1. Equal costs at all levels
Root dominated
Leave dominated
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2. Master method
a subproblems
n/b size of each subproblem
f(n) cost of dividing problem and combining results of subproblems
Competition between:
a number of recursive calls made – bad
b how much problem size decreased each call – good
f(n) determines work outside of recursive call we compare to
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3. We’ll be comparing:
and
Question: is there more work at the root or at the leaves (like in the recursion examples)?
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4. Master theorem case 1:
Let a ≥ 1 and b > 1 be constants, f(n) a function, and let
T(n) be defined on the nonnegative integers by the recurrence
Then T(n) has the following asymptotic bounds:
1: If for some constant e>0 then:
If polynomial larger than f(n): dominates.
leaves dominating in a recursion tree
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5. Master theorem case 2:
Let a ≥ 1 and b > 1 be constants, f(n) a function, and let
T(n) be defined on the nonnegative integers by the recurrence
Then T(n) has the following asymptotic bounds:
2: If then:
If equals f(n):
equal work at each level
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6. Let a ≥ 1 and b > 1 be constants, f(n) a function, and let
Master theorem case 3:
Let a ≥ 1 and b > 1 be constants, f(n) a function, and let
T(n) be defined on the nonnegative integers by the recurrence
Then T(n) has the following asymptotic bounds:
3: If for some constants e>0 and
regularity conditions (ignore for now):
If polynomial smaller than f(n): f(n) dominates.
root dominating in a recursion tree
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7. Master theorem summary – 3 cases
If dominates
leaves dominating in a recursion tree
Like merge sort – equal work at each level
If dominates
root dominating in a recursion tree
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8. Master theorem summary
In all cases we compare to f(n)
Intuition: Either the leaves and recursion process dominate
the cost, or the root dominates the cost, or they are balanced
Proof: we won’t show; but relies on recursion trees and
geometric sums, similar to example cases we looked at
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9. Master theorem example 1
n3 polynomial larger than f(n) = n2
Case 1
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10. Strassen’s method anyone?
Master theorem example 2
Looks similar?
Strassen’s method anyone?
n2.8 polynomial larger than f(n) = n2
Case 1
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11. Master theorem example 3
1 is equal to f(n)=1
Case 2
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12. Strassen’s method anyone?
Master theorem example 4
Looks similar?
Strassen’s method anyone?
Polynomially smaller than f(n)
Case 3
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13. Strassen’s method anyone?
Master theorem example 5
Looks similar?
Strassen’s method anyone?
Polynomially smaller than nlogn
Case 3?
No, not polynomially smaller
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14. Substitution method
Guess a bound (we need a guess!)
Prove correct by induction
Find constants in this process
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15. Example
Prove that is
We need to show that
For appropriate choices of c > 0.
(can’t use big Oh in substitution because of induction,
need to write out definition with constants!)
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16. Induction step: assume
Then
we want
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17. Induction step: assume
Then
we want
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18. Induction step: assume
Then
we want
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19. Induction step: assume
Then
we want
When?
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20. Induction step: assume
Then
we want
Holds for
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21. Induction step: assume
Then
for
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22. Example
Induction needs base condition. For n=1, assume:
Then:
NO!
Asymptotic notation requires only for n>=no
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23. Example
Induction needs base condition.
Asymptotic notation requires only for n>=no
Let’s try n=3, so as not to depend directly on T(1):
Holds for c ≥ 2
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24. Randomized algorithms
Hiring problem
We always want the best hire for a job!
• Using employment agency to send one candidate at a time
• Each day, we interview one candidate
• We must decide immediately if to hire candidate and if so, fire previous
In other words …
• Always want the best hire…
• Cost to interview (low)
• Cost to fire/hire… (expensive)
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