I am doing a question where, given an n x n 2D matrix representing an image, rotate the image by 90 degrees (clockwise).You have to rotate the image in-place, which means you have to modify the input 2D matrix directly. DO NOT allocate another 2D matrix and do the rotation. This my my code:
class Solution {
public void rotate(int[][] matrix) {
int size = matrix.length;
for(int i = 0 ; i < matrix.length; i++){
for(int y = 0 ; y < matrix[0].length ; y++){
matrix[i][y] = matrix[size - y - 1][i];
System.out.println(size - y - 1);
System.out.println(i);
System.out.println("");
}
}
}
}
This is the input and output results:
input matrix: [[1,2,3],[4,5,6],[7,8,9]]
output matrix: [[7,4,7],[8,5,4],[9,4,7]]
expected matrix: [[7,4,1],[8,5,2],[9,6,3]]
I do not really understand why I am getting duplicates in my output such as the number seven 3 times. On my System.out.println statement, I am getting the correct list of indexes :
2
0
1
0
0
0
2
1
1
1
0
1
2
2
What can be wrong?
I have found a solution. I will try my best to explain it.
Let us consider an array of size 4.
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Now lets look at the numbers present only on the outside of the array:
1 2 3 4
5 8
9 12
13 14 15 16
We will proceed by storing the first element 1 in a temporary variable. Next we will replace 1 by 13, 13 by 16, 16 by 4 and at last 4 by 1 (whose value we already stored in the temporary variable).
We will do the same for all the elements of the first row.
Here is a pseudocode if you just want to rotate this outer ring, lets call it an outer ring:
for i = 0 to n-1
{
temp = A[0][i];
A[0][i] = A[n-1-i][0];
A[n-1-i][0] = A[n-1-0][n-1-i];
A[n-1-0][n-1-i] = A[i][n-1-0];
A[i][n-1-0] = temp;
}
The code runs for a total of n times. Once for each element of first row. Implement this code an run it. You will see only the outer ring is rotated. Now lets look at the inner ring:
6 7
10 11
Now the loop in pseudocode only needs to run for 2 times and also our range of indexes has decreased. For outer ring, the loop started from i = 0 and ended at i = n-1. However, for the inner ring the for loop need to run from i = 1 to i = n-2.
If you had an array of size n, to rotate the xth ring of the array, the loop needs to run from i = x to i = n-1-x.
Here is the code to rotate the entire array:
x = 0;
int temp;
while (x < n/2)
{
for (int i = x;i < n-1-x;i++)
{
temp = arr[x][i];
arr[x][i] = arr[n-1-i][x];
arr[n-1-i][x] = arr[n-1-x][n-1-i];
arr[n-1-x][n-1-i] = arr[i][n-1-x];
arr[i][n-1-x] = temp;
}
x++;
}
Here each value of x denotes the xth ring.
0 <= x <= n-1
The reason why the outer loop runs only for x < n/2 times is because each array has n/2 rings when n is even and n/2 + 1 rings if n is odd.
I hope I have helped you. Do comment if face any problems with the solution or its explanation.
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How does the following code work for the problem Uva Live 6823?
I just want an explanation for the algorithm used in this code.
Thanks in advance.
#include <bits/stdc++.h>
using namespace std;
int main()
{
string st;
while(cin>> st){
long long int cnt=0,x=0,arr[]={1,0,0};
for(int a=0;a<st.length();a++){
if(!isdigit(st[a])){
x=0;
arr[0]=1;
arr[1]=arr[2]=0;
}
else{
x=(x+st[a]-48)%3;
cnt+=arr[x];
arr[x]++;
}
}
cout<< cnt <<endl;
}
return 0;
}
This algorithm makes use of the divisibility by 3 rule, namely, if the sum of the digits of a number are divisible by 3, then the number is divisible by 3. It keeps track of the frequency of occurrence for the remainder of the sum of the digits, divided by 3 (eg. 0, 1 or 2). Anytime a new digit is added, the substrings with the same remainder adds another permutation substring that is divisible by 3. Note: the constant used 48 refers to the ASCII value for '0'.
