CSCI 333 The Hard and Impossible

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CSCI 333The Hard and Impossible

• Chapter 15
• 4 and 6 December 2002

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The Worst Sort

• BumbleSort(int A, int Size)
• while (NotSorted(A, Size))
• int i Random(Size)
• unt j Random(Size)
• Swap(AI, Aj)
• Solves the problems in average time of O(n!)
• That is, in centuries for decent sized lists

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The Best Sort

• n! different permutations of the data
• Decision tree could cover all possibilities
• n! leaves
• O(log n!) height
• O(log n!) O(n log n)
• No sort can be better than O(n log n)

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Reduction
Input transformation
Problem A
Problem B
Output transformation
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Whats wrong with this picture?
Input transformation O(n)
Sorting
Pairing O(n)
Output transformation O(n)
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Why some problemsare unsolvable

• More problems than programs
• Programs are countable
• Problems are uncountable
• Halting problem is unsolvable
• Cant write a function Halt such that
• Halt(f) is true if f(0) returns a value
• Halt(f) is false if f(0) gets in a loop

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Post Correspondence Problem

• Given two equal size vectors of strings
• S s1, s2, , sn
• T t1, t2, , tn
• Can you find a sequence of indices
• i1, i2, im
• Such that
• si1 si2 sim ti1 ti2 tim
• Or use the card analogy

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Hard Problems

• Requires ?(cn) for c gt 1

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Examples of proven hard problems

• Few exist
• And they seem rather contrived
• So, whats the problem?
• For the lack of problems

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Knapsack problem

• Given a collection, W, of items of weight
• w1, w2, wn
• And a target weight K
• Is there are subset of W whose sum is K?

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Exponential solution

• bool KnapSack(int W, int N, int K)
• bool ChoicesN
• initPerm(Choices, N) // all false
• foundSolution false
• morePerms false
• while (! foundSolution morePerms) // 2N
• if (exactFit(W, Choices, N, K))
• foundSolution true
• else
• morePerms nextPerm(Choices, N)
• return foundSolution

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Exponential solution part 2

• void initPerm(int Choices, int N)
• for (int i0 iltN i)
• ChoicesN false
• bool nextPerm(int Choices, int N)
• int i
• for (int i0 iltN Choicesi i)
• Choicesi false
• if (i N)
• return false // All permutations are done!
• ChoicesI true
• return true

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Exponential solution part 3

• bool exactFit(int W, bool C,
• int N, int K)
• int sum 0
• for (int i0 iltN i)
• if Choicesi)
• sum sum Wi
• return sum W

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Exponential solution part 3

• bool exactFit(int W, bool C,
• int N, int K)
• int sum 0
• for (int i0 iltN i)
• if Choicesi)
• sum sum Wi
• return sum W

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Non-deterministic solution

• Quija bool KnapSack(int W, int N,
• int K)
• bool ChoicesN
• for (int I0 IltN I)
• ChoicesI Quija(false, true)
• return exactFit(W, Choices, N, K)

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P NP?

• Clearly NP includes P
• Is there a problem that is in NP,
• but not in P?

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NP Completeness

• A Problem is NP complete if
• It is NP
• All other NP programs can be reduced to it
• In polynomial time

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NP Complete Problems

• Knapsack
• Traveling salesman
• Hamiltonian cycle
• Boolean satisfiability
• Class scheduling problem
• Minesweeper consistency problem