CS6014 - Computability and Complexity

Meetings  

• Theme 1 : Computability Theory - 20 meetings

  • Meeting 01 : Mon, Aug 01, 08:00 am-08:50 am

  Administrative announcements. The overview of the course. Historical perspective. Models of computation. Algorithms vs Turing machines. Languages vs Computational Problems.

  References	Lectuer 28 in [K1 Book].
  Exercises
  Reading

  Administrative announcements. The overview of the course. Historical perspective. Models of computation. Algorithms vs Turing machines. Languages vs Computational Problems.
  

  References

  :

  Lectuer 28 in [K1 Book].

  • Meeting 02 : Tue, Aug 02, 12:00 pm-12:50 pm

  The Turing machine model in formal terms. Acceptance. Configurations. Examples. Tape reduction in Turing machines.

  References	Lectuer 28,29,30 in [K1 Book].
  Exercises
  Reading

  The Turing machine model in formal terms. Acceptance. Configurations. Examples. Tape reduction in Turing machines.
  

  References

  :

  Lectuer 28,29,30 in [K1 Book].

  • Meeting 03 : Thu, Aug 04, 11:00 am-11:50 am

  Two computational problems. Total Turing machines. Decidability vs Semi-decidability.

  References	Lecture 29,30 in [K1 Book]
  Exercises
  Reading

  Two computational problems. Total Turing machines. Decidability vs Semi-decidability.
  

  References

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  Lecture 29,30 in [K1 Book]

  • Meeting 04 : Fri, Aug 05, 10:00 am-10:50 am

  Encoding of Turing machines. Two consequences. Universal Turing machines, and Diagonalization. A proof that the halting Problem is undecidable. Variants of diagonalization.

  References	Lecture 31 in [K1 Book]
  Exercises
  Reading

  Encoding of Turing machines. Two consequences. Universal Turing machines, and Diagonalization. A proof that the halting Problem is undecidable. Variants of diagonalization.
  

  References

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  Lecture 31 in [K1 Book]

  • Meeting 05 : Mon, Aug 08, 08:00 am-08:50 am

  HP-42 is also undecidable. Notion of reductions. MP vs HP problem.

  References	Lecture 32-33 in [K1 Book]
  Exercises
  Reading

  HP-42 is also undecidable. Notion of reductions. MP vs HP problem.
  

  References

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  Lecture 32-33 in [K1 Book]

  • Meeting 06 : Tue, Aug 09, 12:00 pm-12:50 pm

  More undecidable problems - EMPTY, TOTAL, REG, CFL, DEC. Abstracting the proof idea. Statement of Rice-Myhill-Shapira theorem.

  References	Lecture 34 in [K1 Book]
  Exercises
  Reading

  More undecidable problems - EMPTY, TOTAL, REG, CFL, DEC. Abstracting the proof idea. Statement of Rice-Myhill-Shapira theorem.
  

  References

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  Lecture 34 in [K1 Book]

  • Meeting 07 : Thu, Aug 11, 11:00 am-11:50 am

  Proof of Rice-Myhill-Shapira theorem. Example applications. REG, DEC, EMPTY, FIN, CFL.

  References	Lecture 34 in [K1 Book]
  Exercises
  Reading

  Proof of Rice-Myhill-Shapira theorem. Example applications. REG, DEC, EMPTY, FIN, CFL.
  

  References

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  Lecture 34 in [K1 Book]

  • Meeting 08 : Fri, Aug 12, 10:00 am-10:50 am

  Beyond semi-decidables The class Co-SD. D = SD intersection Co-SD. FIN is not in SD and FIN is not in CoSD.

  References	Lecture 34 in [K1 Book]
  Exercises
  Reading

  Beyond semi-decidables The class Co-SD. D = SD intersection Co-SD. FIN is not in SD and FIN is not in CoSD.
  

  References

  :

  Lecture 34 in [K1 Book]

  • Meeting 09 : Tue, Aug 16, 12:00 pm-12:50 pm

  SD-completeness of MP. Beyond SD an Co-SD. Rice's Second Theorem about monotone properties of Semi-decidable languages.

