1. Lower Bounds for Sorting, Searching and Selection
Lecture 9
Lower Bounds for
Sorting, Searching and Selection

2. Plan Finish Heaps Lower Bounds
Selection (Find Min): adversary arguments
Sorting: information theory lower bound for comparison based algorithms
Searching: ITLB

3. Finish Heaps
Building a heap from an array, in O(n) time

4. Number of vertices at height h3 2 1
n/2h+1

5. Finish Heaps Building a heap from an array, in O(n) time Idea:
The leaves are already heaps.
Joining two adjacent (sub) heaps with a common root, it suffices to heapify (trickle down from the root). It takes O(h) time (h = distance from the local root to the leaves), for at most n/2h+1 nodes on that level.
Total time: S n h/2h+1 = O(n), because S h/2h <2

6. Recap Linear Time Algorithms: O(n log n) Time Algorithms for sorting:
Compute Sum, Product of n numbers
Find Min/Max of n numbers
Merge 2 arrays of n elements (total)
Partition an array into 2 around a pivot
O(n log n) Time Algorithms for sorting:
Merge Sort
Heap Sort
Quick Sort (on average)
O(log n) Time algorithms:
Binary search

7. Lower Bounds Can we do better? Why not?
Lower bounds prove that we cannot hope for a better algorithm, no matter how smart we are.
Only very few lower bound proofs are known
Most notorious open problems in Theoretical Computer Science are related to proving lower bounds for very important problems
Reading: Ch. 13 textbook

8. Input Lower Bound
Compute the sum of n numbers: all numbers must be looked at, otherwise the answer might not be correct
Adversary argument: assume there is a smart algorithm which computes the sum without looking at all the n inputs. An adversary goes and modifies the input not looked at, then run the algorithm again. It should give the same answer (because it didn’t look at the modified input data), but this is not the correct answer.

9. Adversary Arguments Hmmm… Yes! Is a7 < a9?
Mr. Algorithm: thinks he has a fast way of solving the problem
Ms. Adversary: forces algorithm to work hard by given the worst possible answer

10. Adversary Arguments The answer is 3
This was the input Now try again
The answer is 3

11. Adversary Arguments
Wrong! The input was this time!
The answer is 3
If some questions were not asked, the Adversary tricks the poor Algorithm to try again, on a different input data, with the same answers to the same questions but with a different correct final answer….

12. Adversary Argument for FindMin: need n-1 questions
No.
Is a1 < a2?
a3=7
a4=6
No.
Is a3 < a4?
Is a4 > a5?
a5=5
Yes.
Minimum is a5!
Wrong! It is a2!

13. If less than n-1 questions, the graph of comparisons is disconnected
The adversary can re-arrange the data so that the answer is different

14. Binary Decision Trees ? Yes No
Model algorithms based on successive answers to yes/no questions

15. ?
Yes No ?
Yes No
Answer 1 ?
Yes No
Answer 2
Answer 3
Answer 4

16. Worst case time: depth of tree
A binary tree of depth h has < 2h leaves
A binary tree with N leaves must have depth at least log2N
This gives a lower bound on the worst case time to find an answer
If the number of possible answers is N, then the algorithm MUST ask at least
log N questions

17. Lower Bound for Sorting
Number of possible sorted orders = number of all possible permutations of n elements = n!
Hence any comparison-based algorithm for sorting must take at least log n! = O(n log n) time

18. Lower Bound for Searching
In-class exercise
How many possible answers for the searching question?
What is the log of that?
What is the lower bound for searching?