# GATE | Gate IT 2005 | Question 56

• Difficulty Level : Hard
• Last Updated : 28 Jun, 2021

Let G be a directed graph whose vertex set is the set of numbers from 1 to 100. There is an edge from a vertex i to a vertex j iff either j = i + 1 or j = 3i. The minimum number of edges in a path in G from vertex 1 to vertex 100 is

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(A) 4
(B) 7
(C) 23
(D) 99

Explanation: The task is to find minimum number of edges in a path in G from vertex 1 to vertex 100 such that we can move to either i+1 or 3i from a vertex i.

```Since the task is to minimize number of edges,
we would prefer to follow 3*i.

Let us follow multiple of 3.

1 => 3 => 9 => 27 => 81, now we can't follow multiple
of 3. So we will have to follow i+1. This solution gives
a long path.

What if we begin from end, and we reduce by 1 if
the value is not multiple of 3, else we divide by 3.
100 => 99 => 33 => 11 => 10 => 9 => 3 => 1

So we need total 7 edges.```
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