This is probably best illustrated by example. Consider the string 12321, being processed digit by digit.
// "iteration" #0 (st[a] = '')
x = 0
arr[] = {1,0,0}
cnt = 0
// iteration #1 (st[a]='1'). '1' is not divisible by three.
x = 1
arr[] = {1,1,0}
cnt = 0
// iteration #2 (st[a]='2'). '12' is divisible by three.
x = 0
arr[] = {2,1,0}
cnt = 1
// iteration #3 (st[a]='3'). '123' and '3' are divisible by three
x = 0
arr[] = {3,1,0}
cnt = 3
// iteration #4 (st[a]='2'). No new divisible by three substrings
x = 2
arr[] = {3,1,1}
cnt = 3
// iteration #5 (st[a]='1'). '12321', '321' and '21' are divisible by 3.
// note, the other times when x = 0, st[0..a] = '', '12', '123'
x = 0
arr[] = {4,1,1}
cnt = 6
You could continue adding digits, and see how it works. When it hits a non-digit character, the counting resets, because any non-digit character cannot participate in a permutation.
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Basic question:
for(int i=start;i<=end;i+=step){
System.out.println("Test");
}
start < end
How often is run through the loop, respectively what is the mathematical formula ?
We need to know what the values of start, end, and step are.
if:
start = 0;
end = 10;
step = 1;
It would loop 11 times, each time adding 1 to the previous value of i until it is <= 10. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
if:
start = 0;
end = 10;
step = 2;
It would loop 6 times, each time adding 2 to the previous value of i until it is <= 10. (0, 2, 4, 6, 8, 10)
if:
start = 10;
end = 100;
step = 10;
It would loop 10 times, each time adding 10 to the previous value of i until it is <= 100. (10, 20, 30, 40, 50, 60, 70, 80, 90, 100)
And so on.
Your for loop is shorthand for the following code:
int i = start;
if (i <= end) {
/* loop body */
i += step;
}
To answer your question, it will run ceiling((end - start + 1) / step) times. Walk through the logic on paper to see if you come to the same conclusion.
For a classwork problem I am doing, I am supposed to trace (check for bugs) the following algorithm (in pseudocode):
num <- 2
count <- 1
while count < 5
{
count <- count * num
if count / 2 < 2
print "Hello"
else
while count < 7
{
count <- count + 1
}
print "The count is " + count + "."
}
When i traced this code, I got
num count output
2 1 Hello The count is 1.
My question is, was my trace right? It looks like there is something else I have to add.
When you are tracing the problem, you need to note down all value changes in the program.
In your program, we have 2 variables to trace: count and num. From the program, we can figure out 2 facts:
There is no assignment of num;
All output statements are related to count.
Therefore, we should focus on tracing the changes on count.
Notice that this block:
while count < 7
{
count <- count + 1
}
can be replaced with
if count < 7
{
count = 7
}
The workflow of the program can be depicted in English like below:
Check if count is smaller than 5, YES go to 2, NO program ends;
Double count;
If count / 2 is smaller than 2, YES go to 4, NO go to 5;
Print "Hello", go to 6;
If count is smaller than 7, set count to 7;
Print "The count is +count+.`", go to 1;
Now the task is to use 1 as initial value of count and walk through the work flow until the program terminates.
Let's do it together:
count equals to 1, so go to 2;
Now count equals to 2;
count / 2 equals to 1, which is smaller than 2, so go to 4;
Hello is printed, go to 6;
"The count is 2." is printed, go to 1;
count equals to 2, so go to 2;
Now count equals to 4;
count / 2 equals to 2, which is NOT smaller than 2, so go to 5;
count is set to 7;
"The count is 7." is printed, go to 1;
count equals to 7, so program terminates.
Therefore the output will be:
HelloThe count is 2.The count is 7.
Here is how you should walk through this.
num = 2
count = 1
while 1 < 5
{
2 = 1 * 2
if 2 /2 < 2 //since 1 < 2 print Hello
print "Hello"
else //This is skipped because the if was true
while count < 7
{
count <- count + 1
}
print "The count is " + count + "." //This prints "The Count is 2
}
Then you continue through the while loop with count = 2.