  References	Supplementary Lecture J in [K1 Book]
  Exercises
  Reading

  SD-completeness of MP. Beyond SD an Co-SD. Rice's Second Theorem about monotone properties of Semi-decidable languages.
  

  References

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  Supplementary Lecture J in [K1 Book]

  • Meeting 10 : Thu, Aug 18, 11:00 am-11:50 am

  What languages become decidable if HP is decidable? The example beyond SD and CoSD. MPXOR. Oracle Turing machines. Relative Computability. Turing reductions.

  References
  Exercises
  Reading

  What languages become decidable if HP is decidable? The example beyond SD and CoSD. MPXOR. Oracle Turing machines. Relative Computability. Turing reductions.
  

  References

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  None

  • Meeting 11 : Fri, Aug 19, 10:00 am-10:50 am

  Languages Semi-decidable in HP. Relativizing proofs. The Arithmetic Hierarchy. Diagonalization proof relativizes. Arithmetic hierarchy is strict.

  References
  Exercises
  Reading

  Languages Semi-decidable in HP. Relativizing proofs. The Arithmetic Hierarchy. Diagonalization proof relativizes. Arithmetic hierarchy is strict.
  

  References

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  None

  • Meeting 12 : Mon, Aug 22, 08:00 am-08:50 am

  Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages. Quantifier characterization of AH. Placing the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy.

  References
  Exercises
  Reading

  Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages. Quantifier characterization of AH. Placing the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy.
  

  References

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  None

  • Meeting 13 : Tue, Aug 23, 12:00 pm-12:50 pm

  FIN is complete for the second level(Sigma-2) of the hierarchy.

  References
  Exercises
  Reading

  FIN is complete for the second level(Sigma-2) of the hierarchy.
  

  References

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  None

  • Meeting 14 : Thu, Aug 25, 11:00 am-11:50 am

  Proof of the Quantifier characterization.

  References	Notes sent to the mailing list.
  Exercises
  Reading

  Proof of the Quantifier characterization.
  

  References

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  Notes sent to the mailing list.

  • Meeting 15 : Fri, Aug 26, 10:00 am-10:50 am

  Proof of the Quantifier Characterization of Arithmetic Hierarchy.

  References
  Exercises
  Reading

  Proof of the Quantifier Characterization of Arithmetic Hierarchy.
  

  References

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  None

  • Meeting 16 : Mon, Aug 29, 08:00 am-08:50 am

  Peano's Axioms, Godels Incompleteness Theorem. Computability based non-constructive proof.

  References
  Exercises
  Reading

  Peano's Axioms, Godels Incompleteness Theorem. Computability based non-constructive proof.
  

  References

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  None

  • Meeting 17 : Tue, Aug 30, 12:00 pm-12:50 pm

  Godel's Incompleteness Theorem - details of the proof.

  References
  Exercises
  Reading

  Godel's Incompleteness Theorem - details of the proof.
  

  References

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  None

  • Meeting 18 : Thu, Sep 01, 11:00 am-11:50 am

  Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple sets cannot be SD-complete. Post's construction of Simple sets.

  References
  Exercises
  Reading

  Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple sets cannot be SD-complete. Post's construction of Simple sets.
  

  References

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  None

  • Meeting 19 : Fri, Sep 02, 10:00 am-10:50 am

  Productive Sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive.

  References
  Exercises
  Reading

  Productive Sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive.
  

  References

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  None

  • Meeting 20 : Tue, Sep 06, 12:00 pm-12:50 pm

  Every SD-complete set is co-productive.

  References
  Exercises
  Reading

  Every SD-complete set is co-productive.
  

  References

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  None

• Theme 2 : Resource Bounded Complexity Theory - 35 meetings

  • Meeting 21 : Thu, Sep 08, 11:00 am-11:50 am

  Shorter Lecture : Blum's Axioms. Examples and Non-examples of resources.

