# Google Interview Question

Given a set of N routers in 2-D coordinate, a source router, a destination router. The problem is to find if it is possible to transmit a signal from source to destination under the following constrain-

Each router can transmit messages up to Euclidean distance K.

Each router will shut down after transmitting a message. (In simple words, each router can be used only once.)

N <= 1e5, K <= 10

I have tried to make a graph with nodes as routers and add edges between 2-routers if their Euclidean distance is less than or equal to K. Now, I use normal bfs to see if it is possible to reach the destination. But the problem is The time complexity in this solution is O(N*N). I need a better solution. Does any help please?

Are there any constraints on the coordinates of the routers?

I think this might help Euclidean Minimum Spanning Tree You can also use a quad tree to find the nearest edges instead of generating them all. I think that would be O(n log n)

0-1 BFS should work, K is small. First put the source router in the queue then for each router, try to visit all nearest router such that the Euclidean distance <= K, there can be at most K*K (approx) nearest adjacent routers whose Euclidean distance <= K. Correct me If I am wrong. Time complexity will be O(N*K*K)

Simple and nice solution. Thanks!

How is the nearest adjacent routers whose Euclidean distance <= K be at most K*K. Because for all

ifrom 1 to K euclidean distance, there would bei*ipossible positions.And how would you compute those adjacent?

This problem can be solved with simple BFS if all coordinates are integers.

Store all coordinates in map of sets (key in map is $$$x$$$ coordinate, sets store $$$y$$$). Further you need common bfs. When you need add connected routers to queue you iterate by all suitable $$$x$$$ and find in respectives maps suitable $$$y$$$. After adding new router to the queue you remove it.

Foreach router we iterate by all suitable $$$x$$$. It's $$$O(K)$$$. Foreach $$$x$$$ we find all suitable $$$y$$$. It can be done in $$$O(log(n) + C)$$$, where $$$C$$$ is count of suitable $$$y$$$. Sum of all $$$C \le n$$$, because we remove router when we found it. Total complexity $$$O(N\cdot K \cdot log(n))$$$.

Also this problem can be solved in $$$O(N\cdot log^2(n))$$$ with some more complex data structures for non-integers coordinates or greater $$$K$$$.