1. Lecture 14 Lower Bounds
Decision tree model
Linear-time reduction

2. Lower & Upper Bounds
Upper bounds of a problem A: there is an algorithm for the problem that is O(f(n)).
Lower bounds of a problem A: ALL algorithms for the problem that are Ω(f(n)).
An optimal algorithm for some problem is an algorithm with its upper bound asymptotically equivalent to the lower bound of this problem.

3. Trivial lower bounds
Finding the maximum element of a list of n elements: Ω(n).
Multiplying two n × n matrix: Ω(n2).

4. Roadmap
Decision tree model
Linear-time reduction

5. Decision tree model Definition: Decision Tree Model
Decision tree model is the model of computation in which an algorithm is considered to be a decision tree, i.e., a sequence of branching operations based on comparisons of some quantities.

6. Algorithm ⇔ Decision tree
Linear Search
Depth =
worst time complexity
A[1] = x?
tests 1
A[2] = x? T F 2
A[2] = x? T F n
A[n] = x?
...... T F

7. Algorithm ⇔ Decision tree
Binary Search
A[n/2] ? x
n/2
A[n/4] = x?
A[3n/4] = x?
< =
>
n/4
A[n/8] = x?
< =
>
3n/4
......
A[7n/8] = x?
A[n/n] = x?
......
A[i] = x?
A[j] = x?
A[n] = x?
...
< =
> n j i 1

8. Lower bounds based on Decision Tree Model
How many leaves?
Lower bound = the smallest depth

9. Searching problem ⎡log(n+1)⎤ = Θ(logn) Problem: Search
Input: A array A[1..n] of n elements, and an element a
Output: The index i if A[i]=a, 0 otherwise
Any search algorithm a1 a2 a3 an
......
n+1 leaves
⎡log(n+1)⎤ = Θ(logn)

10. Sorting problem ⎡log(n!)⎤ = Θ(nlogn) Problem: Sorting
Input: An array A[1..n] of n elements
Output: The sorted array of A[1..n]
a1a2...an
Any sort algorithm
a2a1...an
a3a1...an
a4a1...an
anan-1...a1
anan-1...a2
......
n! leaves
⎡log(n!)⎤ = Θ(nlogn)

11. Decision trees Definition: Simple Decision tree model
Every decision is based on the comparison of two numbers within constant time
Definition: Algebra Decision tree model
Every decision is based on some polynomial function:
f(x1,x2,...,xn) = 0?

12. Element uniqueness Theorem
Problem: ElementUniqueness
Input: A set of n numbers
Output: yes, if there does not exist two of them that are equal; no, otherwise
Theorem
Any algorithm base on algebraic decision tree model to determine element uniqueness requires Ω(nlogn) comparisons in the worst case.

13. Roadmap
Decision tree model
Linear-time reduction

14. Polynomial-time reduction
Q is polynomially reduced to Q’
An algorithm for Q
An algorithm for problem Q’
yes
Input of Q
Transform
Input of Q’ no
Review
(P v.s. NP)
Polynomial-time
Q∝p Q’ : Q’ is harder than Q

15. Linear-time reductions
An algorithm for A
Input of A
Transform
Input of B
Linear-time
Any algorithm for problem B
Output of B
Output of A
Transform
f(n) = Ω(n)
A∝nB: A has lower bound Ω(f(n)) implies
B has lower bound Ω(f(n))

16. The convex hull problem
Sorting∝nConvexHull

17. The closest pair problem
ElementUniqueness ∝n ClosestPair

18. The Euclidean minimum spanning tree problem
Sorting ∝n EuclideanMinSpan

19. Conclusion
Theorem
In algebraic decision tree model for computation, any algorithm that solves
sorting problem, or
element uniqueness problem, or
convex hull problem, or
the closest pair problem, or
the Euclidean minimum spanning tree problem
requires Ω(nlogn) operations in the worst case.