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Fake-Coin Problem
Decrease and Conquer Technique
Definition
Best illustrates the decrease-by-a-constant-factor
strategy
Among n identical-looking coins, one is fake.
Among n identical-looking coins, one is fake.
Definition
Among n identical-looking coins, one is fake.
Problem: to design an efficient algorithm for
detecting the fake coin
detecting the fake coin
Assumption: fake coin is known to be lighter than the
genuine one
How to compare?
With a balance scale,
any two sets of coins
any two sets of coins
can be compared
How to compare?
 Based on tipping to the left, to
the right, or staying even, the
Balance scale will tell whether
Balance scale will tell whether
the sets weigh the same or
which of the sets is heavier than
the other but not by how much.
How to compare?
Staying even,
No Fake Coin
No Fake Coin
How to compare?
Tipping to the left,
Fake Coin -
Right
Fake Coin -
Right
How to compare?
Tipping to the right,
Fake Coin -
Left
Fake Coin -
Left
Basic Idea
 If n is odd (n−1) is even
 Keep one extra coin aside
 Split (n−1) coins into two piles and put each half on the balance scale
 If the piles weigh the same, the coin put aside must be fake
 If the balance is not even, then we choose the lightest pile of coins to
be the one containing the fake.
 Proceed in the same manner with the lighter pile - one with the
fake coin
Basic Idea
 If n is even
 Split n coins into two piles and put each half on the balance scale
 If the piles weigh the same, No fake coin
 If the balance is not even, then we choose the lightest pile of
coins to be the one containing the fake.
 Proceed in the same manner with the lighter pile - one with
the fake coin
Fake-Coin Problem

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Fake-Coin Problem

  • 1. Fake-Coin Problem Decrease and Conquer Technique
  • 2. Definition Best illustrates the decrease-by-a-constant-factor strategy Among n identical-looking coins, one is fake. Among n identical-looking coins, one is fake.
  • 3. Definition Among n identical-looking coins, one is fake. Problem: to design an efficient algorithm for detecting the fake coin detecting the fake coin Assumption: fake coin is known to be lighter than the genuine one
  • 4. How to compare? With a balance scale, any two sets of coins any two sets of coins can be compared
  • 5. How to compare?  Based on tipping to the left, to the right, or staying even, the Balance scale will tell whether Balance scale will tell whether the sets weigh the same or which of the sets is heavier than the other but not by how much.
  • 6. How to compare? Staying even, No Fake Coin No Fake Coin
  • 7. How to compare? Tipping to the left, Fake Coin - Right Fake Coin - Right
  • 8. How to compare? Tipping to the right, Fake Coin - Left Fake Coin - Left
  • 9. Basic Idea  If n is odd (n−1) is even  Keep one extra coin aside  Split (n−1) coins into two piles and put each half on the balance scale  If the piles weigh the same, the coin put aside must be fake  If the balance is not even, then we choose the lightest pile of coins to be the one containing the fake.  Proceed in the same manner with the lighter pile - one with the fake coin
  • 10. Basic Idea  If n is even  Split n coins into two piles and put each half on the balance scale  If the piles weigh the same, No fake coin  If the balance is not even, then we choose the lightest pile of coins to be the one containing the fake.  Proceed in the same manner with the lighter pile - one with the fake coin