Summary of Previous Class

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Summary of Previous Class

• Computable functions
• Runtime of a computation and space of a
  computation
• Big-O, small-o notation and growth of functions
• Runtime depends on the model of computation
• All reasonable models of computation are
  polynomially equivalent.

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Execution Times (for a Computer that Executes 109
Instructions per Second )
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The class P

• P ?k TIME(nk)
• P is robust (P is invariant for all models that
  are polynomially equivalent to a single-tape
  Turing machine).
• P roughly corresponds to the class of problems
  that can be efficiently solvable on a computer.

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Polinomial time - class P

• Stable All reasonable models of computation
  are polynomially equivalent. Then if a problem is
  in P with respect to model A (runs in time
  t(n)O(nc)) then it is in P with respect to model
  B (runs in time t(n)k (O(nc))k O(nkc)).
• Tractable (as speed increases we can solve bigger
  instances) problem can be solved in n3. You
  agreed to wait for time M. Max size nM1/3. A
  year later speed doubled. Max size increased to
  .
• Compare problem can be solved in 2n. Max size
  nlog2 M. Speed doubled max size log2 M 1
  increased by one bit !!!!

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Examples of Problems in P Graph Reachability

• Path ltG,s,tgt G is a directed graph that has
  a directed path from s to t
• Mark vertex s.
• Repeat the following until no additional nodes
  are marked
• Scan all edges in G. If there is an edge (a,b)
  with a marked a and an unmarked b, mark node b.
• If t is marked, accept. Otherwise, reject.

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In P or Not in P?

• Graph Connectedness Given a graph G is it
  connected?
• 3 min

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Examples of Problems in P Graph connectedness

• Path ltGgt G is a connected undirected graph
• Mark vertex s.
• Repeat the following until no additional nodes
  are marked
• Scan all edges in G. If there is an edge (a,b)
  with a marked a and an unmarked b, mark node b.
• If all vertexes are marked, accept. Otherwise,
  reject.

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In P or not in P?

• Substring Given strings s1 and s2 such that
  s1lt s2. Is s1 a substring of s2?
• 3 min

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Examples of Problems in P Substring

• Substring lts1,s2gt s1 is a substring of s2
• Compare every substring of s2 that have length
  s1 with the string s1. If there is a substring
  of s2 equal to s1 then accept. Otherwise, reject

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Hamiltonian-Cycle Problem

• Definition A Hamiltonian cycle in a directed
  graph is a cycle that goes through every vertex
  exactly once.
• Hamiltonian-Cycle Problem Given a directed graph
  G, does G have a Hamiltonian cycle?
• Algorithm Design an algorithm for solving the
  Hamiltonian-cycle problem. What is the running
  time of your algorithm?

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Clique Problem

• Definition A clique in an undirected graph is a
  subgraph in which every two vertices are
  connected by an edge. A k-clique is a clique that
  consists of k vertices.
• Clique Problem Given an undirected graph G and
  an integer k, does G have a k-clique?
• Algorithm Design an algorithm for solving the
  clique problem. What is the running time of your
  algorithm?

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Boolean Formulas

• Recursive definition of Boolean formulas
• Boolean variable is a formula
• If F is a Boolean formula then (?F) is also a
  Boolean formula
• If F1 and F2 are Boolean formulas then (F1?F2) is
  also a Boolean formula
• If F1 and F2 are Boolean formulas then (F1?F2) is
  also a Boolean formula.

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Satisfiability Problem

• Definition A Boolean formula is satisfiable if
  some assignment of 0s and 1s to its variables
  makes the formula true.
• Satisfiability Problem Given a Boolean formula
  F, if F is satisfiable or not?
• Algorithm Design an algorithm for solving the
  satisfiability problem. What is the running time
  of your algorithm?

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Verifiability

• Satisfiability Problem Given a Boolean formula
  F, if F is satisfiable or not?
• Verification Suppose someone presents a
  particular assignment A and states that A
  satisfies F.
• Can we verify this statement?
• What time is required for the verification?
• Verifying Versus Finding It is much easier to
  verify that a given assignment A satisfies F than
  to find such an assignment.

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Another Example of Verifiability

• Hamiltonian-Cycle Problem Given a directed graph
  G, does G have a Hamiltonian cycle?
• Verification Suppose someone presents a cycle in
  G and states that A is a Hamiltonian cycle.
• Can we verify this statement?
• What time is required for the verification?

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One More Example of Verifiability

• Clique Problem Given an undirected graph G and
  an integer k, does G have a k-clique?
• Verification Suppose someone presents a subgraph
  in G and states that this subgraph is a k-clique.
• Can we verify this statement?
• What time is required for the verification?

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Verifiers

• Definition
• A verifier for a language L is an algorithm V
  such that
• L s V accepts lts,pgt for some binary string
  p
• The additional string p is called a proof or
  certificate.

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Polynomial-TimeVerifiers

• Input string s
• Proof p
• Definition A verifier is called a
  polynomial-time verifier if it runs in polynomial
  time in the length of s.
• Definition A language L is called a polynomially
  verifiable L has a polynomial-time verifier.

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The Class NP

• Definition NP is the class of languages that
  have polynomial time verifiers.
• Examples of problems in NP
• Hamiltonian-cycle problem
• Clique problem
• Satisfiability problem

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Nondeterministic Turing Machines

• Nondeterministic Turing machines Similar to
  usual Turing machines, but there may be more than
  one instruction with the same left side.
• NTM M decides L if
• for every w?L there exists at least one
  computation that accepts w
• for u?L every computation of M rejects u.
• NTIME(t(n))L Lis a language decided in
  O(t(n)) time by an NTM

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Poly Verifier for L NTM M deciding L

• Theorem A language is in NP if and only if it is
  decided by a nondeterministic polynomial-time
  Turing machine.
• (?) L have a verifier V that runs in time nk.
• On an input x NTM M first generates
  non-deterministically a binary string c on a
  working tape, then it copies it eo the input tape
  and runs V on ltx,cgt.

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Poly Verifier for L NTM M deciding L
(continued).

• (?) NTM M decides L in time nk.
• Verifier V is a modification of M. On an input
  ltx,cgt it executes instructions of M on ltxgt until
  it must make a choice. Then V reads a bit of c V
  takes first possibility on 0, and rejects it on
  1. V continues to the next bit until only one
  possibility left. If so V takes it.
• Let b be a maximal number of choices M has at any
  time when computing on ltxgt. A string of length
  at most bnk is needed to model a computation.

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Exponential-Time Algorithms for Problems in NP

• Theorem Each language in NP can be decided by a
  deterministic exponential-time Turing machine.
• Proof idea NTM M decides L in nk-time. M
  simulate computation of M on ltxgt by walking all
  the computational tree of M on ltxgt. Let b be a
  maximal number of choices M has at any time when
  computing on ltxgt. Then the tree size is at most
  . M walks the tree using BFS M in time