Lecture 12: Recurrences - PowerPoint Presentation, Algorithms PDF Download

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1
CSE 421
Algorithms g
Richard Anderson
Lecture 12
Recurrences
Announcements
•Midterm
– Feb 9, In class, closed book
– Through section 5.2
Divide and Conquer
• Recurrences, Sections 5.1 and 5.2
• Algorithms
– Counting Inversions (5.3)
Cl t P i (5 4) – Closest Pair (5.4)
– Multiplication (5.5)
– FFT (5.6)
Divide and Conquer
Array Mergesort(Array a){
n = a.Length;
if (n <= 1)
t return a;
b = Mergesort(a[0 .. n/2]);
c = Mergesort(a[n/2+1 .. n-1]);
return Merge(b, c);
}
Algorithm Analysis
• Cost of Merge
• Cost of Mergesort
T(n) <= 2T(n/2) + cn; T(1) <= c;
Page 2


1
CSE 421
Algorithms g
Richard Anderson
Lecture 12
Recurrences
Announcements
•Midterm
– Feb 9, In class, closed book
– Through section 5.2
Divide and Conquer
• Recurrences, Sections 5.1 and 5.2
• Algorithms
– Counting Inversions (5.3)
Cl t P i (5 4) – Closest Pair (5.4)
– Multiplication (5.5)
– FFT (5.6)
Divide and Conquer
Array Mergesort(Array a){
n = a.Length;
if (n <= 1)
t return a;
b = Mergesort(a[0 .. n/2]);
c = Mergesort(a[n/2+1 .. n-1]);
return Merge(b, c);
}
Algorithm Analysis
• Cost of Merge
• Cost of Mergesort
T(n) <= 2T(n/2) + cn; T(1) <= c;
2
Recurrence Analysis
• Solution methods
– Unrolling recurrence
– Guess and verify
Plugging in to a “Master Theorem” – Plugging in to a Master Theorem
Unrolling the recurrence
Substitution
Prove T(n) <= cn (log
2
n + 1) for n >= 1
Induction:
Base Case:
Induction Hypothesis:
A better mergesort (?)
• Divide into 3 subarrays and recursively 
sort
• Apply 3-way merge
What is the recurrence?
Unroll recurrence for                
T(n) = 3T(n/3) + dn
T(n) = aT(n/b) + f(n)
Page 3


1
CSE 421
Algorithms g
Richard Anderson
Lecture 12
Recurrences
Announcements
•Midterm
– Feb 9, In class, closed book
– Through section 5.2
Divide and Conquer
• Recurrences, Sections 5.1 and 5.2
• Algorithms
– Counting Inversions (5.3)
Cl t P i (5 4) – Closest Pair (5.4)
– Multiplication (5.5)
– FFT (5.6)
Divide and Conquer
Array Mergesort(Array a){
n = a.Length;
if (n <= 1)
t return a;
b = Mergesort(a[0 .. n/2]);
c = Mergesort(a[n/2+1 .. n-1]);
return Merge(b, c);
}
Algorithm Analysis
• Cost of Merge
• Cost of Mergesort
T(n) <= 2T(n/2) + cn; T(1) <= c;
2
Recurrence Analysis
• Solution methods
– Unrolling recurrence
– Guess and verify
Plugging in to a “Master Theorem” – Plugging in to a Master Theorem
Unrolling the recurrence
Substitution
Prove T(n) <= cn (log
2
n + 1) for n >= 1
Induction:
Base Case:
Induction Hypothesis:
A better mergesort (?)
• Divide into 3 subarrays and recursively 
sort
• Apply 3-way merge
What is the recurrence?
Unroll recurrence for                
T(n) = 3T(n/3) + dn
T(n) = aT(n/b) + f(n)
3
T(n) = T(n/2) + cn
Where does this recurrence arise?
Solving the recurrence exactly
T(n) = 4T(n/2) + cn T(n) = 2T(n/2) + n
2
T(n) = 2T(n/2) + n
1/2
Recurrences
• Three basic behaviors
– Dominated by initial case
– Dominated by base case
All cases equal we care about the depth – All cases equal – we care about the depth