1. Caching Parallel Computational Models Other Topics in Algorithms Wednesday, August 13 th 1

2. Announcements 2 1.PS#6 due tonight at midnight 2.Winners of the Competition: 1.Shir Aharon 2.Rasoul Kabirzadeh 3.Alice Yeh & Marie Feng 3.Extra Office Hours on Thursday & Friday  Semih: Th 10am-12pm (Gates 424)  Billy: Friday 1pm-5pm (Gates B24)  Mike: Th 3pm-5pm (Gates B24)  Yiming: Fr: 5pm-7pm (Gates B24)

3. Outline For Today 1.Caching 2.Other Algorithms & Algorithmic Techniques Beyond CS 161 1.Parallel Algorithms (Finding Min, Bellman-Ford) 2.Linear Programming 3.Other Topics & Classes 3

4. Alan Turing: Father of Computer Science 4

5. Computer Science 5 Studies the powers of machines. Fundamental Question CS asks: What is “computable” by machines? Turing, along with Church and Godel, was the person who made computation something we can mathematically study.

6. Turing’s Answer To What Computation Is (1936) 6 “On Computable Numbers, with an Application to the Entscheidungsproblem”: “We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions”

7. Turing’s Answer To What Computation Is 7 Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book. The behavior of the computer at any moment is determined by the symbols which he is observing, and his " state of mind " at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite.

8. Turing’s Answer To What Computation Is 8 Let us imagine the operations performed by the computer to be split up into "simple operations" which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change of the physical system consisting of the computer and his tape. … We may suppose that in a simple operation not more than one symbol is altered.

9. Turing’s Answer To What Computation Is 9 Besides these changes of symbols, the simple operations must include changes to the observed squares. … I think it is reasonable to suppose that they can only be squares whose distance from the closest of the immediately previously observed squares does not exceed a certain fixed amount.

10. The Turing Machine 10 It’s paradoxical that as humans in our quest to understand what machines can do, we have been studying an abstract machine that in essence imitates a human being.

11. Church-Turing Thesis 11 Central Dogma of Computer Science: **Whatever is computable is computable by the Turing machine.** There is no proof of this claim.

12. Turing In Defense Of His Claim: 12 “All arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The arguments which I shall use are of three kinds. (a) A direct appeal to intuition. (b) A proof of the equivalence of two definitions (in case the new definition has a greater intuitive appeal). (c) Giving examples of large classes of numbers which are computable.”

13. Algorithm = Turing Machine 13 When we say there is an algorithm computing shortest paths of a graph in O(mlog(n)) times we really mean: There is a Turing Machine that computes the shortest paths of a graph in O(mlog(n)) operations.

14. Turing Machine Is A Very Powerful Machine 14 CS tries to understand the limits of TM. We limit/extend TM and try to understand what can be computed by it.  Limit the # times it’s head is allowed to move left/right and it changes states to poly-time. => poly-time algs  What if the machine had access to a random source => randomized algorithms.  What if there were multiple heads on the tape => parallel algorithms  Limit the length of its tape. => space-efficient algs  What if the head was only allowed to move right => streaming algorithms ……

15. Outline For Today 1.Caching 2.Other Algorithms & Algorithmic Techniques Beyond CS 161 1.Parallel Algorithms (Finding Min, Bellman-Ford) 2.Linear Programming 3.Other Topics & Classes 15

16. Online Algorithms 16  Takes as input a possibly infinite stream.  At each point in time t make a decision based on  what has been seen so far  but without knowing the rest of the input  Type of Optimality Analysis: Competitive Ratio  “Worst” (Cost of online algorithm)/(Cost of OPT) ratios against any input stream  Where OPT is the best solution possible if we knew the entire input in advance

17. Caching 17 Slow Disk O.w (miss), send request to disk, put the page into cache. Q: Which page to evict? If page is in cache (hit) reply directly from cache

18. Caching 18 Input: N pages in disk, and stream of infinite page requests. Online Algorithm: Decide which page to evict from cache when it’s full and there’s a miss. Goal: minimize the number of misses. Idea: LRU: Remove the Least Recently Used page

