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Algorithm to generate an array with n length and k number of inversions in O(n log n) time?

I'm writing an algorithm that will return an array with determined length and number of inversions (number pairs, where the left side number is larger than the right side number). I.e. array [3, 1, 4, 2] contains three inversions (3, 1), (3, 2) and (4, 2). So in practice, when given the length of n=3 and number of inversions k=3, the algorithm should generate an array [3, 1, 4, 2] (or another array that fulfills these requirements).
Since the number of inversions is also the number of swaps that has to be made for the array to be sorted in ascending order, I approached this problem by creating an array from 1 to n - 1 and using an insertion sort algorithm in reverse to make k swaps.
This approach works just fine for smaller inputs, but the algorithm should be able to efficiently generate arrays up to n=10^6 and k=n(n-1)/2 and anything in between, so the algorithm should be working in O(n log n) time instead of O(n^2). Below is the code:
import java.util.*;
public class Inversions {
public int[] generate(int n, long k) {
// Don't mind these special cases
if (n == 1) {
int[] arr = {1};
return arr;
}
if (k == 0) {
int[] arr = new int[n];
for (int i = 0; i < n; i++) {
arr[i] = 1;
}
return arr;
}
int[] arr = new int[n];
for (int i = 0; i < n; i++) {
arr[i] = i + 1;
}
int inversions = 0;
int i = 0;
while (inversions < k && i < n) {
int j = i - 1;
while (j >= 0 && arr[j] < arr[j + 1] && inversions < k) {
int helper = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = helper;
inversions++;
j--;
}
i++;
}
return arr;
}
}
And the main class for testing with different input arrays:
public class Main {
public static void main(String[] args) {
Inversions in = new Inversions();
int[] arr1 = in.generate(4,3);
int[] arr2 = in.generate(4,0);
int[] arr3 = in.generate(4,6);
System.out.println(Arrays.toString(arr1)); // [3,1,4,2]
System.out.println(Arrays.toString(arr2)); // [1,1,1,1]
System.out.println(Arrays.toString(arr3)); // [4,3,2,1]
}
}
The algorithm does not return exactly the same arrays as the sample results, but passes all the tests, except the ones where the input size is very large. I have also tried different variations with merge sort, since it's working in O(n log n time) but with no avail.
It would be great if you guys have some ideas. If you are not familiar with Java, doesn't matter, pseudocode or any other kinds of suggestions are more than welcome!

If you reverse the initial m elements in the array, you create m(m-1)/2 inversions.
If you reverse the initial m+1 elements, you create m(m+1)/2 inversions.
The difference between these is only m.
So:
Generate a sorted array
Find the largest m such that m(m-1)/2 <= k
Reverse the first m elements in the array to create m(m-1)/2 inversions
Shift the next element forward k - m(m-1)/2 positions to create the remaining required inversions.
This takes O(n) time, which is better than you require.

Another O(n) algorithm: Start with a sorted array. When you swap the first and last elements, you get x = 2 * (n-2) + 1 inversions. Consider these two elements fixed and work on the remaining array only. If x is too large, consider a smaller array. Repeat this as long as needed.
Untested code:
for (int first=0, last = n-1; remainingInversions>0; ) {
int x = 2 * (last-first-1) + 1;
if (x <= remainingInversion) {
first++;
last--;
remainingInversion -= x;
} else {
last--; // consider a smaller array
}
}

If k >= n - 1, put element n - 1 first in the array, so that it is inverted with n - 1 elements; otherwise put it last in the array, so that it is inverted with 0 elements. Continue this greedy approach to determine where the rest of the elements go.
Here's a solution that implements generate() to run in linear time with a little bit of math.
public class Inversions {
public static int[] generate(int n, long k) {
int[] array = new int[n];
// locate k in various sums of (n-1), (n-2), ..., 1
int a = (int) Math.sqrt((n * (n - 1) - 2 * k)); // between the sum of [(n-1)+...+(n-a)] and the sum of [(n-1)+...+(n-a-1)]
int b = n - 1 - a; // counts of (n-1), (n-2), ..., (n-a)
int c = (int) (k - n * b + (b * b + b) / 2); // spillover = k - [(n-1)+(n-b)]*b/2;
// put elements in the array
for (int i = 0; i < b; i++) {
array[i] = n - 1 - i;
}
for (int i = b; i < n - 1 - c; i++) {
array[i] = i - b;
}
array[n - 1 - c] = n - 1 - b;
for (int i = n - c; i < n; i++) {
array[i] = i - b - 1;
}
return array;
}
public static void main(String[] args) {
int n = Integer.parseInt(args[0]);
long k = Long.parseLong(args[1]);
for (int i = 0; i < n; i++) {
StdOut.print(generate(n, k)[i] + " ");
}
}
}