Start of second iteration.
while 2 < 5
{
4 = 2 * 2
count changes each time through the loop.
I have a list of numbers and I want to add up all the different combinations.
For example:
number as 1,4,7 and 13
the output would be:
1+4=5
1+7=8
1+13=14
4+7=11
4+13=17
7+13=20
1+4+7=12
1+4+13=18
1+7+13=21
4+7+13=24
1+4+7+13=25
Is there a formula to calculate this with different numbers?
A simple way to do this is to create a bit set with as much bits as there are numbers.
In your example 4.
Then count from 0001 to 1111 and sum each number that has a 1 on the set:
Numbers 1,4,7,13:
0001 = 13=13
0010 = 7=7
0011 = 7+13 = 20
1111 = 1+4+7+13 = 25
Here's how a simple recursive solution would look like, in Java:
public static void main(String[] args)
{
f(new int[] {1,4,7,13}, 0, 0, "{");
}
static void f(int[] numbers, int index, int sum, String output)
{
if (index == numbers.length)
{
System.out.println(output + " } = " + sum);
return;
}
// include numbers[index]
f(numbers, index + 1, sum + numbers[index], output + " " + numbers[index]);
// exclude numbers[index]
f(numbers, index + 1, sum, output);
}
Output:
{ 1 4 7 13 } = 25
{ 1 4 7 } = 12
{ 1 4 13 } = 18
{ 1 4 } = 5
{ 1 7 13 } = 21
{ 1 7 } = 8
{ 1 13 } = 14
{ 1 } = 1
{ 4 7 13 } = 24
{ 4 7 } = 11
{ 4 13 } = 17
{ 4 } = 4
{ 7 13 } = 20
{ 7 } = 7
{ 13 } = 13
{ } = 0
The best-known algorithm requires exponential time. If there were a polynomial-time algorithm, then you would solve the subset sum problem, and thus the P=NP problem.
The algorithm here is to create bitvector of length that is equal to the cardinality of your set of numbers. Fix an enumeration (n_i) of your set of numbers. Then, enumerate over all possible values of the bitvector. For each enumeration (e_i) of the bitvector, compute the sum of e_i * n_i.
The intuition here is that you are representing the subsets of your set of numbers by a bitvector and generating all possible subsets of the set of numbers. When bit e_i is equal to one, n_i is in the subset, otherwise it is not.
The fourth volume of Knuth's TAOCP provides algorithms for generating all possible values of the bitvector.
C#:
I was trying to find something more elegant - but this should do the trick for now...
//Set up our array of integers
int[] items = { 1, 3, 5, 7 };
//Figure out how many bitmasks we need...
//4 bits have a maximum value of 15, so we need 15 masks.
//Calculated as:
// (2 ^ ItemCount) - 1
int len = items.Length;
int calcs = (int)Math.Pow(2, len) - 1;
//Create our array of bitmasks... each item in the array
//represents a unique combination from our items array
string[] masks = Enumerable.Range(1, calcs).Select(i => Convert.ToString(i, 2).PadLeft(len, '0')).ToArray();
//Spit out the corresponding calculation for each bitmask
foreach (string m in masks)
{
//Get the items from our array that correspond to
//the on bits in our mask
int[] incl = items.Where((c, i) => m[i] == '1').ToArray();
//Write out our mask, calculation and resulting sum
Console.WriteLine(
"[{0}] {1}={2}",
m,
String.Join("+", incl.Select(c => c.ToString()).ToArray()),
incl.Sum()
);
}
Outputs as:
[0001] 7=7
[0010] 5=5
[0011] 5+7=12
[0100] 3=3
[0101] 3+7=10
[0110] 3+5=8
[0111] 3+5+7=15
[1000] 1=1
[1001] 1+7=8
[1010] 1+5=6
[1011] 1+5+7=13
[1100] 1+3=4
[1101] 1+3+7=11
[1110] 1+3+5=9
[1111] 1+3+5+7=16
Here is a simple recursive Ruby implementation:
a = [1, 4, 7, 13]
def add(current, ary, idx, sum)
(idx...ary.length).each do |i|
add(current + [ary[i]], ary, i+1, sum + ary[i])
end
puts "#{current.join('+')} = #{sum}" if current.size > 1
end
add([], a, 0, 0)
Which prints
1+4+7+13 = 25
1+4+7 = 12
1+4+13 = 18
1+4 = 5
1+7+13 = 21
1+7 = 8
1+13 = 14
4+7+13 = 24
4+7 = 11
4+13 = 17
7+13 = 20
If you do not need to print the array at each step, the code can be made even simpler and much faster because no additional arrays are created:
def add(ary, idx, sum)
(idx...ary.length).each do |i|
add(ary, i+1, sum + ary[i])
end
puts sum
end
add(a, 0, 0)
I dont think you can have it much simpler than that.
