Theorem
Any one-tape DTM accepting the language
Lecture 24:
L = {w 0|w| wR | w ∈ {0, 1}∗ }
A quadratic runtime lower bound for one-tape
DTMs accepting
|w|
{w 0
2
runs Ω(n ) steps on inputs x ∈ L.
w | w ∈ {0, 1} }
R
∗
Corollary (1)
DTIME1-tape (n) ( DTIME1-tape (n2 )
Some interesting topics this course did not cover
Corollary (2)
DTIME1-tape (n) ( DTIME(n)
Lecture 24 : Overview
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Lecture 24 : Run-time lower bound for palindromes
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Definition
Given a one-tape DTM M, input w, and a tape cell
index i the crossing sequence SM (w, i) is the sequence
of states M is in when moving across the boundary
between cells i and i + 1.
i
i+1
Remark
∞
X
|SM (w, i)| = timeM (w)
i=−∞
if the head is not allowed to stay. Otherwise, 6.
q3
Proof: Exercise.
SM (w, i) = q3q1q5q4
q1
q5
q4
Lecture 24 : Run-time lower bound for palindromes
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Lecture 24 : Run-time lower bound for palindromes
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Lemma
Proof of Theorem
Suppose u = u1 u2 and v = v1 v2 are accepted by TM
M. If SM (u, |u1 |) = SM (v, |v1 |), then M also accepts
u1 v2 (and v1 u2 ).
Let M be a one-tape DTM with L(M) = L
Choose input w 0|w| wR and i with |w| 6 i < 2 · |w|
Proof
Head over u1 : M cannot distinguish input from u
Head over v2 : M cannot distinguish input from v,
because same state is reached when entering v2 part
This proceeds for the remainder of M’s run
M always in state it would be in for u or v
⇒ M accepts u1 v2
2
Lecture 24 : Run-time lower bound for palindromes
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For w 0 6= w and |w 0 | = |w| we have (Lemma!)
SM (w 0 0|w | w 0R , i) 6= SM (w 0|w| wR , i)
0
Therefore, w can be uniquely reconstructed from |w|, i,
and the according crossing sequence
Lecture 24 : Run-time lower bound for palindromes
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Proof continued
Algorithm that prints unique w for given m = |w|, i,
and crossing sequence s:
Proof continued
FORALL w ∈ {0, 1}m DO
simulate M on input v = w0m wR and
note crossing sequence at i
IF SM (v, i) = s THEN PRINT w; EXIT; END;
END
Pick w with K(w) > |w|
⇒ |w| 6 K(w) 6 log |w| + C · |s| + D
⇒ |s| > (|w| − log |w| − D)/C,
where constants C and D do not depend on w
Size of algorithm at most: (m and s are constants)
log m + C · |s| + D,
Lecture 24 : Run-time lower bound for palindromes
(C, D > 0 depend on M)
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Lecture 24 : Run-time lower bound for palindromes
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With little modifications the proof also establishes
quadratic time lower bounds for the following languages:
Proof continued
Then for x ∈ L with 3n = |x| and complex initial third:
timeM (x) >
∞
X
|SM (x, i)| >
i=−∞
>
2n−1
X
2n−1
X
{ww | w ∈ {0, 1}∗ }
{wwR | w ∈ {0, 1}∗ }
|SM (x, i)|
i=n
Corollary (3)
(n − log n − D)/C = Ω(|x|2 )
i=n
2
For any one-tape DTM that accepts L the average
runtime on uniformly distributed inputs of length n is
also quadratic.
Proof: Exercise
Lecture 24 : Run-time lower bound for palindromes
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Kolmogorov Complexity Literature
Lecture 24 : Run-time lower bound for palindromes
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Some interesting topics not covered
Logical Characterization of Complexity Classes
I
M. Li and P.M.B. Vitányi: An introduction to
Kolmogorov complexity and its applications,
Springer-Verlag 1993
W. Paul: Kolmogorov complexity and lower bounds.
Foundations of Computation Theory, 1979,
Akademie-Verlag, Berlin, 1979
I
I
I
“FO+TC = NLOG”
“FO+LFP = P”
“∃-SecO = NP”
“SecO+TC = PSPACE”
E.g. COLOR(3) ∼ class of 3-colorable graphs is definable
in ∃-SecO:
∃R1 ∃R2 ∃R3 [∀x(R1 (x) ∨ R2 (x) ∨ R3 (x)) ∧
(∧16i6=j63 ¬∃x Ri (x) ∧ Rj (x)) ∧
∀x∀y (E(x, y) ⇒ ∨i6=j Ri (x) ∧ Rj (y))]
Lecture 24 : Run-time lower bound for palindromes
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Lecture 24 : Run-time lower bound for palindromes
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Q/A & Final Exam
Approximability
I
I
NP-hard optimization problems
How close can we get in polynomial time?
E.g. Traveling Salesman Problem (TSP):
Q/A: Monday Dec. 12, 11-12pm in Ath 332
Given distances dij ∈ N0 and budget b, is there a closed
tour that visits all nodes with total distance 6 b?
Final Exam: Wednesday Dec. 14, 1-4pm in CSC 349
(closed book)
Decision problem TSP(D) is NP-complete
If there is a polynomial-time (ε < 1)–approximation
algorithm (relative error) for TSP(Opt), then P=NP
If the dij obey the triangle inequality, a
polynomial-time 2/3–approximation is known
Lecture 24 : Run-time lower bound for palindromes
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Lecture 24 : Run-time lower bound for palindromes
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