19. LRU with k = 3 19 412153441132451 miss

20. LRU with k = 3 20 412153441132451

21. LRU with k = 3 21 412153441132451 miss

22. LRU with k = 3 22 412153441132451

23. LRU with k = 3 23 412153441132451 miss

24. LRU with k = 3 24 412153441132451

25. LRU with k = 3 25 412153441132451 hit

26. LRU with k = 3 26 412153441132451 miss

27. LRU with k = 3 27 412153441132451

28. LRU with k = 3 28 412153441132451 miss

29. LRU with k = 3 29 412153441132451

30. LRU with k = 3 30 412153441132451 miss

31. LRU with k = 3 31 412153441132451

32. LRU with k = 3 32 412153441132451 hit

33. LRU with k = 3 33 412153441132451 miss

34. LRU with k = 3 34 412153441132451 so and so forth…

35. Competitive Ratio Claim 35 Claim: If the optimal sequence of choices for a size-h cache causes m misses. Then, for the same sequence of requests, LRU for a size-k cache causes misses Interpretation: If LRU had twice as much cache size as an algorithm OPT that knew the future, it would have at most twice the misses of OPT. Note will prove the claim for

36. Proof of Competitive Ratio 36 Recursively break the sequence of inputs into phases. Let t be the time when we see the (k+1)st different request. Phase 1: a 1 … a t-1 Let t` be the time we see the (k+1)st different element starting from a t Phase 2: a t … a t’-1 412153441132451

37. Proof of Competitive Ratio 37 412153441132451 k=3 Phase 1 Phase 2 Phase 3 Phase 4 By construction, each phase has k distinct requests. Q: At most how many misses does LRU have in each phase? A: k b/c even if it evicted everything in the k+1 st item, it would have at most k misses.

38. Proof of Competitive Ratio 38 412153441132451 Phase 1 Phase 2 Phase 3 Phase 4 Q: What’s the minimum misses that any size-h cache must have in any phase? A: k-h b/c k distinct items will be in the cache at different points during the phase, so at least k-h of them must trigger misses. Therefore the CR: k/k-h Q.E.D.

39. Outline For Today 1.Caching 2.Other Algorithms & Algorithmic Techniques Beyond CS 161 1.Parallel Algorithms (Finding Min, Bellman-Ford) 2.Linear Programming 3.Other Topics & Classes 39

40. 40 Parallel Algorithms Question: Which problems are parallelizable, which are inherently sequential?  Parallelizable: Connected components, sorting, selection, many computational geometry problems all have parallel algorithms  (Believed To Be) Inherently Sequential (P-complete):  DFS  Horn-satisfiability  Conway’s Game of Life, and others.

41. 41 2 Common Computational Models  Model 1: Shared Memory (PRAM): Single Machine Memory CPU 1 CPU 2 CPU k …  Each Time Step, each processor:  Can read a location of memory  Can write to a location of memory Q: How much time does it take to solve a computational problem with polynomial # processors?

42. 42 Example 1: Finding the min in an array Memory CPU (1,2) CPU (1,3) CPU (4, 5) …  There are n(n-1) processors, one for each pair (i, j)  Initially allocate an array of size n all 0  Step 1: Each cpu (i, j) compares i and j  If i < j, then write 1 to j o.w. write 1 to location i  Step 2: return the item whose output memory location is 0

43. 43 Example 1: Finding the min in an array Memory CPU (1,2) CPU (1,3) CPU (4, 5) …

44. 44 Model 2: Distributed Memory Machine 1 Machine 2 Machine k Input Partition 1 Input Partition 2 Input Partition k  Each machine  performs local computation  send/receives messages to/from other machines  can be synchronously or asynchronously Q: How much communication is necessary? Q: How many synchronizations is necessary?

