Therefore your loop has 32 iterations and prints 32 bits.
If you wish to display more bits, start the loop with long t = 0x4000000000000000L; (which is equivalent to the binary number starting with 01 and ending with 62 0s).