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Algorithms

In the Josephus problem, Flavius Josephus and his comrades decide to form a circle and kill themselves every third person to avoid capture by the Romans. Can you find the safe place? Discover a faster algorithm to solve the problem.

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Algorithms

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  1. Algorithms 12/19/2019 Che-Rung Lee CS135601 Introduction to Information Engineering

  2. Josephus problem Flavius Josephus is a Jewish historian living in the 1st century. According to his account, he and his 40 comrade soldiers were trapped in a cave, surrounded by Romans. They chose suicide over capture and decided that they would form a circle and start killing themselves using a step of three. As Josephus did not want to die, he was able to find the safe place, and stayed alive with his comrade, later joining the Romans who captured them. CS135601 Introduction to Information Engineering

  3. Can you find the safe place? 40 41 1 2 39 3 38 4 37 5 36 6 35 7 34 8 33 9 32 Safe place 10 31 11 30 Can you find the safe place FASTER? 12 29 13 28 14 27 15 26 16 25 17 24 18 23 19 22 21 20 CS135601 Introduction to Information Engineering

  4. Algorithm • An effective method for solving a problem using a finite sequence of instructions. • It need be able to solve the problem. (correctness) • It can be represented by a finite number of (computer) instructions. • Each instruction must be achievable (by computer) • The more effective, the better algorithm is. • How to measure the “efficiency”? CS135601 Introduction to Information Engineering

  5. Outline • The first algorithm • Model, simulation, algorithm primitive • The second algorithm • Algorithm discovery, recursion • The third algorithm • Solving recursion, mathematical induction • The central role of computer science • Why study math? CS135601 Introduction to Information Engineering

  6. The first algorithm CS135601 Introduction to Information Engineering

  7. A simpler version • Let’s consider a similar problem • There are n person in a circle, numbered from 1 to n sequentially. • Starting from the number 1 person, every 2nd person will be killed. • What is the safe place? • The input is n, the output f(n) is a number between 1 and n. • Ex: f(8) = ? 1 2 8 3 7 4 6 5 CS135601 Introduction to Information Engineering

  8. The first algorithm: simulation • We can find f(n) using simulation. • Simulation is a process to imitate the real objects, states of affairs, or process. • We do not need to “kill” anyone to find f(n). • The simulation needs • (1) a model to represents “n people in a circle” • (2) a way to simulate “kill every 2nd person” • (3) knowing when to stop CS135601 Introduction to Information Engineering

  9. Model n people in a circle • We can use “data structure” to model it. • This is called a “circular linked list”. • Each node is of some “struct” data type • Each link is a “pointer” 1 8 2 7 3 struct PIC { int ID; struct PIC *next; } 4 6 5 CS135601 Introduction to Information Engineering

  10. Kill every 2nd person • Remove every 2nd node in the circular liked list. • You need to maintainthe circular linked structure after removing node 2 • The process can continue until … 1 8 2 7 3 6 4 5 CS135601 Introduction to Information Engineering

  11. Knowing when to stop • Stop when there is only one node left • How to know that? • When the *next is pointing to itself • It’s ID is f(n) • f(8) = 1 1 8 7 3 6 4 5 CS135601 Introduction to Information Engineering

  12. How fast can it compute f(n)? • Since the first algorithm removes one node at a time, it needs n-1 steps to compute f(n) • The cost of each step (removing one node) is a constant time  (independent of n) • So the total number of operations to find f(n) is (n-1) • We can just represent it as O(n). CS135601 Introduction to Information Engineering

  13. The Second Algorithm CS135601 Introduction to Information Engineering

  14. Can we do better? • Simulation is a brute-force method. Faster algorithms are usually expected. • The scientific approach • Observing some cases • Making some hypotheses (generalization) • Testing the hypotheses • Repeat 1-3 until success CS135601 Introduction to Information Engineering

