8/11/2019 Recurrences Hand Out

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Chapter 4: Recurrences

Recurrences

Recurrence Trees

Master Method

Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Examples of Recurrences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3More Examples of Recurrences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Recursion Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Recursion Tree Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Master Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Master Method Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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Recurrences

A recurrence describes a function in terms of its values on smaller inputs.

The general form of a recurrence is:

T(n) = aT(s(n)) +f(n)

where we assume T(1) (1). T(n) = aT(s(n)) +f(n) can be interpreted as:

a= Number of subproblemss(n) = Size of the subproblemsf(n) = Time to divide into subproblems and combine results

CS 5633 Analysis of Algorithms Chapter 4: Slide 2

Examples of Recurrences

Merge-Sort: T(n) = 2T(n/2) +c n2 subproblemsn/2 = subproblem size(n)time to merge results

Insertion-Sort: T(n) = T(n 1) +c nI.e., first sort n 1 elts., then insert nth elt.1 subproblemn 1 = subproblem size(n)time to insert nth element

CS 5633 Analysis of Algorithms Chapter 4: Slide 3

More Examples of Recurrences

Bit-Multiply: T(n) = 3T(n/2) +c n3 subproblems (3 multiplications)n/2 = subproblem size

(n)time to add/sub. results Strassens Alg.: T(n) = 7T(n/2) +c n2

Matrices are n n7 subproblemsn/2 n/2= subproblem size(n2) time to form submatrices and add/sub. results

CS 5633 Analysis of Algorithms Chapter 4: Slide 4

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8/11/2019 Recurrences Hand Out

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Recursion Trees

Recursion tree for T(n) = 2T(n/2) +n

T(n)

n

Tn

2

T

n2

n

n/2 n/2

Tn4

Tn4

Tn4

Tn4

CS 5633 Analysis of Algorithms Chapter 4: Slide 5

Recursion Tree Continued

n

n/2 n/2

n/4 n/4 n/4 n/4

...

log2n

n

n

n

...

Total: (n lg n)

CS 5633 Analysis of Algorithms Chapter 4: Slide 6

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Master Method

For recurrences of the form:

T(n) = aT(n/b) +f(n)

T(n) (f(n))follows immediately. T(n) (nlogba) because the recursion tree has alogbn =nlogba leaves.

Height = logbn.Branching factor = a.

CS 5633 Analysis of Algorithms Chapter 4: Slide 7

Master Method Continued

For recurrences of the form:

T(n) = aT(n/b) +f(n)

T(n) (f(n))iff(n) is poly. larger than n logba (and is regular)

T(n) (nlogba)iff(n) is poly. smaller than n logba

T(n) (nlogba lg n)

iff(n) (nlog

ba

)CS 5633 Analysis of Algorithms Chapter 4: Slide 8

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