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Recursion and Exhaustion

Recursion and Exhaustion. Hong Kong Olympiad in Informatics 2009 Hackson Leung 2009-01-24. Agenda. Pre-requisite Recursion Exhaustion More...?. Pre-requisite. Know something beforehand. Pre-requisite. Function Mathematically, it gives output(s) from input(s)

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Recursion and Exhaustion

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  1. Recursion and Exhaustion Hong Kong Olympiad in Informatics 2009 Hackson Leung 2009-01-24

  2. Agenda • Pre-requisite • Recursion • Exhaustion • More...?

  3. Pre-requisite Know something beforehand

  4. Pre-requisite • Function • Mathematically, it gives output(s) from input(s) • In short, y is the output, aka functionof x • x is the parameter of the function • f gives what x can be related to y • In programming, function can give nothing • void in C/C++ • Procedure in Pascal

  5. Pre-requisite • Function • Simple exercise • Write a function f such that

  6. Pre-requisite • Stack • First In, Last Out (FILO) • Supported operations • Push to the bottom • Pop from the top • Learn more in future training • 2009-4-25 I Object O K H Container (Stack)

  7. Pre-requisite • In the computer • Each function is an object • Start a new function • Push • Return from a function • Pop • Container is the system stack Function int f() Running int main() System Stack

  8. Recursion Playing with functions!

  9. Recursion • Warmup • Mark Six • 6 integers, ranged from 1 to 49 inclusive • Numbers are not repeated • Write a program to generate all possible Mark Six results • 1 2 3 4 5 6 is not the same as 6 5 4 3 2 1 • 6 for loops!? • OK. There is a lottery called Mark Twelve….

  10. Recursion • Recursion • To recur means to happen again • In computer science, we say that a subroutine (function or procedure) is recursive when it calls itself one or more times • We call the programming technique by using recursive subroutine(s) as recursion • The correctness, time complexity and space complexity can be proved by mathematical induction

  11. Recursion • Example • Correct?

  12. Recursion • Example • Problematic • It does not stop! • When to stop? • Everybody knows that • So we do not recur on ! • Base case(s) / Terminating condition(s)

  13. Recursion • Recursion requires two components • Recurrence relation(s) (by howf relates to itself) • Base case(s) (by when should f not to recur)

  14. Recursion • Common recurrence relations • Factorial • Combinations • More on Combinations • Permutations • Integral powers

  15. Recursion • Case Study 1 • Give all permutations of the string ``ABC ’’ • ABC, ACB, BCA, BAC, CAB, CBA • Before deriving... • Parameter(s)? • Stage k • What does f(k) mean? • f(k) depends on...?

  16. Recursion • Case Study 1 • Give all permutations of the string ``ABC ’’ • ABC, ACB, BCA, BAC, CAB, CBA • Solution 1 • f(k) depends on f(k-1) for sure! • f(k) will add for each results from f(k-1) an unused character from ``ABC’’ and generate a new result • e.g. In f(2), place ``B’’ to ``A’’ and ``C’’, which are generated from f(1) • Call f(3) to finish the task • Base case(s)?

  17. Recursion • Case Study 1 • Give all permutations of the string ``ABC ’’ • ABC, ACB, BCA, BAC, CAB, CBA • Solution 1 • f(k) will add for each results from f(k-1) an unused character from ``ABC’’ and generate a new result • Not easy to implement, because... • You need to remember all results from f(k-1) • As well as the occurrence from each of them

  18. Recursion • Case Study 1 • Give all permutations of the string ``ABC ’’ • ABC, ACB, BCA, BAC, CAB, CBA • Solution 2 • In f(k), add an unused character into the current result, and directly go to f(k+1) • Call f(0) • Base case(s)...?

  19. Recursion • Case Study 1 • Give all permutations of the string ``ABC ’’ • ABC, ACB, BCA, BAC, CAB, CBA • Solution 2 • A more intuitive approach, in coding • No extra memory for storing result strings and occurence (``Grow-on-fly’’) • Note that the general idea is identical to solution 1

  20. Recursion • Case Study 2 • Calculate • Given: • Solution 1 • Simple! • What does f(k) mean? • Recurrence relation(s)? • Base case(s)? • Call f(?)?

