Find kthsmallest element from two sorted arrays - arrays

Question

I have implemented a working function that does it. But it is not very efficient because it copies a new copy in each call. I am having trouble converting it to using only a_start, a_end, b_start, b_end. I have tried a couple of ways to convert it, but none of them are working for all cases. How can I convert it so that it takes in a start and end pointers for both array a and b?
I have tried the following and modified k-i-1 and k-j-1 so that it only takes in k, but did not work.
int m = a_right-a_left, n=b_right-b_left;
int i = (a_left+a_right)/2;
or int i = (int)((m* (k-1)) / (m+n) );
Below is my working code using a new copy of array each call.
public static int kthSmallest(int[] a, int[] b, int k) {
if (a.length==0)
return b[k-1];
else if (b.length==0)
return a[k-1];
else if (b.length<a.length)
return kthSmallest(b, a, k);
// make sure i + j = k - 1
int m = a.length, n=b.length;
int i = (int)((double)m / (m+n) * (k-1)); // make sure i won't be out of bounds
int j = k - 1 - i;
int bj_1 = 0, ai_1 = 0;
if (i==0) { ai_1 = Integer.MIN_VALUE; } // in case i = 0, outOfBound
else { ai_1 = a[i-1]; }
if (j==0) { bj_1 = Integer.MIN_VALUE; } // in case j = 0, outOfBound
else { bj_1 = b[j-1]; }
if (bj_1 < a[i] && a[i] < b[j]) // kth smallest found, b[j-1] < a[i] < b[j]
return a[i];
if (ai_1 < b[j] && b[j] < a[i]) // kth smallest found, a[i-1] < b[j] < a[i]
return b[j];
if ( a[i] < b[j] ) // if true, exclude a's lower bound (if 2 arrays merged, a's lower bound must
// reside before kth smallest, so also update k.
// also exclude b's upper bound, since they are all greater than kth element.
return kthSmallest(Arrays.copyOfRange(a, i+1, a.length), Arrays.copyOfRange(b, 0, j), k-i-1);
else
return kthSmallest(Arrays.copyOfRange(a, 0, i), Arrays.copyOfRange(b, j+1, b.length), k-j-1);
}

Answer 1

Here's O(log a.length + log b.length) algorithm from the answer to "How to find the kth smallest element in the union of two sorted arrays?" question. It is a direct port from C++ recursive implementation to Java:
public static int ksmallest(int[] a, int[] b,
int a1, int a2, int b1, int b2,
int k) {
int lena = a2 - a1;
int lenb = b2 - b1;
assert (0 <= k && k < (lena + lenb));
if (lena == 0) {
return b[b1 + k];
}
if (lenb == 0) {
return a[a1 + k];
}
int mida = lena / 2;
int midb = lenb / 2;
int ma = a[a1 + mida];
int mb = b[b1 + midb];
if ((mida + midb) < k) {
return (mb < ma) ?
ksmallest(a, b, a1, a2, b1 + midb + 1, b2, k - (midb + 1)) :
ksmallest(a, b, a1 + mida + 1, a2, b1, b2, k - (mida + 1));
}
else {
return (mb < ma) ?
ksmallest(a, b, a1, a1 + mida, b1, b2, k) :
ksmallest(a, b, a1, a2, b1, b1 + midb, k);
}
}
There is also C++ iterative implementation with the same time complexity (without recursion). It could be ported to Java the same way as the recursive version.
Sanity check for the recursive version:
/** concatenate a, b arrays */
public static int[] concatenate(int[] a, int[] b) {
int lena = a.length;
int lenb = b.length;
int[] c = new int[lena + lenb];
System.arraycopy(a, 0, c, 0, lena);
System.arraycopy(b, 0, c, lena, lenb);
return c;
}
public static void main(String[] args) {
int a[] = {0, 3, 7, 8};
int b[] = {0, 2, 3};
int c[] = concatenate(a, b);
Arrays.sort(c);
for (int n = 0; n < (a.length + b.length); n++) {
int k = ksmallest(a, b, 0, a.length, 0, b.length, n);
if (k != c[n]) {
System.out.println(n + ": expected " + c[n] + " got " + k);
}
}
}
On success, it prints nothing.