  15. Observing a case • Let’s checkout the case n = 8. • What have you observed? • All even numbered people die • This is true for all kinds of n. • The starting point is back to 1 • This is only true when n is even • Let’s first consider the case when n is even 1 2 8 3 7 4 6 5 CS135601 Introduction to Information Engineering

  16. 1 4 2 3 n is even • For n=8, after the first “round”, there are only 4 people left. • If we can solve f(4), can weuse it to solve f(8)? • If we renumber the remaining people, 11, 32, 53, 74, it becomes the n=4 problem. • So, if f(4)=x, f(8) =2x – 1 1 2 8 3 7 4 6 5 CS135601 Introduction to Information Engineering

  17. 4 1 3 2 n is odd • Let’s checkout the case n=9 • The starting point becomes 3 • If we can solve f(4), can weuse it to solve f(9)? • If we renumber the remaining people, 31, 52, 73, 94,it becomes the n=4 problem. • So, if f(4)=x, f(9) =2x+1 1 9 2 8 3 7 4 6 5 CS135601 Introduction to Information Engineering

  18. Recursive relation • Hypothesis: If f(n)=x, f(2n)=2x–1. If f(n)=x, f(2n+1)=2x+1. • How to prove or disprove it? • This is called a recursive relation. • You can design a recursive algorithm • Compute f(8) uses f(4); • Compute f(4) uses f(2); • Compute f(2) uses f(1); • f(1) =1. Why? CS135601 Introduction to Information Engineering

  19. Recursive function call Josephus(9) int Josephus (n) { /* base case */ if (n == 1) return 1; if (n%2 == 0) /*n is even*/ return 2*Josephus (n/2)-1; else/*n is odd*/ return 2*Josephus((n–1)/2)+1; } output fn fn=3 call Josephus(4) fn  2*fn+1 fn=1 call Josephus(2) fn  2*fn–1 fn=1 call Josephus(1) fn  2*fn–1 fn=1 if (n = 1) return 1 CS135601 Introduction to Information Engineering

  20. How fast can it compute f(n)? • n reduces at least half at each step. • Need log2(n) steps to reach n=1. • Each step needs a constant number of operations . • The total number of operations is log2(n) CS135601 Introduction to Information Engineering

  21. The Third Algorithm CS135601 Introduction to Information Engineering

  22. Can we do better? • Can we solve the recursion? • If we can, we may have a better algorithm to find f(n) • Remember: • observation, hypotheses, verification CS135601 Introduction to Information Engineering

  23. 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 Let’s make more observations • n = 6, f(6) = ? • n = 7, f(7) = ? • n = 8, f(8) = ? • n = 9, f(9) = ? 5 • n = 2, f(2) = ? • n = 3, f(3) = ? • n = 4, f(4) = ? • n = 5, f(5) = ? 1 3 7 1 1 3 3 Have you observed the pattern of f(n)? CS135601 Introduction to Information Engineering

  24. What are the patterns? • f(1)=f(2)=f(4)=f(8)=f(16)=1. What are they in common? • If we group the sequence [1], [2,3], [4,7],[8,15], f(n) in each group is a sequence of consecutive odd numbers starting from 1. • Let k = n – the first number in n’s groupWhat is the pattern of k? They are power of 2, 2m. f(n)=2k+1 CS135601 Introduction to Information Engineering

  25. Guess the solution • f(n) = 2k+1 • k = n – the first number in n’s group • The first number in n’s group is 2m. 2m  n < 2m+1 • How fast can you find m? • (1) use “binary search” on the bit pattern of n • Since n has log2(n) bits, it takes log2(log2(n)) steps. • (2) use “log2” function and take its integer part • It takes constant time only. (independent of n.) CS135601 Introduction to Information Engineering

  26. -4 10 -6 10 -8 10 3 4 5 6 7 8 10 10 10 10 10 10 Running time of three algorithms 2 10 Algorithm 1 Running time Algorithm 2 0 10 Algorithm 3 -2 10 n CS135601 Introduction to Information Engineering

