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## PowerPoint Slideshow about 'Algorithms' - oshin

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exact

approximate

fast

Speed

slow

AlgorithmsTradeoff: Execution speed vs. solution quality

“Short & sweet”

“Quick & dirty”

“Slowly but surely”

“Too little, too late”

Idea: Some intractable problems can be efficiently approximated within close to optimal!

Fast:

- Simple heuristics (e.g., greed)
- Provably-good approximations

Slower:

- Branch-and-bound approaches
- Integer Linear Programming relaxation

Minumum vertex cover problem: Given a graph, find a minimum set of vertices such that each edge is incident to at least one vertex of these vertices.

Example:

Input graph

Heuristic solution

Optimal solution

Applications: bioinformtics, communications,

civil engineering, electrical engineering, etc.

- One of Karp’s original NP-complete problems

Theorem: The minimum vertex cover problem is NP-complete (even in planar graphs of max degree 3).

Theorem: The minimum vertex cover problem can be solved exactly within exponential time nO(1)2O(n).

Theorem: The minimum vertex cover problem can not be approximated within £ 1.36*OPT unless P=NP.

Theorem: The minimum vertex cover problem can be approximated (in linear time) within 2*OPT.

Idea: pick an edge, add its endpoints, and repeat.

y

Approximate Vertex Cover

Algorithm: Linear time 2*OPT approximation for the minimum vertex cover problem:

- Pick random edge (x,y)
- Add {x,y} to the heuristic solution
- Eliminate x and y from graph
- Repeat until graph is empty

Best approximation

bound known for VC!

Idea: one of {x,y} must be in any optimal solution.

Þ Heuristic solution is no worse than 2*OPT.

B

A

B

A

B

E

C

E

E

D

C

C

D

D

Maximum Cut

Maximum cut problem: Given a graph, find a partition

of the vertices maximizing the # of crossing edges.

Example:

cut size = 2

cut size = 4

cut size = 5

Input graph

Heuristic solution

Optimal solution

Applications: VLSI circuit design, statistical physics,

communication networks.

- One of Karp’s original NP-complete problems.

Theorem [Karp, 1972]: The minimum vertex cover

problem is NP-complete.

Theorem: The maximum cut problem can be solved

in polynomial time for planar graphs.

Theorem: The maximum cut problem can not

be approximated within £ 17/16*OPT unless P=NP.

Theorem: The maximum cut problem can be

approximated in polynomial time within 2*OPT.

Theorem: The maximum cut problem can be

approximated in polynomial time within 1.14*OPT.

=1.0625*OPT

B

E

C

D

A

B

A

B

E

E

C

C

D

D

Maximum Cut

Algorithm: 2*OPT approximation for maximum cut:

Start with an arbitrary node partition

- If moving an arbitrary node across the partition

will improve the cut, then do so

- Repeat until no further improvement is possible

cut size = 2

cut size = 3

cut size = 5

Input graph

Heuristic solution

Optimal solution

Idea: final cut must contain at least half of all edges.

Þ Heuristic solution is no worse than 2*OPT.

Approximate Traveling Salesperson

Traveling salesperson problem: given a pointset, find

shortest tour that visits every point exactly once.

2*OPT metric TSP heuristic:

- Compute MST
- T = Traverse MST
- S = shortcut tour
- Output S

Analysis:

S < T

=

2*

MST < OPT TSP

2*

triangle inequality!

T covers minimum spanning tree twice

TSPminus an edge is a spanning tree

- NP transformations typically do not preserve the

approximability of the problem!

- Some NP-complete problems can be approximated

arbitrarily close to optimal in polynomial time.

Theorem: Geometric TSP can be approximated in

polynomial time within (1+e)*OPT for any e>0.

- Other NP-complete problems can not be approximated

within any constant in polynomial time (unless P=NP).

Theorem: General TSP can not be approximated

efficiently within K*OPT for any K>0 (unless P=NP).

Definition: two graphs G1=(V1,E1) and G2=(V2,E2) are

isomorphic iff $ bijection ƒ:V1®V2such that

"vi,vjÎV1 (vi,vj)ÎE1Û (ƒ(vi),ƒ(vj))ÎE2

Isomorphism ºedge-preserving vertex permutation

Problem: are two given graphs isomorphic?