Mathematica solution:
{#, Total##}& /# Subsets[{1, 4, 7, 13}] //MatrixForm
Output:
{} 0
{1} 1
{4} 4
{7} 7
{13} 13
{1,4} 5
{1,7} 8
{1,13} 14
{4,7} 11
{4,13} 17
{7,13} 20
{1,4,7} 12
{1,4,13} 18
{1,7,13} 21
{4,7,13} 24
{1,4,7,13} 25
This Perl program seems to do what you want. It goes through the different ways to choose n items from k items. It's easy to calculate how many combinations there are, but getting the sums of each combination means you have to add them eventually. I had a similar question on Perlmonks when I was asking How can I calculate the right combination of postage stamps?.
The Math::Combinatorics module can also handle many other cases. Even if you don't want to use it, the documentation has a lot of pointers to other information about the problem. Other people might be able to suggest the appropriate library for the language you'd like to you.
#!/usr/bin/perl
use List::Util qw(sum);
use Math::Combinatorics;
my #n = qw(1 4 7 13);
foreach my $count ( 2 .. #n ) {
my $c = Math::Combinatorics->new(
count => $count, # number to choose
data => [#n],
);
print "combinations of $count from: [" . join(" ",#n) . "]\n";
while( my #combo = $c->next_combination ){
print join( ' ', #combo ), " = ", sum( #combo ) , "\n";
}
}
You can enumerate all subsets using a bitvector.
In a for loop, go from 0 to 2 to the Nth power minus 1 (or start with 1 if you don't care about the empty set).
On each iteration, determine which bits are set. The Nth bit represents the Nth element of the set. For each set bit, dereference the appropriate element of the set and add to an accumulated value.
ETA: Because the nature of this problem involves exponential complexity, there's a practical limit to size of the set you can enumerate on. If it turns out you don't need all subsets, you can look up "n choose k" for ways of enumerating subsets of k elements.
PHP: Here's a non-recursive implementation. I'm not saying this is the most efficient way to do it (this is indeed exponential 2^N - see JasonTrue's response and comments), but it works for a small set of elements. I just wanted to write something quick to obtain results. I based the algorithm off Toon's answer.
$set = array(3, 5, 8, 13, 19);
$additions = array();
for($i = 0; $i < pow(2, count($set)); $i++){
$sum = 0;
$addends = array();
for($j = count($set)-1; $j >= 0; $j--) {
if(pow(2, $j) & $i) {
$sum += $set[$j];
$addends[] = $set[$j];
}
}
$additions[] = array($sum, $addends);
}
sort($additions);
foreach($additions as $addition){
printf("%d\t%s\n", $addition[0], implode('+', $addition[1]));
}
Which will output:
0
3 3
5 5
8 8
8 5+3
11 8+3
13 13
13 8+5
16 13+3
16 8+5+3
18 13+5
19 19
21 13+8
21 13+5+3
22 19+3
24 19+5
24 13+8+3
26 13+8+5
27 19+8
27 19+5+3
29 13+8+5+3
30 19+8+3
32 19+13
32 19+8+5
35 19+13+3
35 19+8+5+3
37 19+13+5
40 19+13+8
40 19+13+5+3
43 19+13+8+3
45 19+13+8+5
48 19+13+8+5+3
For example, a case for this could be a set of resistance bands for working out. Say you get 5 bands each having different resistances represented in pounds and you can combine bands to sum up the total resistance. The bands resistances are 3, 5, 8, 13 and 19 pounds. This set gives you 32 (2^5) possible configurations, minus the zero. In this example, the algorithm returns the data sorted by ascending total resistance favoring efficient band configurations first, and for each configuration the bands are sorted by descending resistance.