In fact, every time you exchange the last element with the one before it, the number of inversions increments. Here is a java solution:
public static int[] generate(int n, long k) {
int[] arr = new int[n];
for(int i = 0; i < n; i++) {
arr[i] = i;
}
long inversions = 0;
int j = (n-1);
int s = 0;
while(inversions < k) {
int temp = arr[j];
arr[j] = arr[j-1];
arr[j-1] = temp;
inversions++;
j--;
if(j == s) {
j = (n-1);
s++;
}
}
return arr;
}

I got an implementation in Python with O(n) complexity.
It is based on two rules.
Reversing an array of size m gives m*(m-1)/2 inversions.
Shifting an element by m positions, creates m inversions.
def get_m(k):
m=0
while m*(m-1)/2<=k:
m+=1
else:
m-=1
return m
def generate(l, k):
"""
Generate array of length l with k inversions.
"""
# Generate a sorted array of length l
arr = list(range(0,l))
# If no inversions are needed, return sorted array.
if k==0:
return arr
# Find largest m such that m*(m-1)/2 <= k
m=get_m(k)
# Reverse first m elements in the array which will give m*(m-1)/2 inversions
arr = arr[m-1::-1]+arr[m:]
# Calculate for any remaining inversions
remaining_k = k-(m*(m-1)/2)
# For remaining inversions, move the last element to its left by remaining_k
if remaining_k>0:
arr.insert(int(len(arr)-remaining_k - 1), arr[-1])
arr = arr[:-1]
return arr
if __name__ == '__main__':
l = int(sys.argv[1])
k = int(sys.argv[2])
arr = generate(l, k)
print(arr)

There's a very easy way to create n inversions...
That is to move the last element to the front.
It's not exactly efficient due to the additional memory used, but I would do something like this:
Create an array that is twice the length n.
Fill it from the start to the middle with a sentinel (i.e. null) if we use an Integer[] instead of int[].
Fill it from the middle, ascending.
Then do something like the below...
I'm sure I have off by one errors and other bugs but the general idea is captured in the below code.
int start = 0;
int mid = arr.length / 2;
int end = arr.length - 1;
while (v > 0)
{
if (v < (end - mid))
{
arr[start++] = arr[mid + v];
arr[mid + v] = null;
}
else
{
arr[start++] = arr[end];
v -= (end - mid);
end--;
}
}
So we have an array filled with the starting values, a bunch of nulls, then the original incremental values, with one that may have become null, and an "end" pointer that points to the middle of the original zone.
So the final step is to copy from 0 -> endPos, ignoring the nulls, to the final array.

The logic is not much difficult. For example, we have 10 numbers [0,1,2,3,4,5,6,7,8,9] say, to generate like 18 inversions. Firstly, insert 9 before 0, --->[9,0,1,2,3,4,5,6,7,8], which generates 9 inversions. Still 9 inversions left, so we insert 8 before 0, ---->[9,8,0,1,2,3,4,5,6,7], so we get additional 8 inversions. Finally, 1 inversions left, we insert 7 before 6----->[9,8,0,1,2,3,4,5,7,6]. I only use arrays in this case. This program works in O(n) complexity. The following code only considering n numbers (0,1,2.....n-1) and their inversions.
public static int[] generate(int n, long k) {
int[] a = new int[n];
int[] b = new int[n];
for (int i = 1; i < n; i++) {
a[i] = 1 + a[i - 1];
}
if (n == 0 || k == 0) return a;
else {
int i = 0;
while (k > 0) {
if (k > n - i - 1) {
b[i] = a[n - 1 - i];
}
else {
//auxilary array c to store value
int[] c = new int[(int) (k + 1)];
for (int j = i; j < n - 1 - k; j++) {
b[j] = j - i;
}
for (int j = (int) (n - 1 - k); j < n; j++) {
c[j - (int) (n - 1 - k)] = j - i;
}
b[(int) (n - 1 - k)] = c[(int) k];
for (int j = (int) (n - k); j < n; j++) {
b[j] = c[j - (int) (n - k)];
}
break;
}
k = k - (n - 1 - i);
i++;
}
return b;
}
}

