1. CSCI 2670 Introduction to Theory of Computing
November 16, 2005

2. Announcement Homework due next Tuesday (11/22)
7.3 a, 7.4, 7.6 (union only), 7.9, 7.12
1st edition
7.3a, 7.4, 7.6 (union only), 7.10, 7.12
November 16, 2005

3. Last class Relationship of time complexity for different TM models
If a problem can be solved in O(t(n)) time on a multi-tape TM, it can be solved in O(t2(n)) time on a single-tape TM
If a problem can be solved in O(t(n)) time on a nondeterministic TM, it can be solved in 2O(t(n)) time on a deterministic TM
November 16, 2005

4. Polynomial vs. exponential time
We distinguish between algorithms that have polynomial running time and those that have exponential running time
Assume a single tape deterministic TM
Polynomial functions – even ones with large exponents – grow less quickly than exponential functions
We can only process large data sets with polynomial running time algorithms
November 16, 2005

5. Polynomial equivalence
Two algorithms A1 and A2 are polynomially equivalent if we can simulate A2 using A1 with only a polynomial increase in running time
November 16, 2005

6. The class P
P is the class of languages that are decidable in polynomial time on a single-tape Turing machine
P = k TIME(nk)
P “roughly corresponds” to the problems that are realistically solvable on a computer
November 16, 2005

7. Size of input: Important consideration
The running time is measured in terms of the size of the input
If we increase the input size can that make the problem seem more efficient
E.g., if we represent integers in unary instead of binary
We consider only reasonable encodings
November 16, 2005

8. A problem in class P Binary tree query
Given a binary search tree T and a key k, find the node in T with key(node) = k
How do we show this problem is in class P?
Write an algorithm and show that the algorithm has running time O(nk) for some k
November 16, 2005

9. Binary search M = “On input <G,k> Let node = root(G)
Do while key(node) <> k
If key(node) < k
If right(node) == NIL
return NIL
Else
node = right(node)
If left(node) == NIL
node = left(node)
Return node
November 16, 2005

10. Execution time Worst case running time? O(|nodes|)
Occurs if tree is unbalanced
Is this O(n)?
Yes … any reasonable encoding will have an entry for each node
November 16, 2005

11. Path problem
PATH = {<G,s,t> | G is a directed graph that has a path from s to t>}
PATH  P
How would we show this?
Mark all nodes that can be reached from s
If t gets marked, accept
If t doesn’t get marked, reject
November 16, 2005

12. Solution to path problem
M = “On input <G,s,t>, where G is a digraph with nodes s and t:
Place a mark on node s
Repeat the following until no additional nodes are marked
Scan all edges of G
If an edge (a,b) is found where a is marked and b is unmarked, mark b
If t is marked accept
Else reject”
November 16, 2005

13. Verify solution is in P Is M a decider?
Yes
How long will it take M to run?
Loop takes O(|edges|) to execute each time
Loop is executed O(|nodes|) times
Total time is O(|edges|x|nodes|)
Is this polynomial in the length of the input?
This is o(n2) where n = lentgh of input
November 16, 2005