This is not the code to generate the sums, but it generates the permutations. In your case:
1; 1,4; 1,7; 4,7; 1,4,7; ...
If I have a moment over the weekend, and if it's interesting, I can modify this to come up with the sums.
It's just a fun chunk of LINQ code from Igor Ostrovsky's blog titled "7 tricks to simplify your programs with LINQ" (http://igoro.com/archive/7-tricks-to-simplify-your-programs-with-linq/).
T[] arr = …;
var subsets = from m in Enumerable.Range(0, 1 << arr.Length)
select
from i in Enumerable.Range(0, arr.Length)
where (m & (1 << i)) != 0
select arr[i];
You might be interested in checking out the GNU Scientific Library if you want to avoid maintenance costs. The actual process of summing longer sequences will become very expensive (more-so than generating a single permutation on a step basis), most architectures have SIMD/vector instructions that can provide rather impressive speed-up (I would provide examples of such implementations but I cannot post URLs yet).
Thanks Zach,
I am creating a Bank Reconciliation solution. I dropped your code into jsbin.com to do some quick testing and produced this in Javascript:
function f(numbers,ids, index, sum, output, outputid, find )
{
if (index == numbers.length){
var x ="";
if (find == sum) {
y= output + " } = " + sum + " " + outputid + " }<br/>" ;
}
return;
}
f(numbers,ids, index + 1, sum + numbers[index], output + " " + numbers[index], outputid + " " + ids[index], find);
f(numbers,ids, index + 1, sum, output, outputid,find);
}
var y;
f( [1.2,4,7,13,45,325,23,245,78,432,1,2,6],[1,2,3,4,5,6,7,8,9,10,11,12,13], 0, 0, '{','{', 24.2);
if (document.getElementById('hello')) {
document.getElementById('hello').innerHTML = y;
}
I need it to produce a list of ID's to exclude from the next matching number.
I will post back my final solution using vb.net
v=[1,2,3,4]#variables to sum
i=0
clis=[]#check list for solution excluding the variables itself
def iterate(lis,a,b):
global i
global clis
while len(b)!=0 and i<len(lis):
a=lis[i]
b=lis[i+1:]
if len(b)>1:
t=a+sum(b)
clis.append(t)
for j in b:
clis.append(a+j)
i+=1
iterate(lis,a,b)
iterate(v,0,v)
its written in python. the idea is to break the list in a single integer and a list for eg. [1,2,3,4] into 1,[2,3,4]. we append the total sum now by adding the integer and sum of remaining list.also we take each individual sum i.e 1,2;1,3;1,4. checklist shall now be [1+2+3+4,1+2,1+3,1+4] then we call the new list recursively i.e now int=2,list=[3,4]. checklist will now append [2+3+4,2+3,2+4] accordingly we append the checklist till list is empty.
set is the set of sums and list is the list of the original numbers.
Its Java.
public void subSums() {
Set<Long> resultSet = new HashSet<Long>();
for(long l: list) {
for(long s: set) {
resultSet.add(s);
resultSet.add(l + s);
}
resultSet.add(l);
set.addAll(resultSet);
resultSet.clear();
}
}
public static void main(String[] args) {
// this is an example number
long number = 245L;
int sum = 0;
if (number > 0) {
do {
int last = (int) (number % 10);
sum = (sum + last) % 9;
} while ((number /= 10) > 0);
System.err.println("s = " + (sum==0 ? 9:sum);
} else {
System.err.println("0");
}
}