#zhong yang: It works nicely in the expected range 0 <= k <= n(n-1)/2 but it should be better to throw either an exception or null if k is out of this range instead of returning some array!

Dynamic Programming (Codility Q: NumberSolitaire)

This is the question:
codility.com/programmers/task/number_solitaire
and below link is my result (50% from Codility):
https://codility.com/demo/results/training8AMJZH-RTA/
My code (at the first, I tried to solve this problem using Kadane's Algo):
class Solution {
public int solution(int[] A) {
int temp_max = Integer.MIN_VALUE;
int max = 0;
int k = 1;
if(A.length == 2) return A[0] + A[A.length-1];
for(int i = 1; i < A.length-1; i++) {
if(temp_max < A[i]) temp_max = A[i];
if(A[i] > 0) {
max += A[i];
temp_max = Integer.MIN_VALUE;
k = 0;
} else if(k % 6 == 0) {
max += temp_max;
temp_max = Integer.MIN_VALUE;
k = 0;
}
k++;
}
return A[0] + max + A[A.length-1];
}
And below is the solution (100% from Codility result) that I found from web:
class Solution {
public int solution(int[] A) {
int[] store = new int[A.length];
store[0] = A[0];
for (int i = 1; i < A.length; i++) {
store[i] = store[i-1];
for (int minus = 2; minus <= 6; minus++) {
if (i >= minus) {
store[i] = Math.max(store[i], store[i - minus]);
} else {
break;
}
}
store[i] += A[i];
}
return store[A.length - 1];
}
}
I have no idea what is the problem with my code:(
I tried several test cases but, nothing different with the solution & my code
but, codility test result shows mine is not perfectly correct.
(https://codility.com/demo/results/training8AMJZH-RTA/)
please anyone explain me the problem with my code~~

Try this test case[-1, -2, -3, -4, -3, -8, -5, -2, -3, -4, -5, -6, -1].
you solution return -4 (A[0],A[1],A[length-1],Here is the mistake), but the correct answer is -6 (A[0],A[6],A[length-1]).
Here is a my solution,easy to understand:
public int solution(int[] A) {
int lens = A.length;
int dp[] = new int[6];
for (int i = 0; i < 6; i++) {
dp[i] = A[0];
}
for (int i = 1; i < lens; i++) {
dp[i%6] = getMax6(dp) + A[i];
}
return dp[(lens-1)%6];
}
private int getMax6(int dp[]){
int max = dp[0];
for (int i = 1; i < dp.length; i++) {
max = Math.max(max, dp[i]);
}
return max;
}

Readable solution from Java:
public class Solution {
public static void main(String[] args) {
System.out.println(new Solution().solution(new int[]{1, -2, 0, 9, -1, -2}));
}
private int solution(int[] A) {
int N = A.length;
int[] dp = new int[N];
dp[0] = A[0];
for (int i = 1; i < N; i++) {
double sm = Double.NEGATIVE_INFINITY;
for (int j = 1; j <= 6; j++) {
if (i - j < 0) {
break;
}
double s1 = dp[i - j] + A[i];
sm = Double.max(s1, sm);
}
dp[i] = (int) sm;
}
return dp[N-1];
}
}

Here is a solution similar to #0xAliHn but using less memory. You only need to remember the last 6 moves.
def NumberSolitaire(A):
dp = [0] * 6
dp[-1] = A[0]
for i in range(1, len(A)):
maxVal = -100001
for k in range(1, 7):
if i-k >= 0:
maxVal = max(maxVal, dp[-k] + A[i])
dp.append(maxVal)
dp.pop(0)
return dp[-1]

