dush1729's blog

How to solve SPOJ problem YAPP?
By dush1729, history, 10 years ago, In English

Problem link

Why is the answer 2 ^ ( n — 1 ) ?

  • Vote: I like it
  • 0
  • Vote: I do not like it

Comments Comments (5)
| Write comment?
»
10 years ago, hide # |
 
Vote: I like it 0 Vote: I do not like it

Suppose the correct answer for f(n) = k and we want to find out f(n+1)

One could either place the highest number i.e. (n+1) either towards the rightmost end in the permutation or towards the leftmost end to satisfy constraints "maximum element between the indices [i..j] is either present at index i, or at index j" to satisfy for [1...n]. The other inner constraints would automatically be satisfied as we know f(n) is true. So there are 2 possible choices. f(n+1) = 2*f(n)

Now the base case n = 1. There is only 1 choice so f(1) = 1. Kind of Mathematical Induction Proof :P

So a recurrence relation f(n) = 2*f(n-1) when n > 1. Solving this recurrence f(n) = 2**n-1

→ Reply
  • »
    »
    10 years ago, hide # ^ |
     
    Vote: I like it 0 Vote: I do not like it

    I cannot understand why is f(n+1) = 2*f(n) ?

    → Reply
    • »
      »
      »
      10 years ago, hide # ^ |
      ← Rev. 2  
      Vote: I like it 0 Vote: I do not like it

      Emm. Suppose the ans for n = 3 has these possibilities:

      1 2 3
3 1 2
2 1 3
3 2 1

      4 possible answers

      Now i need to find ans for n = 4. So _ _ _ _. I can either place it at 1st index or 4th index to satisfy the property "maximum element between the indices [i..j] is either present at index i, or at index j" for range [1..4]

      So the ans will be:

      1 2 3 4
3 1 2 4
2 1 3 4
3 2 1 4

4 1 2 3
4 3 1 2
4 2 1 3
4 3 2 1

      4 + 4 = 8 possible answers

      → Reply