Recursion and Exhaustion

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Recursion and Exhaustion

• Hong Kong Olympiad in Informatics 2009
• Hackson Leung
• 2009-01-24

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Agenda

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

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Pre-requisite

• Know something beforehand

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Pre-requisite

• Function
• Mathematically, it gives output(s) from input(s)
• In short, y is the output, aka function of 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

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Pre-requisite

• Function
• Simple exercise
• Write a function f such that

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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)
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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()
int main()
System Stack
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Recursion

• Playing with functions!

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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.

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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

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Recursion

• Example
• Correct?

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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)

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Recursion

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

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Recursion

• Common recurrence relations
• Factorial
• Combinations
• More on Combinations
• Permutations
• Integral powers

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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...?

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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)?

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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

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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(k1)
• Call f(0)
• Base case(s)...?

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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

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Recursion

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

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Recursion

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

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Recursion

• Case Study 2
• Calculate
• Given
• Solution 2
• Efficient enough!

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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!

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Exhaustion

• Try your ...... BEST

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Exhaustion

• ??/??/???
• 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

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Exhaustion

• Analogy
• Consider you are selling Broadband service
• You want to promote it in a fixed building
• Known Facts
• You dont know who live inside are interested in
  your service
• You dont 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

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Exhaustion

• Exhaustion
• You dont know who live inside are interested in
  your service
• Yes, you dont know the direct answers in the
  problem
• You dont want to promote to the same person
  twice
• You dont waste time on checking same candidate
• Still, you want to find a way to promote your
  service and want all potential users to subscribe
• Thats what describes exhaustion

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Exhaustion

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

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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

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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

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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

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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

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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?

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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?

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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?

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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

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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 3050), 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

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Extras

• Only for attended trainee

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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)

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Tasks

• Prove to me that you really learnt something

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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