Based on the solutions posted, I made nice readable code. Not best performance.
int[] mark = new int[A.length];
mark[0] = A[0];
IntStream.range(1, A.length)
.forEach(i -> {
int max = Integer.MIN_VALUE;
mark[i] = IntStream.rangeClosed(1, 6)
.filter(die -> i - die >= 0)
.map(r -> Math.max(mark[i - r] + A[i], max))
.max().orElse(max);
});
return mark[A.length - 1];

Because you are not using dynamic programming, you are using greedy algorithm. Your code will fail when the max number in a range will not be the right choice.
function solution(A) {
// This array contains a maximal value at any index.
const maxTill = [A[0]];
// It's a dynamic programming so we will choose maximal value at each
// Index untill we reach last index (goal)
for (let i = 1; i < A.length; i++) {
// Step 1 . max value of each index will be atleast equal to or greater than
// max value of last index.
maxTill[i] = maxTill[i - 1];
// For each index we are finding the max of last 6 array value
// And storing it into Max value.
for (let dice = 1; dice <= 6; dice++) {
// If array index is itself less than backtrack index
// break as you dont have 6 boxes in your left
if (dice > i) {
break;
} else {
// The most important line .
// Basically checking the max of last 6 boxes using a loop.
maxTill[i] = Math.max(
maxTill[i - dice],
maxTill[i]
);
}
}
// Until this point maxStill contains the maximal value
// to reach to that index.
// To end the game we need to touch that index as well, so
// add the value of the index as well.
maxTill[i] += A[i];
}
return maxTill[A.length - 1];
}
console.log(solution([-1, -2, -3, -4, -3, -8, -5, -2, -3, -4, -5, -6, -1]));

This is my solution. I try to make the code easy to apprehend. It might not save space as much as it can.
private static int solution(int A[])
{
// N // N is an integer within the range [2..100,000];
// A[] // each element of array A is an integer within the range [−10,000..10,000].
int N = A.length;
int[] bestResult = new int[N]; // record the current bestResult
Arrays.fill(bestResult, Integer.MIN_VALUE); // fill in with the smallest integer value
// initialize
bestResult[0] = A[0];
for (int i = 0;i < A.length;i++) {
// calculate six possible results every round
for (int j = i + 1; (j < A.length) && (i < A.length) && j < (i + 1) + 6; j++) {
// compare
int preMaxResult = bestResult[j]; // the max number so far
int nowMaxResult = bestResult[i] + A[j]; // the max number at bestResult[i] + A[j]
bestResult[j] = Math.max(preMaxResult, nowMaxResult);
}
}
return bestResult[bestResult.length-1];
}

Here is the simple Python 3 solution:
import sys
def solution(A):
dp = [0] * len(A)
dp[0] = A[0]
for i in range(1, len(A)):
maxVal = -sys.maxsize - 1
for k in range(1, 7):
if i-k >= 0:
maxVal = max(maxVal, dp[i-k] + A[i])
dp[i] = maxVal
return dp[len(A)-1]

100% c++ solution(
results)
#include <climits>
int solution(vector<int>& A) {
const int N = A.size();
if (N == 2)
return A[0] + A[1];
vector<int> MaxSum(N, INT_MIN);
MaxSum[0] = A[0];
for (int i = 1; i < N; i++) {
for (int dice = 1; dice <= 6; dice++) {
if (dice > i)
break;
MaxSum[i] = max(MaxSum[i], A[i] + MaxSum[i - dice]);
}
}
return MaxSum[N-1];
}

100% python solution
with the help of the answers above and https://sapy.medium.com/cracking-the-coding-interview-30eb419c4c57
def solution(A):
# write your code in Python 3.6
# initialize maxUntil [0]*n
n = len(A)
maxUntil = [0 for i in range(n)]
maxUntil[0]=A[0]
# fill in maxUntil, remember to chack limits
for i in range(1, n): # for each
maxUntil[i] = maxUntil [i-1]
# check the max 6 to the left:
# for 1,..,6:
for dice in range(1,7):
if dice > i: # if dice bigger than loc - we are out of range
break
#else: check if bigger than cur elem, if so - update elem
maxUntil[i] = max(maxUntil[i],maxUntil[i-dice])
# add the current jump:
maxUntil[i] +=A[i]
# must reach the last sq:
return maxUntil[n-1]