  21. Recursion • Case Study 2 • Calculate • Given: • Solution 1 • Efficient enough? • Consider Y can be as large as 240...

  22. Recursion • Case Study 2 • Calculate • Given: • Solution 2 • Efficient enough!

  23. Recursion • Summary • Usually a (nice and) well defined function can be easily implemented by recursion • Different thinking can also lead to different complexity in coding • Different thinking can even lead to different complexity in time • NOTE: Recursion does not mean SLOW!

  24. Exhaustion Try your ...... BEST

  25. Exhaustion • 窮舉/窮尋/暴力法 • Also called Brute Force • Anyway it is not about violence... • Sometimes for finding the answers, you need to try all possible candidates and see if they are the answers

  26. Exhaustion • Analogy • Consider you are selling Broadband service • You want to promote it in a fixed building • Known Facts • You don’t know who live inside are interested in your service • You don’t want to promote to the same person twice • Still, you want to find a way to promote your service and want all potential users to subscribe

  27. Exhaustion • Exhaustion • You don’t know who live inside are interested in your service • Yes, you don’t know the direct answers in the problem • You don’t want to promote to the same person twice • You don’t waste time on checking same candidate • Still, you want to find a way to promote your service and want all potential users to subscribe • That’s what describes exhaustion

  28. Exhaustion • Exhaustion related problem • Constraints Satisfaction Problem (CSP) • Given all constraints, give any/all solution(s) that satisfy the constraint(s) • E.g. Sudoku

  29. Exhaustion • Case Study 1 • Irreversible Transform • Given a transform H, you can calculate y = H(x) • But given y, you cannot easily calculate x such that y = H(x), we call H is irreversible • Given y, tell me how many x can be transformed to y • Suppose x and y are 32bit signed integers

  30. Exhaustion • Case Study 1 • There is no explicit information about H, you can only try all possible x values • If the transformation is not complicated, the time complexity is still acceptable • Example transformation: Game of Life

  31. Exhaustion • Case Study 2 • Narrow Range Broadband • Given all clients’ positions as well as the profits that can be made from each of them • You can only setup one server station with limited transmission distance • Give the best possible position and the profit for the company

  32. Exhaustion • Case Study 2 • Give the best possible position and the profit for the company • Any definite answer, first? • The more the clients it covers, the better? • Map size: at most 100 x 100 (w x h) • Number of clients: at most 1000 (n) • Maximum distance: 200 (d) • Manhatten Distance:

  33. Exhaustion • Case Study 2 • Give the best possible position and the profit for the company • Solution 1 • Each position can be the best • For each position • Find from its reachable distance • Add profit if that position is a client • Complexity?

  34. Exhaustion • Case Study 2 • Give the best possible position and the profit for the company • Solution 1 • Complexity? • A bit slow • Any definite NON answer?

  35. Exhaustion • Case Study 2 • Give the best possible position and the profit for the company • Solution 2 • A client is served if a server can reach it within d units • Similarly, if a client can reach the server d units, it is served • By exploiting the fact that distance is symmetric, one can achieve an algorithm • Any faster algorithm?

  36. Exhaustion • Case Study 3 • You have a lock with password of length N • Each character is A to Z inclusive • You only know that the password contains distinct characters • Target: Unlock it • Discussion

  37. Exhaustion • Summary • If you cannot figure out fast way to solve a given problem (may or may not exist), try brute force • For small case (usually 30%~50%), they are designed for brute force purposes • Unless you proved that the only way to solve is to try all possible cases (HKOI2009 Dictionary), this method is bound to fail most data intensive cases

  38. Extras Only for attended trainee

  39. Extras • Recursion • Know the complexity – Master Theorem • Interesting problems that are solved by recursion • Solving some simple recurrence relations • Exhaustion • More classic examples • Not really slow – Prunning • Can be even faster – A.I. Thinking (Heuristics)

  40. Tasks Prove to me that you really learnt something

  41. Tasks • HKOJ • 2021 Lovely String • 2031 Narrow Range Broadband • 2062 Sudoku • 2076 SOS • 2086 Storage Box • 4013 Mahjong • 20750 8 Queen Chess Problem • IOI • 94 Day 2 Problem 1 The Clocks

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