45. Recap: Bellman-Ford 45 ∀ v, and for i={1, …, n} P (v, i) : shortest s v path with ≤ i edges (or null) L (v, i) : w(P (v, i) ) (and + ∞ for null paths) L (v, i) = min L (v, i-1) min u: ∃ (u,v) ∈ E : L (u, i-1) + c (u,v)

46. 46 Example: Distributed Bellman-Ford A C D F B E 5 3 -6 2 2

47. 47 Example: Distributed Bellman-Ford ∞ ∞ 0 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ A C D F B E 5 3 -6 2 2

48. 48 Example: Distributed Bellman-Ford ∞ ∞ 0 0 5 5 ∞ ∞ ∞ ∞ ∞ ∞ A C D F B E 5 3 -6 2 2

49. 49 Example: Distributed Bellman-Ford ∞ ∞ 0 0 5 5 8 8 7 7 A C D F B E 5 3 -6 2 2

50. 50 Example: Distributed Bellman-Ford 1 1 0 0 5 5 8 8 7 7 A C D F B E 5 3 -6 2 2

51. 51 Example: Distributed Bellman-Ford 1 1 0 0 5 5 0 0 7 7 A C D F B E 5 3 -6 2 2

52. 52 Example: Distributed Bellman-Ford 1 1 0 0 5 5 0 0 7 7 A C D F B E 5 3 -6 2 2 Termination When No Vertex Value Changes

53. 53 Parallel Algorithms  Parallel Computing: CS 149  A systems course.  Teaches parallel technologies but also algorithms.

54. Outline For Today 1.Caching 2.Other Algorithms & Algorithmic Techniques Beyond CS 161 1.Parallel Algorithms (Finding Min, Bellman-Ford) 2.Linear Programming 3.Other Topics & Classes 54

55. 55 Linear Programming (CS 261/361)  Optimization Problem of following structure: maximize x 1 + x 2 subject to x 1 + 2x 2 ≤ 1 2x 1 + x 2 ≤ 1 x 1 ≥ 0 x 2 ≥ 0

56. 56 Geometric Interpretation Constraint 2= 2x 1 + x 2 ≤ 1 x1x1 x2x2 Constraint 1= x 1 + 2x 2 ≤ 1 Feasible Solution Set **Opt Solution**

57. 57 Linear Programming Applications  Tons! Lots and lots of problems can be solved or approximated with LP!  Vertex Cover  Set Cover  Load Balancing  Lots of problems in manufacturing/operations research/finance, etc..  Covered in CS 261/361 **Invented By George Dantzig.**

58. Outline For Today 1.Caching 2.Other Algorithms & Algorithmic Techniques Beyond CS 161 1.Parallel Algorithms (Finding Min, Bellman-Ford) 2.Linear Programming 3.Other Topics & Classes 58

59. 59  Approximation Techniques (including LP, SP: Semidefinite Programming)  Many more Approximation Algorithms Approximation Algorithms (CS 261/CS 361)

60. 60  Commonly appearing general randomization techniques.  Probabilistic Method  Markov Chains  **Chernoff Bounds**  Very very cool/elegant algorithms! Randomized Algorithms (CS 365)

61. 61  Competitive Ratio  Average-Case Analysis  Instance Optimality  Smoothed Analysis, and others… Beyond Worst-Case (CS 369)

62. 62  Resource Lower Bounds on Computational Problems  Algorithms study what can be computed and with how much resource?  Complexity studies what cannot be computed with how much resource?  Turing Machines/P-NP/Circuit- Complexity/Randomized-Complexity/Polynomial Hierarchy/Space Complexity/PCP/One-way Functions Complexity Theory (CS 254)

63. 63  Power of Quantum Computers over Classic Computers  Quantum Teleportation  Quantum Circuits  Quantum Algorithms: Fourier Transform/Factoring/Search, etc…  Quantum Error Correction Quantum Computing (CS 259Q)

64. 64  Sequencing  Genome assembly  Gene sampling  Gene finding  Gene comparison  Gene regulation, etc… Computational Genomics (CS 262)

65. 65  Geometric modeling of physical objects  Search in high-dimensional spaces  Triangulation of a set of points  Image Processing/Graphics/Vision Computational Geometry (CS 268)

66. 66  Design and Analysis of Computational Problems In Strategic Environments  Key Analysis Concept: “Price of Anarchy”  Also Studies Complexities of Finding Equilibria  Auctions  Traffic Routing  Multi-agent Systems, and others… Algorithmic Game Theory (CS 364)

67. 67  Randomized Algorithms (CS 365) (Chernoff Bounds)  Linear Programming (CS 261/361)  Complexity Theory (CS 254) My Recommendations For Future Classes

68. 68  Tim Roughgarden  Keith Schwarz  Billy, Mike & Yiming Acknowledgements

69. 69 Thank you! Good luck in the final!