1. Complexity
Introduction
Complexity ©D.Moshkovitz

2. Introduction Objectives: Overview: Tea parties and computations
To introduce some of the basic concepts of complexity theory.
Overview:
Tea parties and computations
Growth rate and reasonability
Reducibility
… And more…
Complexity ©D.Moshkovitz

3. The Crazy Tea Party
Problem To seat all guests at a round table, so people who sit in adjacent seats like each other.
John
Mary
Bob
Jane
Alice 
Complexity ©D.Moshkovitz

4. Solution for the Example
Problem To seat all guests at a round table, so people who sit an adjacent seats like each other.
John
Jane
Mary
Alice
Bob
Complexity ©D.Moshkovitz

5. Naive Algorithm For each ordering of the guests around the table
Verify each guest likes the guest sitting in the next seat.
Complexity ©D.Moshkovitz

6. How Much Time Should This Take? (worse case)
guests
steps n
(n-1)! 5 24
say our computer is capable of 1010 instructions per second, this will still take  3·10138 years! 15
100
9·10155
Complexity ©D.Moshkovitz

7. Problem Plan a trip that visits every site exactly once.
Tours
Problem Plan a trip that visits every site exactly once.
Complexity ©D.Moshkovitz

8. Solution for the Example
Problem Plan a trip that visits every site exactly once.
Complexity ©D.Moshkovitz

9. Naive Algorithm (Backtracking)
For each site
Try out all reachable sites, which have not been visited yet
Backtrack and retry
Repeat the process until stuck
Complexity ©D.Moshkovitz

10. How Much Time Should This Take?
If our computer can carry out 10,000 instructions per second, than it will take us 4 years!
sites
steps n n! 5
120 15
100
9·10157
Complexity ©D.Moshkovitz

11. Is a Problem Tractable? YES! And here’s the efficient algorithm for it
NO! and I can prove it
and what if neither is the case?
Complexity ©D.Moshkovitz

12. Food for Thought…
Can you design an efficient algorithm for the tour problem if the sites’ map contains no cycles?
Complexity ©D.Moshkovitz

13. Growth Rate: Sketch n! =2O(n lg n) 2n n2 10n time input length
Complexity ©D.Moshkovitz

14. The World According to Complexity
reasonable
unreasonable
polynomial  nO(1)
exponential  2nO(1)
Complexity ©D.Moshkovitz

15. Could One be Fundamentally Harder than the Other?
Seating
Tour
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16. Relations Between Problems
Assuming an efficient procedure for problem A, there is an efficient procedure for problem B
B cannot be radically harder than A
Complexity ©D.Moshkovitz

17. Reductions B A p
B cannot be radically harder than A In other words: A is at least as hard as B
Complexity ©D.Moshkovitz

18. Which One is Harder? ?
Seating
Tour
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19. “directly reachable from…”
Reduce Tour to Seating First Observation: The problems aren’t so different
site
guest
“directly reachable from…”
“liked by…”
Complexity ©D.Moshkovitz

20. Reduce Tour to Seating Second Observation: Completing the circle
Let’s invite to our party a very popular guest,
i.e one who can sit next to everyone else.
Complexity ©D.Moshkovitz

21. Reduce Tour to Seating
If there is a tour, there is also a way to seat all the imagined guests around the table.
popular guest
. . .
. . .
Complexity ©D.Moshkovitz

22. Reduce Tour to Seating
If there is a seating, we can easily find a tour path (no tour, no seating).
popular guest
. . .
. . .
Complexity ©D.Moshkovitz

23. Bottom Line
The seating problem is at least as hard as the tour problem
Complexity ©D.Moshkovitz

24. Discussion
Although we couldn’t come up with an efficient algorithm for the problems
Nor to prove they don’t have one,
We managed to show a very powerful claim regarding the relation between their hardness
Complexity ©D.Moshkovitz

25. Discussion
Interestingly, we can also reduce the seating problem to the tour problem.
Moreover, there is a whole class of problems, which can be pairwise efficiently reduced to each other.
Complexity ©D.Moshkovitz

26. Food for Thought…
Can you reduce the seating problem to the tour problem?
Complexity ©D.Moshkovitz

27. NPC NPC  ? exponential algorithms efficient algorithms
Contains thousands of distinct problem
NPC
each reducible to all others 
exponential algorithms
efficient algorithms ?
Complexity ©D.Moshkovitz

28. How Can Studying Complexity Make You a Millionaire?
P vs NP is the most fundamental open question of computer science.
Resolving it would get you great honor
… as well as substantial fortune…
Philosophical: if P=NP
Human ingenuity unnecessary
No need for mathematicians!!
Is nature nondeterministic?
Complexity ©D.Moshkovitz

29. What More?
Let’s review some more questions to be discussed in this course.
Complexity ©D.Moshkovitz

30. Generalized Tour Problem
Add prices to the roads of the tour problem
Ask for the least costly tour $3
$17
$13 $8
$19
$10
$13
$12
Complexity ©D.Moshkovitz

31. Approximation How about approximating the optimal tour?
I.e – finding a tour which costs, say, no more than twice as much as the least costly. $3
$17
$13 $8
$19
$10
$13
$12
Complexity ©D.Moshkovitz

32. Is Running Time the Only Precious Resource?
How about memory (space)?
Are there other resources?
Complexity ©D.Moshkovitz

33. Games
Each player picks a word, which begins with the letter that ended the previous word.
Girl
Apple
Lord
Yard
Deer
Egg
Dog
Guy
Complexity ©D.Moshkovitz

34. Dictionary
Apple
Egg
Girl
Ear
Lord
Dog
Red
Complexity ©D.Moshkovitz

35. Game Tree Apple Egg Girl Ear Lord Dog Red Egg Girl Dog Apple Ear Lord
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36. Winning Games How can we find a winning strategy for this game?
How much space would it take to compute it using game trees?
Complexity ©D.Moshkovitz

37.  Summary We got to know two problems:
The seating problem (HAMILTONIAN-CYCLE)
The tour problem (HAMILTONIAN-PATH)
Although we could say very little about them, we managed to show their computational hardness are related.
Moreover, we claimed they are part of a large class of problems, called NPC.
Complexity ©D.Moshkovitz

38. 
Summary
We also mentioned some of the topics we will discuss later in the course:
Approximation
Space-bounded computations
Complexity ©D.Moshkovitz