  27. The Central Role of Computer Science Why study math? CS135601 Introduction to Information Engineering

  28. Algorithms we studied so far • In chap 1, binary and decimal conversion, 2’s complement calculation, data compression, error correction, encryption… • In chap 2, we studied how to use computer to implement algorithms • In homework 4, we have problems for pipeline, prefix sum (parallel algorithm), and virtual memory page replacement (online algorithm) CS135601 Introduction to Information Engineering

  29. In chap 3, we see some examples of using algorithms to help manage resources • Virtual memory mapping, scheduling, concurrent processes execution, etc. • In chap 4, we learned protocols(distributed algorithms, randomized algorithms) • CSMA/CD, CSMA/CA, routing, handshaking, flow control, etc. • In chap 5, we learned some basic algorithmic strategies, such as simulation, recursion, and how to analyze them CS135601 Introduction to Information Engineering

  30. Learning algorithms • Every class in CS has some relations with algorithms • But some classes are not obviously related to algorithms, such as math classes. • 微積分,離散數學,線性代數,機率,工程數學,… • Why should we study them? • They are difficult, and boring, and difficult… CS135601 Introduction to Information Engineering

  31. Why study math? • Train the logical thinking and reasoning • Algorithm is a result of logical thinking: correctness, efficiency, … • Good programming needs logical thinking and reasoning. For example, debugging. • Learn some basic tricks • Many beautiful properties of problems can be revealed through the study of math • Standing on the shoulders of giants CS135601 Introduction to Information Engineering

  32. Try to solve those problems • Given a network of thousands nodes, find the shortest path from node A to node B • The shortest path problem (離散數學) • Given a circuit of million transistors, find out the current on each wire • Kirchhoff's current law (線性代數) • In CSMA/CD, what is the probability of collisions? • Network analysis (機率) CS135601 Introduction to Information Engineering

  33. 微積分 • Limits, derivatives, and integrals of continuous functions. • Used in almost every field • 網路分析, 效能分析 • 訊號處理, 影像處理, 類比電路設計 • 科學計算, 人工智慧, 電腦視覺, 電腦圖學 • … • Also the foundation of many other math • 機率, 工程數學 CS135601 Introduction to Information Engineering

  34. 離散數學 • Discrete structure, graph, integer, logic, abstract algebra, combinatorics • The foundation of computer science • Every field in computer science needs it • Particularly, 演算法, 數位邏輯設計, 密碼學, 編碼理論, 計算機網路, CAD, 計算理論 • Extended courses • Special topics on discrete structure, graph theory, concrete math CS135601 Introduction to Information Engineering

  35. 線性代數 • Vectors, matrices, vector spaces, linear transformation, system of equations • Used in almost everywhere when dealing with more than one variables • 網路分析, 效能分析 • 訊號處理, 影像處理 • 科學計算, 人工智慧, 電腦視覺, 電腦圖學 • CAD, 電路設計 CS135601 Introduction to Information Engineering

  36. 機率 • Probability models, random variables, probability functions, stochastic processes • Every field uses it • Particularly, 網路分析, 效能分析, 訊號處理, 影像處理, 科學計算, 人工智慧, 電腦視覺… • Other examples: randomized algorithm, queue theory, computational finance, performance analysis CS135601 Introduction to Information Engineering

  37. 工程數學 • A condensed course containing essential math tools for most engineering disciplines • Partial differential equations, Fourier analysis, Vector calculus and analysis • Used in the fields that need to handle continuous functions • 網路分析, 效能分析, 訊號處理, 影像處理, 科學計算, 人工智慧, 電腦視覺, CAD, 電路設計… CS135601 Introduction to Information Engineering

  38. Reference • The Josephus problem: • Graham, Knuth, and Patashnik, “Concrete Mathematics”, section 1.3 • http://en.wikipedia.org/wiki/Josephus_problem • http://mathworld.wolfram.com/JosephusProblem.html • Algorithm representation • Textbook: 5.2 • The related courses are those listed in the last part of slides, and of course, 演算法設計,高等程式設計實作 CS135601 Introduction to Information Engineering

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