Answer 2

An algorithm with O(n) but simple to understand;
//both arrays are sorted
private int getKthSmallestElement(int[] array1, int[] array2, int k) {
int elem=-1;
int index1=0,index2=0;
while(k != 0 && (index1<array1.length) && (index2 < array2.length))
{
if(array1[index1] < array2[index2])
{
index1++;
}
else
index2++;
k--;
}
if((index1<array1.length) && (index2 < array2.length))
return array1[index1] > array2[index2] ? array2[index2] :array1[index1] ;
else
{
if(index1 >= array1.length)
{
return array2[index2+k];
}
else{
return array1[index1+k];
}
}
}

Answer 3

Here is a O((logn)^2) simple recursive solution without copying :-
public class KthElement {
public static int binSearch(int[] arr,int low,int high,int key) {
int mid=0;
while(high>=low) {
mid = (high+low)/2;
if(arr[mid]==key) {
return(mid);
}
else if(arr[mid]<key) {
low = mid+1;
}
else {
high = mid-1;
}
}
if(arr[mid]>key) {
return(mid-1);
}
return(mid);
}
public static int kthElement(int[] arr1,int[] arr2,int s1,int h1,int s2,int h2,int k) {
int len1 = (h1-s1+1);
int len2 = (h2-s2+1);
if(len1<=0) {
return(arr2[s2+k-1]);
}
if(len2<=0) {
return(arr1[s1+k-1]);
}
if(k>(len1+len2)) {
return(-1);
}
int mid = (s1+h1)/2;
int i = binSearch(arr2,s2,h2,arr1[mid]);
int size = mid+i-s1-s2+2;
//System.out.println(mid+" "+i+" "+size);
if(size==k){
return(arr1[mid]);
}
if(size>k) {
return(kthElement(arr1, arr2, s1,mid-1, s2,i, k));
}
else {
return(kthElement(arr1, arr2, mid+1,h1,i+1,h2, k-size));
}
}
public static void main(String[] args) {
int[] arr1 = {1,3,5,7,9,11,13};
int[] arr2 = {2,4,6,8,10,12};
int k = 6;
System.out.println(k+"th Element : "+kthElement(arr1, arr2,0,arr1.length-1, 0,arr2.length-1, k));
}
}

Answer 4

Here is JAVA iterative solution :
O( log A.length + log B.length ) time complexity
public static int kthsmallest(int []A, int []B, int K){
int begin = Math.max(0,K-B.length); // binary search begin index
int end = Math.min(A.length,K); // binary search end end index
while(begin < end){
// search until mid = k
int mid = begin +(end-begin)/2;
if(mid<A.length && K-mid>0 && A[mid]<B[K-mid-1]){
begin = mid+1;
}else if( mid > 0 && K-mid <B.length && A[mid-1]>B[K-mid]){
end = mid;
}else{
begin=mid;
break;
}
}
if(begin ==0){
return B[K-1];
}else if(begin == K){
return A[K-1];
}else{
return Math.max(A[begin -1],B[K-begin-1]);
}
}

Answer 5

Same logic as mentioned in the question, but changed it a little bit to not create new arrays for every call.
//alow, ahigh are used to maintain the size of a array we are using instead
//of creating a new array. imilarly for blow, bhigh
public static int search(int[] a, int[] b, int alow, int ahigh, int blow, int bhigh, int k){
if (alow>ahigh) // if "a" array is of zero length, then take kth element from "b"
return b[blow+k-1];
else if (blow>bhigh || bhigh-blow == 0) // Similarly, // if "b" array is of zero length, then take kth element from "a"
return a[alow+k-1];
else if ((bhigh-blow)<(ahigh-alow)) //always make sure that a's length is lower length
return search(b, a,blow,bhigh,alow,ahigh, k);
// make sure i + j = k - 1
int m = ahigh-alow+1, n=bhigh-blow+1;
int i = (int)((double)m / (m+n) * (k-1)); // make sure i won't be out of bounds
int j = Math.min(k - 1 - i,n-1);
int bj_1 = 0, ai_1 = 0;
if (i==0) { ai_1 = Integer.MIN_VALUE; } // in case i = 0, outOfBound
else { ai_1 = a[i-1]; }
if (j==0) { bj_1 = Integer.MIN_VALUE; } // in case j = 0, outOfBound
else { bj_1 = b[j-1]; }
if (bj_1 < a[i] && a[i] < b[j]) // kth smallest found, b[j-1] < a[i] < b[j]
return a[i];
if (ai_1 < b[j] && b[j] < a[i]) // kth smallest found, a[i-1] < b[j] < a[i]
return b[j];
if ( a[i] < b[j] ) // if true, exclude a's lower bound (if 2 arrays merged, a's lower bound must
// reside before kth smallest, so also update k.
// also exclude b's upper bound, since they are all greater than kth element.
return search(a, b,alow+i+1,ahigh, blow,j-1, k-i-1);
else
return search(a,b,alow, i-1, blow+j+1,bhigh, k-j-1);
}