This document discusses the egg dropping problem, where the goal is to determine the minimum number of attempts needed to find the highest floor that an egg can be dropped from without breaking, given a number of eggs and floors. It presents different strategies for solving the problem, including using intervals of floors to test, changing the interval sizes, and modeling the problem with dynamic programming to find identical minimum attempts for any number of eggs and floors. The optimal strategy uses decreasing interval sizes calculated from a formula to generalize the solution.

1. THE EGG DROPPING PROBLEM
MOHAMMAD ALETAHA
SHARIF UNIVERSITY OFTECHNOLOGY
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2. WHYTHIS PROBLEM?
Known as a Google and Microsoft interview test
The question maker is unknown
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3. THE PROBLEM
We have 2 eggs and a 100 floors
What is the min attempt? (worst case)
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4. DEFINITION
We have k numbers of eggs and a building with n floors.
What is the least number of egg-dropping attempts that
is guaranteed to find the highest floor from witch the
egg won’t break on dropping?
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5. NOTES
What strategy minimizes the number of attemps?
We can break the eggs during attempts
No tipping point exist! (mathematical sol’n)
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6. TRIVIAL PROBLEMS
given one egg and ten floors
What is min attempt? (worst case)
Ans: 10
given ten egg and one floors
What is min attempt? (worst case)
Ans: 1
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7. SUBPROBLEMS
Given many eggs
Idea: binary tree
e.g. 100 floors
Max attemps: 7
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8. BACKTO MAIN
2 eggs and a 100 floors
Idea: 10 floors interval
Worst case?
19 attempt for 99
10,20,…,100 (10 att) + 91,…,99 (9 att)
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9. ANOTHER STRATEGY
Idea: bigger intervals
e.g. size of intervals = 12
Worst case?
19 attempts for 95
12,24,...,84,96 (8 att) + 84,…,95 (11 att)
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10. NOTE
In two last strategy:
Attempts are not identical for different numbers
e.g. 2 eggs, 100 floors:
for 24, 8 att & for 80, 9 att
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11. BETTER STRATEGY
Idea: change intervals
Decreasing from x to 1 (optimizing x)
x + (x-1) + (x-2) + (x-3) + (x-4) + … + 1 >= n
For 10 floors x is 4
For 100 floors x is 14
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12. INDENTICAL ATTEMPTS
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13. GENERALIZING
Function: Drop(total floors, total eggs)
We have to complete such a table:
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14. DYNAMIC PROGRAMMING
We call Drop(n,k) recursively
Basic cases:
- Drop(1,x)=1
- Drop(x,1)=x
floor we are currently attempting to drop from is
represented by f
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15. DYNAMIC PROGRAMMING
If we drop an egg from floor f, one of two events happens:
- Case 1: egg breaks:
our problem is reduced to a tower of height f-1, and we now
have k-1 eggs
- Case 2: egg doesn’t break:
we need to check n-f floors and we still have k eggs
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16. EXAMPLE
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17. FORMULATION
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18. C++
IMPLEMENTATION
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20. Thanks for your attention!
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