1. Lecture 6 More Divide-Conquer and Paradigm #4 Data Structure.
Talk at U of Maryland
Lecture 6 More Divide-Conquer and Paradigm #4 Data Structure.
Today, we do more “divide-and-conquer”.
And, we do Algorithm design paradigm # 4: invent or augment a data structure.

2. More divide and conquer: powering a number
Problem: Compute an , where n is an integer.
Naïve algorithm Θ(n).
Divide-and-conquer:
an = an/2 an/ if n is even.
= a(n-1)/2 a(n-1)/2 a if n is odd
Note, another kind of divide and conquer.
Time complexity: T(n) = T(n/2) + Θ(1) = Θ(logn)

3. Back to Fibonacci numbers
Recursion: not careful would give exponential time algorithm.
Plain linear bottom up computation: Θ(n).
Can we do better?
Theorem: Fn+1 Fn
Fn Fn
I will prove this in class.
Then you can compute this similar to powering of an integer in Θ(logn) steps. n

4. VLSI tree layout
Problem: Embed a complete binary tree with n leaves in a grid using minimal area.
W(n)
H(n)
H(n) = Θ(lg n) W(n) = Θ(n)
Area = Θ(n lg n)

5. How do we improve this embedding?
If we wish to have an O(n) solution for the area, perhaps we wish to have √n for L(n) and W(n).
What recursion scheme would get us there?
The Master theorem Case 1 says if b=4, a=2, then log4 2 =1/2. So if we have something like
T(n) = 2T(n/4) + o(√n)
we would get O(√n) solution for L(n) and W(n).

6. H-tree embedding scheme
L(n/4)
L(n)
L(n) = 2L(n/4) + Θ(1) = Θ(√n)
Area = Θ(n)

7. Paradigm #4 Use a data structure.

8. Talk at U of Maryland
Example 1. Heapsort
We invent a data structure (the heap) to support sorting.

9. Talk at U of Maryland
Example 2. Subrange sum
We want to maintain an array of integers a[1..n], and support the following operations:
A. Increment a[i] by b; that is, set a[i] += b.
B. Subrange sum: compute the sum of the elements a[c], a[c+1], ..., a[d].
Augment array as follows:
a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8]
a[1..2] a[3..4] a[5..6] a[7..8]
a[1..4] a[5..8]
a[1..8]
Operations A and B are both in O(logn) time.