8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method.

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Analysis of Algorithms

Lecture: Solving recurrence by

recursion-tree method

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Problem of the day

• How many multiplications do you need to compute 316?

316 =38 x 38

38 =34 x 34

34 =32 x 32

32 =3 x 3

316 =3 x 3 x 3 …. x 3 Answer: 15

Answer: 4

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Outline

• Review of last lecture – analyzing recursive algorithms– Defining recurrence relation

• Today: analyzing recursive algorithms– Solving recurrence relation

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Solving recurrence

1. Recursion tree or iteration method- Good for guessing an answer

2. Substitution method- Generic method, rigid, but may be hard

3. Master method- Easy to learn, useful in limited cases only

- Some tricks may help in other cases

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Recurrence for merge sort

T(n) =(1) if n = 1;

2T(n/2) + (n) if n > 1.

We will usually ignore the base case, assuming it is always a constant (but not 0).

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Recursion tree for merge sort

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant.

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T(n)

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T(n/2) T(n/2)

dn

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dn

T(n/4) T(n/4) T(n/4) T(n/4)

dn/2 dn/2

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

dn

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

dn

dn

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

dn

dn

dn

…

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

dn

dn

dn

#leaves = n (n)

…

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dn

dn/4 dn/4 dn/4 dn/4

dn/2 dn/2

(1)

…

h = log n

dn

dn

dn

#leaves = n (n)

Total(n log n)

…

Later we will usually ignore d

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Recurrence for computing powerint pow (b, n)

m = n >> 1;p=pow(b,m)*pow(b,m);if (n % 2)

return p * b;else

return p;

int pow (b, n)

m = n >> 1;

p = pow (b, m);

p = p * p;

if (n % 2)

return p * b;

else

return p;

T(n) = T(n/2)+(1) T(n) = 2T(n/2)+(1)

Which algorithm is more efficient asymptotically?

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Time complexity for Alg1

Solve T(n) = T(n/2) + 1

• T(n) = T(n/2) + 1

= T(n/4) + 1 + 1

= T(n/8) + 1 + 1 + 1

= T(1) + 1 + 1 + … + 1

= Θ (log(n)) log(n)

Iteration method

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Solve T(n) = 2T(n/2) + 1.

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Solve T(n) = 2T(n/2) + 1.

T(n)

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T(n/2) T(n/2)

1

Solve T(n) = 2T(n/2) + 1.

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1

T(n/4) T(n/4) T(n/4) T(n/4)

1 1

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

1

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

1

2

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

1

2

4

…

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

1

2

4

#leaves = n (n)

…

Solve T(n) = 2T(n/2) + 1.

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1

1 1 1 1

1 1

(1)

…

h = log n

1

2

4

#leaves = n (n)

…

Solve T(n) = 2T(n/2) + 1.

Total(n)

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More iteration method examples

• T(n) = T(n-1) + 1

= T(n-2) + 1 + 1

= T(n-3) + 1 + 1 + 1

= T(1) + 1 + 1 + … + 1

= Θ (n) n - 1

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More iteration method examples

• T(n) = T(n-1) + n

= T(n-2) + (n-1) + n

= T(n-3) + (n-2) + (n-1) + n

= T(1) + 2 + 3 + … + n

= Θ (n2)

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3-way-merge-sort3-way-merge-sort (A[1..n])

If (n <= 1) return;3-way-merge-sort(A[1..n/3]);3-way-merge-sort(A[n/3+1..2n/3]);3-way-merge-sort(A[2n/3+1.. n]);Merge A[1..n/3] and A[n/3+1..2n/3];Merge A[1..2n/3] and A[2n/3+1..n];

•Is this algorithm correct?•What’s the recurrence function for the running

time?•What does the recurrence function solve to?

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Unbalanced-merge-sort

ub-merge-sort (A[1..n])if (n<=1) return;

ub-merge-sort(A[1..n/3]);

ub-merge-sort(A[n/3+1.. n]);

Merge A[1.. n/3] and A[n/3+1..n].

•Is this algorithm correct?•What’s the recurrence function for the running

time?•What does the recurrence function solve to?

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More recursion tree examples

• T(n) = 3T(n/3) + n

• T(n) = T(n/3) + T(2n/3) + n

• T(n) = 3T(n/4) + n

• T(n) = 3T(n/4) + n2

8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method.

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