  References
  Exercises
  Reading

  Shorter Lecture : Blum's Axioms. Examples and Non-examples of resources.
  

  References

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  None

  • Meeting 22 : Fri, Sep 09, 10:00 am-10:50 am

  More Examples. Complexity classes based on a resource bound.

  References
  Exercises
  Reading

  More Examples. Complexity classes based on a resource bound.
  

  References

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  None

  • Meeting 23 : Thu, Sep 15, 11:00 am-11:50 am

  Borodin's Gap Theorem.

  References
  Exercises
  Reading

  Borodin's Gap Theorem.
  

  References

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  None

  • Meeting 24 : Fri, Sep 16, 10:00 am-10:50 am

  Shorter Lecture : Tape reduction and time and space. Tape Compression Theorem.

  References
  Exercises
  Reading

  Shorter Lecture : Tape reduction and time and space. Tape Compression Theorem.
  

  References

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  None

  • Meeting 25 : Mon, Sep 19, 08:00 am-08:50 am

  Linear Speedup Theorem. Constructibility of functions (time and space). Time Hierarchy Theorem. Space Hierarchy Theorem. The ideas of the proofs.

  References
  Exercises
  Reading

  Linear Speedup Theorem. Constructibility of functions (time and space). Time Hierarchy Theorem. Space Hierarchy Theorem. The ideas of the proofs.
  

  References

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  None

  • Meeting 26 : Tue, Sep 20, 12:00 pm-12:50 pm

  Detailed proof of the hierarchy theorem. Importance of the tape reduction. Hennie-Stearns improved Tape reduction theorem for 2 tapes. Optimality of the Hartmani-Stearns Tape reduction.

  References
  Exercises
  Reading

  Detailed proof of the hierarchy theorem. Importance of the tape reduction. Hennie-Stearns improved Tape reduction theorem for 2 tapes. Optimality of the Hartmani-Stearns Tape reduction.
  

  References

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  None

  • Meeting 27 : Thu, Sep 22, 11:00 am-11:50 am

  Optimality of k-tape to 1-tape reduction via Crossing Sequence Arguments. An application to show that TMs using space less than loglog n accept only regular languages.

  References
  Exercises
  Reading

  Optimality of k-tape to 1-tape reduction via Crossing Sequence Arguments. An application to show that TMs using space less than loglog n accept only regular languages.
  

  References

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  None

  • Meeting 28 : Fri, Sep 23, 10:00 am-10:50 am

  Complexity classes. Notion of Efficiency. Edmond's proposition. Union Theorem. Complexity classes P, E and EXP. Space Complexity classes L, PSPACE, and EXPSPACE. From time bound to space bound and from a space bound to a time bound.

  References
  Exercises
  Reading

  Complexity classes. Notion of Efficiency. Edmond's proposition. Union Theorem. Complexity classes P, E and EXP. Space Complexity classes L, PSPACE, and EXPSPACE. From time bound to space bound and from a space bound to a time bound.
  

  References

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  None

  • Meeting 29 : Mon, Sep 26, 08:00 am-08:50 am

  CLIQUE, EXACTCLIQUE, REACH, GI - four computational problems. EXP time algorithm for all of them. PSPACE algorithms for all of them. Poly time algorithm for REACH. EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE.

  References
  Exercises
  Reading

  CLIQUE, EXACTCLIQUE, REACH, GI - four computational problems. EXP time algorithm for all of them. PSPACE algorithms for all of them. Poly time algorithm for REACH. EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE.
  

  References

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  None

  • Meeting 30 : Tue, Sep 27, 12:00 pm-12:50 pm

  Non-determinism, definition of time and space. The Simulation and Containments.

  References
  Exercises
  Reading

  Non-determinism, definition of time and space. The Simulation and Containments.
  

  References

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  None

  • Meeting 31 : Thu, Sep 29, 11:00 am-11:50 am

  Quest for structure within NP. A structural observation about the NP algorithm for Clique. Can that be exploited? No, it seems to be there for every NP algorithm - quantifier characterization for NP.

