# Minimum number of jumps

• There is a row of seats. Assume that it contains N seats adjacent to each other. There is a group of people who are already seated in that row randomly. i.e. some are sitting together & some are scattered.

An occupied seat is marked with a character 'x' and an unoccupied seat is marked with a dot ('.')

Now your target is to make the whole group sit together i.e. next to each other, without having any vacant seat between them in such a way that the total number of hops or jumps to move them should be minimum.

Example

Here is the row having 15 seats represented by the String (0, 1, 2, 3, ......... , 14) -

``````          . . . . x . . x x . . . x . .
``````

Now to make them sit together one of approaches is -
. . . . . . x x x x . . . . .

Following are the steps to achieve this -
1 - Move the person sitting at 4th index to 6th index -
Number of jumps by him = (6 - 4) = 2

2 - Bring the person sitting at 12th index to 9th index -
Number of jumps by him = (12 - 9) = 3

So now the total number of jumps made =
( 2 + 3 ) % MOD =
5 which is the minimum possible jumps to make them seat together.

There are also other ways to make them sit together but the number of jumps will exceed 5 and that will not be minimum.

For example bring them all towards the starting of the row i.e. start placing them from index 0.
In that case the total number of jumps will be
( 4 + 6 + 6 + 9 )%MOD
= 25 which is very costly and not an optimized way to do this movement

• This post is deleted!

• First let's solve this problem using brute force, let's group all the k people together and form a window size of k. Move the window and compare to the original positions, then count the hops and get minimum value. Time complexity is O((n - k) * n), we can optimize to O((n - k) * k), n is the number of seats, and k is the number of people.

Brute force:

``````public int minMoves(char[] seats) {
int n = seats.length;

// step 1. count people
int k = 0;
for (int i = 0; i < n; i++) {
if (seats[i] == 'X') {
k++;
}
}

// step 2. move the window of size k and count the hops
int min = Integer.MAX_VALUE;

for (int j = 0; j < n - k; j++) {
int hops = 0, p = j;

for (int i = 0; i < n; i++) {
if (seats[i] == 'X') {
hops += Math.abs(i - p++);
}
}

min = Math.min(min, hops);
}

return min;
}
``````

Actually we can use greedy algorithm, the idea is simple, every time we consider the people groups from left and right sides, pick the group that has less people and move towards center, for example we have the following input:

. . X X . . . X X . . X X . . X X X . . .

there are two people in the far left group and three people in the far right group, so let's merge the people group from left, and we get the following:

. . . . . `X X` X X . . X X . . X X X . . .

now merge the far right group and we get:

. . . . . X X X X . . X X `X X X` . . . . .

finally we get:

. . . . . . . `X X X X` X X X X X . . . . .

Time complexity is O(n + k), where k is the number of group of people initially.

Efficient method:

``````public int minMoves(char[] seats) {
// step 1. add all groups to deque
Deque<int[]> dq = new ArrayDeque<>();

int i = 0, j = 0, n = seats.length;

while (j < n) {
// skip '.'
while (j < n && seats[j] == '.') j++;
if (j == n) break;
// go through 'X'
for (i = j; j < n && seats[j] == 'X'; j++) {}
}

// step 2. merge groups from both ends
int count = 0;

while (dq.size() > 1) {
int[] left = dq.peekFirst();
int[] right = dq.peekLast();

int lenLeft = left[1] - left[0] + 1;
int lenRight = right[1] - right[0] + 1;

if (lenLeft <= lenRight) {
// merge left two groups
left = dq.pollFirst();
count += lenLeft * (dq.peekFirst()[0] - left[1] - 1);
dq.peekFirst()[0] -= lenLeft;
} else {
// merge right two groups
right = dq.pollLast();
count += lenRight * (right[0] - dq.peekLast()[1] - 1);
dq.peekLast()[1] += lenRight;
}
}

return count;
}
``````

• @jeantimex
Hey, there.. Your brute force method doesn't work for the case when we have a single group pf people,, consider the string, str = ".xxx". Here, the answer should be 0, but the code depicted in your brute force approach gives 3.. What you can do is, in the outer j loop, you can have j iterating from 0 to (n-k),i.e., something like this :
for(j=0;j<=(n-k);j++)

• @jeantimex I really liked your idea and used it to solve the problem. I was able to simplify the solution to this only ( O(1) space) :

``````int Solution::seats(string A) {
int ans = 0;
int count = 0;
int space = 0;
int n = 0;
for(int i = 0; i<A.size();i++){
if(A[i]=='x') n++; // n is total number of occupied seats
}
for(int i = 0; i<A.size();i++){
if(A[i]=='x'){
if(space!=0){
ans += min(n-count,count)*space;  //either you can move left occupied seats to right or right occupied seats to left choose the one in which you have to move less number of elements(for example ***....******.... in this case of course we would like to move left part to right and not vice versa.
space = 0;
}
count++;
}else{
space++;
}
}
return ans;
}
``````

• @jeantimex Can you please share the proof of this algo?

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