≈

≈

Note: Graph isomorphism ÎNP, but not known to be in P

G1

≈

G

Zero-Knowledge Proofs

Idea: proving graph isomorphism without disclosing it!

Premise: Everyone knows G1 and G2 but not ≈

≈must remain secret!

≈

Create random G ≈ G1

Note: ≈ is≈(≈)

Broadcast G

Verifier asks for≈or≈

Broadcast≈or≈

Verifier checks G≈G1 or G≈G2

Repeat k times

ÞProbability of cheating: 2-k

≈

≈

Approximating a “proof”!

G1

χ

Zero-Knowledge Proofs

Idea: prove graph 3-colorable without disclosing how!

Premise: Everyone knows G1 but not its 3-coloring χ

which must remain secret!

χ'

Create random G2≈ G1

Note: 3-coloring χ'(G2) is ≈(χ(G1))

Broadcast G2

Verifier asks for≈or χ'

Broadcast≈or χ'

Verifier checks G1≈G2 or χ'(G2)

Repeat k times

ÞProbability of cheating: 2-k

≈

χ

Interactive proof!

Zero-Knowledge Caveats

- Requires a good random number generator
- Should not use the same graph twice
- Graphs must be large and complex enough

Applications:

- Identification friend-or-foe (IFF)
- Cryptography
- Business transactions

Idea: prove that a Boolean formula P is satisfiable

without disclosing a satisfying assignment!

Premise: Everyone knows P but not its secret

satisfying assignment V !

P = (x+y+z)(x'+y'+z)(x'+y+z')

ƒ

Convert P into a graph 3-colorability

instance G =ƒ(P)

Publically broadcastƒ and G

Use zero-knowledge protocol

to show that G is 3-colorable

Þ P is satisfiable iff G is 3-colorable

Þ P is satisfiable with probability1-2-k

G =

Interactive proof!

- Prover has unbounded power and may be malicious
- Verifier is honest and has limited power

Completeness: If a statement is true, an honest verifier

will be convinced (with high prob) by an honest prover.

Soundness: If a statement is false, even an omnipotent

malicious prover can not convince an honest verifier that

the statement is true (except with a very low probability).

- The induced complexity class depends on the verifier’s

abilities and computational resources:

Theorem: For a deterministic P-time verifier, class is NP.

Def: For a probabilistic P-time verifier, induced class is IP.

Theorem [Shamir, 1992]: IP = PSPACE

Concepts, Techniques, Idea & Proofs

- 2-SAT
- 2-Way automata
- 3-colorability
- 3-SAT
- Abstract complexity
- Acceptance
- Ada Lovelace
- Algebraic numbers
- Algorithms
- Algorithms as strings
- Alice in Wonderland
- Alphabets
- Alternation
- Ambiguity
- Ambiguous grammars
- Analog computing
- Anisohedral tilings
- Aperiodic tilings
- Approximate min cut
- Approximate TSP

- Approximate vertex cover
- Approximations
- Artificial intelligence
- Asimov’s laws of robotics
- Asymptotics
- Automatic theorem proving
- Autonomous vehicles
- Axiom of choice
- Axiomatic method
- Axiomatic system
- Babbage’s analytical engine
- Babbage’s difference engine
- Bin packing
- Binary vs. unary
- Bletchley Park
- Bloom axioms
- Boolean algebra
- Boolean functions
- Bridges of Konigsberg
- Brute fore

- Busy beaver problem
- C programs
- Canonical order
- Cantor dust
- Cantor set
- Cantor’s paradox
- CAPCHA
- Cardinality arguments
- Cartesian coordinates
- Cellular automata
- Chaos
- Chatterbots
- Chess-playing programs
- Chinese room
- Chomsky hierarchy
- Chomsky normal form
- Chomskyan linguistics
- Christofides’ heuristic
- Church-Turing thesis
- Clay Mathematics Institute