  References
  Exercises
  Reading

  Quest for structure within NP. A structural observation about the NP algorithm for Clique. Can that be exploited? No, it seems to be there for every NP algorithm - quantifier characterization for NP.
  

  References

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  None

  • Meeting 32 : Fri, Sep 30, 10:00 am-10:50 am

  Quantifier characterization of NP. CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Quest for structural similarities between problems. CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.

  References
  Exercises
  Reading

  Quantifier characterization of NP. CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Quest for structural similarities between problems. CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.
  

  References

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  None

  • Meeting 33 : Mon, Oct 03, 08:00 am-08:50 am

  Details of Cook-Levin Theorem. Reductions to CIRCUIT SAT. From CIRCUIT SAT to 3CNFSAT.

  References
  Exercises
  Reading

  Details of Cook-Levin Theorem. Reductions to CIRCUIT SAT. From CIRCUIT SAT to 3CNFSAT.
  

  References

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  None

  • Meeting 34 : Tue, Oct 04, 12:00 pm-12:50 pm

  Reduction from SAT to VC.

  References
  Exercises
  Reading

  Reduction from SAT to VC.
  

  References

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  None

  • Meeting 35 : Thu, Oct 06, 11:00 am-11:50 am

  Relationship between NP, CoNP, P and PSAPCE. Two generalizations of NP. 1) Based on oracle access. Definition of Polynomial Hierarchy, PSPACE upper bound. 2) Characterization of PH in terms of Quantifiers. Examples. EXACT-CLIQUE, and MIN-CKT problems as examples.

  References
  Exercises
  Reading

  Relationship between NP, CoNP, P and PSAPCE. Two generalizations of NP. 1) Based on oracle access. Definition of Polynomial Hierarchy, PSPACE upper bound. 2) Characterization of PH in terms of Quantifiers. Examples. EXACT-CLIQUE, and MIN-CKT problems as examples.
  

  References

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  None

  • Meeting 36 : Fri, Oct 07, 10:00 am-10:50 am

  NP-Intermediate languages? Ladner's Theorem. Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem.

  References
  Exercises
  Reading

  NP-Intermediate languages? Ladner's Theorem. Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem.
  

  References

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  None

  • Meeting 37 : Thu, Oct 13, 11:00 am-11:50 am

  Impagliazzo's proof of Ladner's theorem.

  References
  Exercises
  Reading

  Impagliazzo's proof of Ladner's theorem.
  

  References

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  None

  • Meeting 38 : Fri, Oct 14, 10:00 am-10:50 am

  Impagliazzo's proof of Ladner's theorem.

  References
  Exercises
  Reading

  Impagliazzo's proof of Ladner's theorem.
  

  References

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  None

  • Meeting 39 : Tue, Oct 18, 12:00 pm-12:50 pm

  Relativization. Baker-Gill-Solovoy theorem.

  References
  Exercises
  Reading

  Relativization. Baker-Gill-Solovoy theorem.
  

  References

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  None

  • Meeting 40 : Thu, Oct 20, 11:00 am-11:50 am

  References
  Exercises
  Reading

  
  

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  None

  • Meeting 41 : Fri, Oct 21, 10:00 am-10:50 am

  References
  Exercises
  Reading

  
  

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  None

  • Meeting 42 : Mon, Oct 24, 08:00 am-08:50 am

  References
  Exercises
  Reading

  
  

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  None

  • Meeting 43 : Tue, Oct 25, 12:00 pm-12:50 pm

  References
  Exercises
  Reading

  
  

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  None

  • Meeting 44 : Thu, Oct 27, 11:00 am-11:50 am

  References
  Exercises
  Reading

  
  

  References

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  None

  • Meeting 45 : Fri, Oct 28, 10:00 am-10:50 am

  Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. Mahaney's theorem (Statement).

  References
  Exercises
  Reading

  Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. Mahaney's theorem (Statement).
  

