CSCI 2670 Introduction to Theory of Computing

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CSCI 2670Introduction to Theory of Computing
November 17, 2005
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Agenda

• Today
• Finish Section 7.2
• Start Section 7.3

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Announcement

• Homework due next Tuesday (11/22)
• 7.3 a, 7.4, 7.6 (union only), 7.9, 7.12
• 1st edition
• 7.3a, 7.4, 7.6 (union only), 7.10, 7.12

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Last class

• Introduced the class P
• P Uk TIME(nk)
• Proved two languages are in P
• Binary tree query
• PATH

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Another problem in P

• RELPRIME ltx,ygt x and y are relatively prime
• RELPRIME ? P
• How can we show this?
• We cannot find prime factorization of x and y and
  compare
• Use Euclidean algorithm for finding gcd(x,y)
• ltx,ygt?RELPRIME iff gcd(x,y) 1

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Euclidean algorithm

• E On input ltx,ygt, where x and y are natural
  numbers in binary
• Repeat until y 0
• Assign x ? x mod y
• Exchange x and y
• Output x

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Solution to RELPRIME

• R On input ltx,ygt, where x and y are natural
  numbers in binary
• Run E on ltx,ygt
• If result is 1, accept
• Else reject

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Verify solution is in P

• Is R a decider?
• Yes
• How long will it take R to run?
• Execution of R is dominated by E
• Note that if x gt y, then (x mod y) lt x/2
• If x 2y, then x mod y lt y x/2
• If x lt 2y, then x mod y x y lt x/2
• Therefore loop in E is executed O(log2x) times
• Is this polynomial in the length of the input?
• log2x O(n), where n length of input

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Solving vs. verifying

• What if we dont know how to solve the problem in
  O(nk) time?
• Given a problem and a potential solution, can we
  verify the solution is correct?

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Example

• The bin-packing problem
• Given a set of n items with weights w1, w2, ,
  wn, and k bins that can hold a maximum weight of
  1, can we place these items the bins?
• There is no known O(nk) solution to this problem
• What if we have a potential solution
• b1, b2, , bn 1? bi ? k
• Can we verify it in O(nk) time?

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Verifier

• M On input ltw1, , wn, b1, , bn, kgt
• Initialize s1, s2, , sk to 0
• For i 1, , n
• if bi ? 1, 2, , k reject
• sb_i sb_i wi
• if sb_i gt 1 reject
• Next i
• Accept

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The class NP

• Definition A verifier for a language A is an
  algorithm V, where
• AwV accepts ltw,cgt for some string c
• The string c is called a certificate of
  membership in A.
• Definition NP is the class of languages that
  have polynomial-time verifiers.

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Why NP?

• The N in NP stands for non-deterministic
• Any language in NP can be non-deterministically
  solved in polynomial time using the verifier
• Guess the certificate
• Verify

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Other problems in NP

• The vertex cover problem
• Given a graph G ltV,Egt and a number k in N, does
  there exist a subset V of V such that
• V k
• For every (u,v)?E, either u?V or v?V
• There is no known polynomial solution to this
  problem

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Vertex cover

• Can we verify vertex cover in polynomial time?
• Yes
• What should the certificate be?
• The subset V
• How do we verify?
• Check V k
• Test that each (u,v)?E has u?V or v?V
• Takes O(Ek) time

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Example 1

• Find a verifier for the subset-sum problem
• Given a finite set S ? N and a target t ? N, does
  there exist a subset S ? S such that the sum of
  all elements in S is equal to t?
• V takes input S
• Checks S ? S
• O(S x S) O(S2)
• Check Ss?S s t
• O(S) O(S)

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Example 2

• Find a verifier for the traveling-salesman
  problem
• Given a weighted graph G (i.e., a graph where
  each edge has a associated weight) and a distance
  d, does there exist a cycle through the graph
  that visits each vertex exactly once (except for
  the start/end vertex) and has a total distance d?

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Example 2

• Find a verifier for the traveling-salesman
  problem
• V takes input ltvi1,vi2,,vingt
• Check the input is a permutation of the nodes of
  the graph
• O(V2)
• Check S1kltnvik d
• O(ExV)