Concepts, Techniques, Ideas & Proofs

- Clique problem
- Cloaking devices
- Closure properties
- Cogito ergo sum
- Colorings
- Commutativity
- Complementation
- Completeness
- Complexity classes
- Complexity gaps
- Complexity Zoo
- Compositions
- Compound pendulums
- Compressibility
- Computable functions
- Computable numbers
- Computation and physics
- Computation models
- Computational complexity

- Computational universality
- Computer viruses
- Concatenation
- Co-NP
- Consciousness and sentience
- Consistency of axioms
- Constructions
- Context free grammars
- Context free languages
- Context sensitive grammars
- Context sensitive languages
- Continuity
- Continuum hypothesis
- Contradiction
- Contrapositive
- Cook’s theorem
- Countability
- Counter automata
- Counter example
- Cross- product

- Crossing sequences
- Cross-product construction
- Cryptography
- DARPA Grand Challenge
- DARPA Math Challenges
- De Morgan’s law
- Decidability
- Deciders vs. recognizers
- Decimal number system
- Decision vs. optimization
- Dedekind cut
- Denseness of hierarchies
- Derivations
- Descriptive complexity
- Diagonalization
- Digital circuits
- Diophantine equations
- Disorder
- DNA computing
- Domains and ranges

Concepts, Techniques, Ideas & Proofs

- Dovetailing
- DSPACE
- DTIME
- EDVAC
- Elegance in proof
- Encodings
- Enigma cipher
- Entropy
- Enumeration
- Epsilon transitions
- Equivalence relation
- Euclid’s “Elements”
- Euclid’s axioms
- Euclidean geometry
- Euler’s formula
- Euler’s identity
- Eulerian tour
- Existence proofs
- Exoskeletons
- Exponential growth

- Exponentiation
- EXPSPACE
- EXPSPACE
- EXPSPACE complete
- EXPTIME
- EXPTIME complete
- Extended Chomsky hierarchy
- Fermat’s last theorem
- Fibonacci numbers
- Final states
- Finite automata
- Finite automata minimization
- Fixed-point theorem
- Formal languages
- Formalizations
- Four color problem
- Fractal art
- Fractals
- Functional programming
- Fundamental thm of Algebra

- Fundamental thm of Arith.
- Gadget-based proofs
- Game of life
- Game theory
- Game trees
- Gap theorems
- Garey & Johnson
- General grammars
- Generalized colorability
- Generalized finite automata
- Generalized numbers
- Generalized venn diagrams
- Generative grammars
- Genetic algorithms
- Geometric / picture proofs
- Godel numbering
- Godel’s theorem
- Goldbach’s conjecture
- Golden ratio
- Grammar equivalence

Concepts, Techniques, Ideas & Proofs

- Grammars
- Grammars as computers
- Graph cliques
- Graph colorability
- Graph isomorphism
- Graph theory
- Graphs
- Graphs as relations
- Gravitational systems
- Greibach normal form
- “Grey goo”
- Guess-and-verify
- Halting problem
- Hamiltonian cycle
- Hardness
- Heuristics
- Hierarchy theorems
- Hilbert’s 23 problems
- Hilbert’s program
- Hilbert’s tenth problem

- Historical perspectives
- Household robots
- Hung state
- Hydraulic computers
- Hyper computation
- Hyperbolic geometry
- Hypernumbers
- Identities
- Immerman’s Theorem
- Incompleteness
- Incompressibility
- Independence of axioms
- Independent set problem
- Induction & its drawbacks
- Infinite hotels & applications
- Infinite loops
- Infinity hierarchy
- Information theory
- Inherent ambiguity
- Initial state

- Intelligence and mind
- Interactive proofs
- Intractability
- Irrational numbers
- JFLAP
- Karp’s paper
- Kissing number
- Kleene closure
- Knapsack problem
- Lambda calculus
- Language equivalence
- Law of accelerating returns
- Law of the excluded middle
- Lego computers
- Lexicographic order
- Linear-bounded automata
- Local minima
- LOGSPACE
- Low-deg graph colorability
- Machine enhancements

Concepts, Techniques, Ideas & Proofs

- Machine equivalence
- Mandelbrot set
- Manhattan project
- Many-one reduction
- Matiyasevich’s theorem
- Mechanical calculator
- Mechanical computers
- Memes
- Mental poker
- Meta-mathematics
- Millennium Prize
- Minimal grammars
- Minimum cut
- Modeling
- Multiple heads
- Multiple tapes
- Mu-recursive functions
- MAD policy
- Nanotechnology
- Natural languages

