Theory1

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Theory Moon Jung Chung
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Parallel Minimum Spanning Tree (Deterministics)

• Each node is a super node.
• Repeat until only one super node
• For each super node,
• among edges which connects to another super
  node,
• select an edge with minimum
• merge two super nodes into one super node
• How many phase? --gt O(logn) phase.
• each phase O(logn) time in CRCW.
• Actually, with priority CRCW, O(1) time.
• Complexity O(logn) time with O(m) PEs with
  priority CRCW, where m is the number of edges.

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Parallel Minimum Spanning Tree (Deterministic
detailed)

• Repeat until there is only one super node,
• for each edge (x,y), if x and y are different
  component,
• component (x) y
• component (y) x
• For each node with priority -CW, accept the
  minimum value of component.
• Merge two super nodes into a single super node.
• Complexity O(logn) time with O(m) PEs with
  priority CRCW, where m is the number of edges.
• How to avoid priority-CR? gt If tree is a
  spanning tree?

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Parallel Spanning Tree (Probablistic)

• For each edge, if it connects two different super
  nodes, add the edge in a spanning tree, and merge
  two super node as a single node.
• For two super nodes, two edges may be selected at
  the same time connecting them.
• How about cycle?
• To prevent these troubles,
• For each super node, select an edge randomly
  which connects to other super node.
• Verify if two different super nodes selected the
  same edge
• Verify if there is no cycles
• If the selected edge is OK, include the edge in a
  spanning tree, and merge two super nodes.
• How many phase? --gt O(logn) phase in average.
• each phase O(1) time in average
• Complexity O(logn) time with O(m) CREW PEs.
• Parallel Connected Components in EREW gt
• Use matrix multiplication O(log2n) time using
  O(n2) PEs.

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Parallel Models

• (i) Shared Memory (PRAM) -- deterministic
• how about probablistic? example minimum
  spanning tree
• (ii) Circuit depth and size
• (iii) Alternating Turing Machine
• Brent Theorem
• Any depth-d size-n combinational circuit with
  bounded fan-in can be
• simulated by p-processor CREW algorithm in O(n/p
  d) time.
• proof store inputs to the combinational circuit
  in the PRAM
• Each gate evaluate its output if all inputs are
  ready.
• If there are not enough PEs, evaluate gates in
  the order of depth.
• (depth of a gate longest path from the primary
  inputs)
• Complexity Let ni be number of gates at depth i.
  
• The simulation takes ? ni/p ? for the gates at
  the depth i
• total time sum of ?i ?ni/p? ??i ( ni/p 1)
  n/p d.

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Parallel Models

• Brent Theorem for EREW
• Any depth-d size-n combinational circuit with
  bounded fan-in, fan-out can be
• simulated by p-processor EREW algorithm in O(n/p
  d) time.
• proof For exclusive reading, output values are
  copied to all gates where it is used.
• With bounded fan-in, fan-out, it takes constant
  time.
• Reading them one by one also takes constant time.

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Uniform Circuit

• L be a language.
• Circuit complexity of L?
• Definition1
• f(n) number of gates of a circuit accepting
  strings of length in L.
• Def. 1 may not be acceptable one
• L 0n n-th TM accepts n-th input
• L is not even recursively enumerable. But L has
  circuit complexity 1

two candidate circuits accepting a string of
length.
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Uniform Circuit

• Let Ln w w is in L and w n
• There is a family of circuits Cn, and
  generating Cn can be done using polynomial time
  using O(logn) space. Each gate has bounded fan-in
  degree.
• Example of non-uniform Division circuit gt
  O(logn) time, but generation of it will require
  polynomial size space!
• NCk L there is a uniform circuit of poly
  size and (logn)k depth
• NC ?k NCk
• Note SCk L there is a TM with time poly and
  (logn)k space
• Relationship between SC and NC?

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Alternating TM

• TM forks at each state.
• Subprocesses cannot communicate!
• TM has two types of states
• universal
• existential
• At Universal all branches must be accepted.
• Existential one branch should lead to accepting
  state
• That is, each computation can be represented as a
  computation tree.
• Depth of computation tree time complexity.
• Note
• Deterministic TM a path
• Parallel random access machine processes can
  communicate.
• ASPACE (logn) P

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Parallel Computation Thesis

• parallel computation thesis
• parallel time is polynomially equivalent to
  sequential space.
• example parallel time of vector machine is
  equivalent to sequential space
• ATIME (f(n)) and DSPACE (f(n)).
• ATM (S(n), T(n)) Language accepted by ATM with
  space S(n), time T(n).
• Theorem ATM (logn, (logn)k) NCk

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NC-algorithm and P-complete problems

• Let f be a function
• Input input of f
• Output compute f
• NC1 reducible from f to g using oracles of g, we
  can construct NC1 circuits computing f.
• oracle gate is counted as depth logn, size n.
• Division
• input x and y
• output x/y
• Reciprocal
• input x
• output x-1
• Powering
• Input x
• Output xi expressed in n2 bits

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NC-algorithm and P-complete problems

• Special case of function language recognition.
• Let A, B be languages
• A is NC1 reducible to B, if using oracle, A can
  be solved in NC1.
• log space reduction is NC reduction.
• A ?NCB, and B is in NC, then A is also in NC.
• A is complete for P ltgt for any B in P, A ltlogn
  B.
• Theorem Let A be a P-complete problem (with
  respect to log space reduction). If A is in NC,
  then P ? NC.

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P-complete and hard problems to make parallel

• Examples of P-complete problems
• Monotone Circuit Value problem
• Input Monotone circuit (and, or gates, but
  without not gate, and values to primary input.
• Question Is the output of circuit 0 with the
  given primary input values?
• Generating lexiographically smallest depth
  first search tree
• These P-complete problems may not be
  parallelizable!
• Open question
• Perfect matching, depth first search (directed,
  undirected), integer GCD,
• modular exponentiation