  References

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  None

  • Meeting 46 : Mon, Oct 31, 08:00 am-08:50 am

  Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP.

  References
  Exercises
  Reading

  Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP.
  

  References

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  None

  • Meeting 47 : Tue, Nov 01, 12:00 pm-12:50 pm

  Space Complexity. Complexity of Reductions. Composition of Logspace reductions. Savitch's Theorem.

  References
  Exercises
  Reading

  Space Complexity. Complexity of Reductions. Composition of Logspace reductions. Savitch's Theorem.
  

  References

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  None

  • Meeting 48 : Wed, Nov 02, 05:00 pm-06:00 pm

  Immerman-Szelepsinyi Theorem. Inductive Counting.

  References
  Exercises
  Reading

  Immerman-Szelepsinyi Theorem. Inductive Counting.
  

  References

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  None

  • Meeting 49 : Thu, Nov 03, 11:00 am-11:50 am

  QBF is PSPACE-complete.

  References
  Exercises
  Reading

  QBF is PSPACE-complete.
  

  References

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  None

  • Meeting 50 : Fri, Nov 04, 10:00 am-10:50 am

  CVP is P-complete. Story of Satisfiability 2SAT is in NP.

  References
  Exercises
  Reading

  CVP is P-complete. Story of Satisfiability 2SAT is in NP.
  

  References

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  None

  • Meeting 51 : Sat, Nov 05, 12:30 pm-01:30 pm

  2SAT is NP-hard, HornSAT is P-complete. Story of Reachability. Randomized Algorithms.

  References
  Exercises
  Reading

  2SAT is NP-hard, HornSAT is P-complete. Story of Reachability. Randomized Algorithms.
  

  References

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  None

  • Meeting 52 : Mon, Nov 07, 08:00 am-08:50 am

  Two consequences of Success Probability Amplification. The class P/poly. BPP is in P/poly. Every language in P/poly Turing Reduces to sparse sets. If NP is contained in BPP then PH collapses. The class RP, CoRP and structural simulations.

  References
  Exercises
  Reading

  Two consequences of Success Probability Amplification. The class P/poly. BPP is in P/poly. Every language in P/poly Turing Reduces to sparse sets. If NP is contained in BPP then PH collapses. The class RP, CoRP and structural simulations.
  

  References

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  None

  • Meeting 53 : Tue, Nov 08, 12:00 pm-12:50 pm

  BPP is in Sigma^2.

  References
  Exercises
  Reading

  BPP is in Sigma^2.
  

  References

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  None

  • Meeting 54 : Thu, Nov 10, 11:00 am-11:50 am

  Randomized Logspace. Reachability in Undirected graphs. Significance of spectral gap. Every graph has a spectral gap. Algebraic Expanders. Combinatorial Expanders and equivalence. Reachability in Graphs with combinatorial expander components.

  References
  Exercises
  Reading

  Randomized Logspace. Reachability in Undirected graphs. Significance of spectral gap. Every graph has a spectral gap. Algebraic Expanders. Combinatorial Expanders and equivalence. Reachability in Graphs with combinatorial expander components.
  

  References

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  None

  • Meeting 55 : Fri, Nov 11, 10:00 am-10:50 am

  Graph Powering, Replacement product and their effects in the algebraic expansion. Details of Reingold's algorithm.

  References
  Exercises
  Reading

  Graph Powering, Replacement product and their effects in the algebraic expansion. Details of Reingold's algorithm.
  

  References

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  None

• Evaluation Meetings - 3 meetings

  • Meeting 56 : Tue, Sep 13, 10:00 am-11:15 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

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  None

  • Meeting 57 : Mon, Oct 17, 08:00 am-08:50 am

  Quiz 1 (Syllabus : Lectures 1-21

  References
  Exercises
  Reading

  Quiz 1 (Syllabus : Lectures 1-21
  

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  • Meeting 58 : Tue, Nov 15, 12:00 pm-12:50 pm

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

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  None