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Robbing a House with Greedy Algorithms. Casey O’Brien. Knapsack Problem. Your Knapsack Max Weight: 10 pounds. Goal: Maximize total value Constraint: Total weight cannot exceed 10 pounds. Play The Robber!. Final Knapsack: Final Value: $14. Play The Robber!. Final Knapsack: - PowerPoint PPT Presentation
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Robbing a House withGreedy Algorithms
Casey O’Brien
Knapsack Problem
Your Knapsack
Max Weight: 10 pounds
Item
Item Value
13
5
3
6
9
Item Value Weight
13 8
5 5
3 0.6
6 1.5
9 3
Goal: Maximize total value
Constraint: Total weight cannot exceed
10 pounds
Item Value Weight
13 8
5 5
3 0.6
6 1.5
9 3
Item Value Weight Value/Weight
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Play The Robber!
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
12 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $14
Play The Robber!
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $19
Final Knapsack:
Final Value: $16
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $14
Final Knapsack:
Final Value: $20
Final Knapsack:
Final Value: $17
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $15
Final Knapsack:
Final Value: $18
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Greedy Algorithms
Greedy by Largest Value
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $19
Greedy by Smallest Weight
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $18
Greedy by Largest Ratio
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Item Value Weight Value/Weight
Knapsack
13 8 1.6
5 5 1
3 0.6 5
6 1.5 4
9 3 3
Final Knapsack:
Final Value: $18
Problem:Greedy Algorithms do not guarantee optimal
solution
Exhaustive Enumeration
Exhaustive Enumeration
Try All Possibilities
How Many Possibilities?
Final Knapsack:
Final Value: $20
Problem:Takes Too Long!
40 Items
40 Items
~1 Trillion Possibilities
40 Items
~1 Trillion Possibilities
~35,000 Years to Compute
What Can We Do?
Settle For Less Than Optimal
Recall Greedy Algorithms:
By Value: $19By Weight: $18By Ratio: $18
Optimal: $20
Moral of the Story:
We can use Greedy Algorithms to
approximate solutions to the knapsack problem quickly.
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