I would like to explain the algorithm I have come up with and then show you the implementation in C++.
Create a hash for the max sums. We only need to store the elements within reach, so the last 6 elements. This is because the dice can only go back so much.
Initialise the hash with the first element in the array for simplicity since the first element in this hash equals to the first element of the inputs.
Then go through the input elements from the second element.
For each iteration, find the maximum values from the last 6 indices. Add the current value to that to get the current max sum.
When we reach the end of the inputs, exit the loop.
Return the max sum of the last element calculated. For this, we need clipping with module due to the space optimisation
The runtime complexity of this dynamic programming solution is O(N) since we go through element in the inputs. If we consider the dice range K, then this would be O(N * K).
The space complexity is O(1) because we have a hash for the last six elements. It is O(K) if we does not consider the number of dice faces constant, but K.
int solution(vector<int> &A)
{
vector<int> max_sums(6, A[0]);
for (size_t i = 1; i < A.size(); ++i) max_sums[i % max_sums.size()] = *max_element(max_sums.cbegin(), max_sums.cend()) + A[i];
return max_sums[(A.size() - 1) % max_sums.size()];
}

Here's my answer which gives 100% for Kotlin
val pr = IntArray(A.size) { Int.MIN_VALUE }
pr[0] = A.first()
for ((index, value) in pr.withIndex()) {
for (i in index + 1..min(index + 6, A.lastIndex)) {
pr[i] = max(value + A[i], pr[i])
}
}
return pr.last()
I used forwarded prediction, where I fill the next 6 items of the max value the current index can give.

how to find the sum of two elements in an array closest to zero

How to find two elements from an array whose sum is closest to zero but not zero(note: -1 is closest to zero than +2).I tried this...
int a[]=new int[n];
int b[]=new int[2];
int prev=0;
for(int i=0;i<n;a[i++]=in.nextInt());
for(int i=0;i<a.length;i++){
for(int j=i+1;j<a.length;j++){
int sum=a[i]+a[j];
if(prev==0)
prev=sum;
if(sum > 0){
if(sum < prev ){
prev=sum;
b[0]=a[i];
b[1]=a[j];
}
}
else if(sum < 0){
if(-sum < prev){
prev=sum;
b[0]=a[i];
b[1]=a[j];
}
}
}
}
Sop(b[0]+" "+b[1]);

I have a few remarks, you are using 3 for loops, which can be improved to just 2 nested for loops (the outer loop for selecting the current element and the inner loop to compare with the other elements).
Also you have multiple if tests to check if the sum is now closer to zero then the previous sum. However these if tests can be reduced to just one if test, by taking the absolute value of the sum instead of testing on sum > 0 and sum < 0, which is fine for the readability.
This is what i came up with :
int array[] = new int[5];
array[0] = -3; array[1] = -2; array[2] = -1; array[3] = 1; array[4] = 2; // Fill array
int idx[] = new int[2]; // Will store the result (index of the two elements that need to be added)
double lowest_sum = Double.POSITIVE_INFINITY; // Of type double to be able to use infinity
for(int i = 0; i < array.length; i++) {
// Outer loop --> Uses a current (array[i]) from left to right
int current = array[i];
for(int j = i+1; j < array.length; j++) {
// Inner loop --> Check all elements we didn't used as current till now
int compare_with = array[j];
if((Math.abs(current + compare_with) < lowest_sum) && ((current + compare_with) != 0)) {
// We found two elements whose sum is closer to zero
lowest_sum = Math.abs(current + compare_with);
idx[0] = i; // Index of the first element to add
idx[1] = j; // Index of second element to add
}
}
}
int res_idx1 = idx[0];
int res_idx2 = idx[1];
System.out.println("The first element to add is : " + array[res_idx1] + "\nThe second element to add is : " + array[res_idx2]);
Input : array = [-3, -2, -1, 1, 2] , Output : The first element to add is : -3,
The second element to add is : 2
Note that this code will print a solution and not all solutions (if multiple solutions exists). It should be fairly trivial to edit the code such that it returns all solutions.

