Google Interview Question for Software Engineers

Country: United States
Interview Type: Phone Interview

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Brute-force solution: Maintain disjoint sets. Compute Euclidean distance between each pair of points and update the parent of each point’s disjoint set (using union by rank and path compression). Time complexity: O(N^2). Space: O(N)

/**Optimal Solution: Use closest points algorithm to solve this problem in O(NlogN) time and O(N) space**/

class Point {
	int x;
	int y;

float euclidDistance(Point a, Point b) {}

int find(int[] ds, int idx) {
	Stack<Integer> stack = new Stack<Integer>();
	while (ds[idx] != idx) {
			idx = ds[idx];
	while (!stack.isEmpty()) {
		int top = stack.pop():
		ds[top] = idx;
	return idx;

int pointGroups(Point[] pts, float k) {
	int[] sets = new int[pts.length]//disjoint sets
	for (int i = 0; i < sets.length; i++) {
		sets[i] = i;

	// Sort points by ascending x coordinate
	pts.sort(new Comparator() {
		public int compare(Point first, Point sec) {
			return first.x - sec.x

	pointsGroupHelp(pts,0,pts.length - 1, k, ds);
	//Apply path compression and count the total number of groups
	int count = 0;
	boolean[] seenGroup = new boolean[ds.length];
	for (int i = 0; i < ds.length; i++) {
		int p = find(ds,i);
		if (!seenGroup[p]) {
			seenGroup[p] = true;
	return count;	

void bruteForce(Point[] pts, int start, int end, int[] ds, float k) {
	for (int i = start; i <= end; i++) {

		for (int j = i + 1; j <= end, j++) {
				float d = euclidDistance(pts[i],pts[j]);
				if (d <= k) {

						int minParent = Math.min(ds[i],ds[j]);
						int maxParent = Math.max(ds[i],ds[j]);
						ds[maxParent] = minParent;

void pointsGroupHelp(Point[] pts,int start,,int end, float k, int[] ds) {
	if ((end - start + 1) <= 3) {

	int mid = start + (end - start) / 2;
	int midX = pts[mid].x; //mid line

	pointsGroupHelp(pts,start,mid,k,ds); // group together all points <= k distance that lie to the left of mid line
	pointsGroupHelp(pts,mid + 1, end, k, ds); // group together all points <= k distance that lie to the right of mid line

	// Find subarray of points that are within <=k units away from midline and group pairs whose distance is <= k
	int stripStart = -1;
	int stripEnd = stripStart;
	for (int i = start; i <= end; i++) {
		if (Math.abs(pts[i].x - midX) <= k) {
			stripStart = stripStart == -1 ? i : stripStart;
			stripEnd = i;

- divm01986 December 29, 2019 | Flag Reply
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You can not do better than N^2 but some quick optimizations can be done which are also practical during the interview. Like one I am doing below is that list of points is sorted based on their point.x values and comparing each point to the other points from left to right, if point2.x - point1.x > k that there is no need to compare any other values in rest of the list as all the other values will have x values equal to, or bigger than point2.x value. Note: The code below is not tested

List<List<Point>> getSets(Point [] points, int k){
		List<List<Point>> listOfLists = new ArrayList<List<Point>>();
		int [] set = new int[points.length];
		Arrays.fill(set, -1);
		Arrays.sort(points, (a,b)-> a.x - b.x);
		HashMap<Integer, List<Point>> map = new HashMap<>();
		for(int i = 0; i < points.length; i++){
			for(int j = i + 1; j < points.length; j++){
				if(points[j].x - points[i].x > k)break;
				int distance = getDistance(points[i], points[j]);
				if(distance > k)break;
				int rootOfI = find(set, i);
				int rootOfJ = find(set, j);
		for(int i = 0; i < set.length; i++){
			int root = find(set, i);
			map.putIfAbsent(root, new ArrayList<Point>());
		for(Map.Entry<Integer, List<Point>> entry: map.entrySet()){
			if(entry.getValue().size() > 1)
		return listOfLists;
	int getDistance(Point i, Point j){
		double xVal = Math.pow(Math.abs(i.x - j.x), 2);
		double yVal = Math.pow(Math.abs(i.y - j.y), 2);
		double distance = Math.sqrt(xVal + yVal);
		return (int)distance;
	int find(int [] set, int i){
		if(set[i] < 0)return i;
		return set[i] = find(set,set[i]);
	void union(int [] set, int root1, int root2){
		if(set[root1] < set[root2]){
			set[root2] = root1;
			if(set[root1] == set[root2])set[root2]--;
			set[root1] = root2;

- Raminder March 20, 2020 | Flag Reply
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of 1 vote


- khkh December 02, 2019 | Flag Reply
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of 1 vote

The idea is to map the 2d points to some sort of fixed reference and only compare where it make sense, instead of comparing each point with all the other point, which makes it a N2 algorithm. So here is my take on this. Divide the entire grid into a chunk of k x k grid squares.
e.g. 1st squre(0,0)(0,k)(k,0)(k,k)
2nd squre (0,k)(k,k)(0,2k)(k.2k)
Then iterate over each point and assign a parent grid
CEIL(i/k) * k, CEIL(j/k)*k

Once we have the squres assigned, iterate over the points which belongs only to adjacent 8 sqaures. Rest will lay out of bound for any 2d points inside this grid. Then like minimum spanning tree algorithm, keep chaining the points.

- Confused Aatma December 03, 2019 | Flag Reply
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Order : O(N)
However, if all the 2d points are packed with in these 9 squres, then a further division will reduce the number of iteration within the points.

- Confused Aatma December 03, 2019 | Flag
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of 1 vote

You can find this in O(N) approach. Find the min from the array and the max from the array.
Now you should have your buckets like this:
bucket1: [min, min + k]
bucket2: [min + k + 1, min + k + 1 + k]
bucketN/K: [max - k,max]
Now, you start iteration over the array and find out which bucket the current number belongs to and add it in the corresponding bucket.
When the iteration is over, just check for those buckets whose count of items inserted is greater then 1.

- Abhishek J December 11, 2019 | Flag Reply
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of 1 vote

As you said O(N^2) answer is pretty straight forward, for each element we check rest of the elements to see if their difference is less than K

One way to get time complexity down to O(N) or O(NlogN) by using a sliding window.

O(N) Time complexity if the input is sorted, else we sort the input and that gives O(NlogN) complexity.

The sliding window will work like below

input: {1,2,5,10} k = 5

We start with the first 2 elements.

i = 0, j = 1

If the difference between the first and last element (j - i) of the slide is <= k, we increase the sliding window to right (j++)
When the difference becomes > k, we increment i (i++)

Every time a new digit gets added to the window, it adds (j-i) number of additional groups to the total.


input: {1,2,5,10} K = 5
sliding window : [1,2] .  total Groups: 1 [[1,2]]
since 2 - 1 <= 5 ==> j++;
sliding window : [1,2, 5] .  total Groups: 3 [[1,2], [2,5],[1,5]]
since 5 - 1 <= 5 ==> j++;
sliding window : [1,2, 5, 10] .  10-1 > k so we increment i
sliding window : [2, 5, 10] .  10-2 > k so we increment i
sliding window : [5, 10] .  total Groups: 3 [[1,2], [2,5],[1,5], [5,10]]

- Saurabh January 07, 2020 | Flag Reply
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In question its mentioned 2d points.. on what basis you will sort your array..What would be the reference ...((1,2),(4,10),(3,11)) cant be sorted like (1,2,3,4, 5) :)

- freak March 22, 2020 | Flag

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