- Navier-Stokes equations
- Neural networks
- Newtonian mechanics
- NLOGSPACE
- Non-approximability
- Non-closures
- Non-determinism
- Non-Euclidean geometry
- Non-existence proofs
- NP
- NP completeness
- NP-hard
- NSPACE
- NTIME
- Occam’s razor
- Octonions
- One-to-one correspondence
- Open problems
- Oracles
- P completeness

- P vs. NP
- Parallel postulate
- Parallel simulation
- Dovetailing simulation
- Parallelism
- Parity
- Parsing
- Partition problem
- Paths in graphs
- Peano arithmetic
- Penrose tilings
- Physics analogies
- Pi formulas
- Pigeon-hole principle
- Pilotless planes
- Pinwheel tilings
- Planar graph colorability
- Planarity testing
- Polya’s “How to Solve It”
- Polyhedral dissections

Concepts, Techniques, Ideas & Proofs

- Polynomial hierarchy
- Polynomial-time
- P-time reductions
- Positional # system
- Power sets
- Powerset construction
- Predicate calculus
- Predicate logic
- Prime numbers
- Principia Mathematica
- Probabilistic TMs
- Proof theory
- Propositional logic
- PSPACE
- PSPACE completeness
- Public-key cryptography
- Pumping theorems
- Pushdown automata
- Puzzle solvers
- Pythagorean theorem

- Quantifiers
- Quantum computing
- Quantum mechanics
- Quaternions
- Queue automata
- Quine
- Ramanujan identities
- Ramsey theory
- Randomness
- Rational numbers
- Real numbers
- Reality surpassing Sci-Fi
- Recognition and enumeration
- Recursion theorem
- Recursive function theory
- Recursive functions
- Reducibilities
- Reductions
- Regular expressions
- Regular languages

- Rejection
- Relations
- Relativity theory
- Relativization
- Resource-bounded comput.
- Respect for the definitions
- Reusability of space
- Reversal
- Reverse Turing test
- Rice’s Theorem
- Riemann hypothesis
- Riemann’s zeta function
- Robots in fiction
- Robustness of P and NP
- Russell’s paradox
- Satisfiability
- Savitch’s theorem
- Schmitt-Conway biprism
- Scientific method
- Sedenions

Concepts, Techniques, Ideas & Proofs

- Self compilation
- Self reproduction
- Set cover problem
- Set difference
- Set identities
- Set theory
- Shannon limit
- Sieve of Eratosthenes
- Simulated annealing
- Simulation
- Skepticism
- Soundness
- Space filling polyhedra
- Space hierarchy
- Spanning trees
- Speedup theorems
- Sphere packing
- Spherical geometry
- Standard model
- State minimization

- Steiner tree
- Stirling’s formula
- Stored progam
- String theory
- Strings
- Strong AI hypothesis
- Superposition
- Super-states
- Surcomplex numbers
- Surreal numbers
- Symbolic logic
- Symmetric closure
- Symmetric venn diagrams
- Technological singularity
- Theory-reality chasms
- Thermodynamics
- Time hierarchy
- Time/space tradeoff
- Tinker Toy computers
- Tractability

- Tradeoffs
- Transcendental numbers
- Transfinite arithmetic
- Transformations
- Transition function
- Transitive closure
- Transitivity
- Traveling salesperson
- Triangle inequality
- Turbulance
- Turing complete
- Turing degrees
- Turing jump
- Turing machines
- Turing recognizable
- Turing reduction
- Turing test
- Two-way automata
- Type errors
- Uncomputability

Concepts, Techniques, Ideas & Proofs

- Uncomputable functions
- Uncomputable numbers
- Uncountability
- Undecidability
- Universal Turing machine
- Venn diagrams
- Vertex cover
- Von Neumann architecture
- Von Neumann bottleneck
- Wang tiles & cubes
- Zero-knowledge protocols

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“Make everything as simple

as possible, but not simpler.”

- Albert Einstein (1879-1955)

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