you can try:
int a[]=new int[n];
int b[]=new int[2];
int prev=0;
for(int i=0;i<n;a[i++]=in.nextInt());
for(int i=0;i<a.length;i++){
for(int j=i+1;j<a.length;j++){
int sum=a[i]+a[j];
if(prev==0)
prev=sum;
if(Math.abs(sum)>0 && Math.abs(sum)<Math.abs(prev)){
prev=sum;
b[0]=a[i];
b[1]=a[j];
}
}
}
Sop(b[0]+" "+b[1]);

This problem can be solved in O(N*log(N)). The most expensive operation in this case will be sorting your array. If your domain allows you to use non-comparative sorts, such as counting sort then you'll be able to reduce time complexity of the whole solution to linear time.
The idea is that in sorted array, you can iterate elements in ascending and descending order in parallel and thus find all pairs with minimal/maximal sum in linear time. The only disadvantage of such approach in application to your task is that you need to find minimal absolute value of the sum, that means finding minimum among positive sums and maximum among negative sums. This will require two linear passes.
My solution is below. It is verified on randomized data against the bruteforce O(N^2) solution.
// note: mutates argument!
static Solution solve(int a[]) {
Arrays.sort(a);
int i = 0;
int j = a.length - 1;
// -1 indicates uninitialized min value
int minI = -1;
int minJ = -1;
int min = 0;
// finding maximal sum among negative sums
while (i < j) {
int cur = a[i] + a[j];
if (cur != 0 && (minI == -1 || Math.abs(cur) < Math.abs(min))) {
min = cur;
minI = i;
minJ = j;
}
// if current sum is below zero, increase it
// by trying the next, larger element
if (cur < 0) {
i++;
} else { // sum is already non-negative, move to the next element
j --;
}
}
i = 0;
j = a.length - 1;
// finding minimal sum among positive sums
while (i < j) {
int cur = a[i] + a[j];
if (cur != 0 && (minI == -1 || Math.abs(cur) < Math.abs(min))) {
min = cur;
minI = i;
minJ = j;
}
if (cur > 0) {
j--;
} else {
i ++;
}
}
if (minI >=0) {
return new Solution(minI, minJ, min);
//System.out.printf("a[%d]=%d, a[%d]=%d, sum=%d", minI, minJ, a[minI], a[minJ], min);
} else {
return null;
//System.out.println("No solution");
}
}
I just realized that sorting messes the indices, so minI and minJ will not correspond to the indices in the original non-sorted array. The fix is simple — original array should be converted to the array of pairs (value, original_index) before sort. Though I will not implement this fix in my example snippet, as it will further affect readability.

Count and print Triplets in a sorted array

Given a sorted array of n integers, display triplets such that a[i] < a[j] < a[k].
My code is
public static void countTriplets(int arr[], int index, int arr1[], int position)
{
if (position == 3)
{
System.out.println(Arrays.toString(arr1));
return;
}
for (int i = index; i < arr.length; i++)
{
arr1[position] = arr[i];
countTriplets(arr, index + 1, arr1, position + 1);
}
}
However it prints all possible triplets.Where am i going wrong ?

Count the number of unique elements in the array. Let it be 'N'. Then the answer is n * (n - 1) * (n - 2) / 6.
The reasoning is as follows: for any three distinct numbers a, b, c, we can form a tuple of sorted elements such that say b < c < a. Since we don't want repetitions, we have to count the number of unique elements.
For example, consider {1, 2, 3, 3, 4, 5, 5, 6}
Number of unique elements = 6. The answer is (6 * 5 * 4) / 6 = 20.
Some code in C++:
#include <stdio.h>
int count_triplets(int *a, int n)
{
int counter = 0;
if (n < 3) {
return 0;
}
for (int i = 0; i < n; i++) {
// jump to the last of the repeated values
if ((i < n - 1) && (a[i] == a[i + 1])) {
continue;
}
for (int j = i + 1; j < n; j++) {
// jump to the last of the repeated values
if ((j < n - 1) && (a[j] == a[j + 1])) {
continue;
}
for (int k = j + 1; k < n; k++) {
// jump to the last of the repeated values
if ((k < n - 1) && (a[k] == a[k + 1])) {
continue;
}
printf("[%d, %d, %d]\n", a[i], a[j], a[k]);
counter ++;
}
}
}
return counter;
}
int main(int argc, char *argv[])
{
printf("Enter the number of elements:");
int n = 0;
scanf("%d", &n);
printf("Enter the elements:\n");
int a[100] = {0};
for (int i = 0; i < n; i++) {
scanf("%d", &a[i]);
}
int triplets = count_triplets(a, n);
printf("Number of triplets = [%d]\n", triplets);
return 0;
}
This is not the most efficient but should lead you to more efficient solutions.

The simple way to do this is with nested loops:
for (int i = 0; i < arr.length-2; i++)
{
for (int j = i+1; j < arr.length-1; j++)
{
for (int k = j+1; k < arr.length; k++)
{
// output the triplet arr[i], arr[j], arr[k]
++numTriplets;
}
}
}
The code above will do what you're asking. It does not take into account the possibility of duplicates in the source array. Given the array [1, 2, 3, 4, 5], it outputs:
1,2,3
1,2,4
1,2,5
1,3,4
1,3,5
1,4,5
2,3,4
2,3,5
2,4,5
3,4,5
Update
The general solution to this problem is one of creating combinations. That is, selecting combinations of n items from a list of size m. The math tells us that the number of combinations is equal to:
m!
---------
(n!)(m-n)!
Substituting numbers for your example, we have:
c = 5!/((3!) * (5-3)!)
= 120/(6 * 2)
= 120/12
= 10
So you can compute the number of combinations in O(1) easily enough (if you use an approximation for the factorial function), but if you want to enumerate them your time complexity approaches O(m!) (for sufficiently large values of m).
You certainly can't enumerate all the combinations in O(n) or O(n log n). That would be kind of like asking for an algorithm that can enumerate all n-digit numbers in O(n) or O(n log n).

the answer can be reduced to selection of three numbers from n numbers which is nc3 i.e
return (n*(n-1)*(n-2))/3! where n>=3 else return 0

Count the Number of Swaps in Selection Sort

I've read through all the discussions trying to find an answer but none of the answers have worked for me so I'm trying it this way.
public static int SelectionSort(long[] num)
{
int i, j, first;
long temp;
int swap = 0;
int pass = 0;
int count = 0;
boolean Mini = false;
for (i = num.length - 1; i > 0; i--)
{
for(int k = 0; k < num.length; k++)
{
System.out.println(" k = " + k
+ " \t X[i] = " + num[k] + " swap count: " + swap);
}
System.out.println("");
first = 0; //initialize to subscript of first element
for(j = 1; j <= i; j ++) //locate smallest element between positions 1 and i.
{
if(num[j] < num[first])
{
first = j;
//Mini = true;
}
}
//if(Mini){
// swap++;
//}
temp = num[first]; //swap smallest found with element in position i.
num[first] = num[i];
num[i] = temp;
}
return swap;
}
Using a simple array as my test case:
long[] X = {1, 4, 3, 2, 5};
The number of swaps should only equate to 1 because it's swapping the first and last elements only. However, it isn't working. While I know my if condition doesn't work, I can't think of what would. I can't seem to work the logic that it increments a swap when items are actually swapped.

Why not increment the counter when you actually perform a swap?
//swap smallest found with element in position i.
swap++
temp = num[first];
num[first] = num[i];
num[i] = temp;
EDIT:
Good point in the comment. The current code still performs a superfluous swap if the array is sorted (i.e., first and i are the same):
if (first != i) {
swap++
temp = num[first];
num[first] = num[i];
num[i] = temp;
}

With your implementation you will always have n swaps (where n is the number of elements in your array).
What I think you want is to only perform a swap when it actually makes a difference ... so when "first" and "i" have different values. Otherwise you switch the element with itself.
if (first != i) {
temp = num[first];
num[first] = num[i];
num[i] = temp;
swapp++;
}