CSE 494: Electronic Design Automation

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CSE 494 Electronic Design Automation

• Lecture 3

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Basic Terminology

• A graph is a pair of sets G(V,E) where V is the
  set of vertices, and E (u,v) is a set of pair
  of distinct vertices called edges.
• A vertex u is adjacent to v, if (u,v) belongs to
  (or is in) E.
• Set of vertices adjacent to u are denoted by
  Adj(u).
• An edge (u,v) is incident on both u, and v.
• The degree of a vertex u is the number of edges
  incident on u.

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Basic Terminology

• A complete graph on n vertices is a graph in
  which every vertex is adjacent to every other
  vertex. (Denoted by K_n
• A graph G (V,E) is a subgraph of G iff V is
  a subset of V, and E is a subset of E.
• A walk P of a graph G is defined as a finite
  alternating sequence P v_0, e_1,,e_k,v_k.
• A tour is walk in which all edges are distinct.
• A walk is an open walk if the terminal vertices
  are distinct.
• A path is an open walk in which no vertex appears
  more than once.

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Basic Terminology

• The length of a path is the number of edges in
  it.
• A path is a (u,v) path if u, and v are the
  terminal vertices.
• A cycle is a path (v_0,v_k) of length k (k gt 2)
  where v_0 v_k.
• Odd cycle if k is odd, Even cycle if k is even.
• Two vertices u and v are connected if G has a
  (u,v) path.
• A connected component of G is a subgraph of G
  that has a path from each vertex to every other
  vertex.
• An edge e in E is called a cut edge in G if its
  removal from G increases the number of connected
  components of G by atleast one.

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Basic Terminology

• A tree is a connected subgraph with no cycles.
• A clique is a complete subgraph of G.

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Basic Terminology

• A directed graph is a pair of sets (V,E) where E
  is a set of ordered pairs of distinct vertices,
  called directed edges.
• A directed edge e(u,v) is an in-edge of v, and
  out-edge of u.
• The in-degree of u is the number of in-edges of v
  (out-degree similarly defined).
• A vertex u is an ancestor of v if there exists a
  directed path from u to v in G.
• A directed acyclic graph (DAG) is a directed
  graph with no cycles.

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Basic Terminology

• Hypergraph is a pair (V,E) where V is a set of
  vertices, and E is a set of sets of vertices.
• Each e in E denoted by v_0, v_1,,v_k is called
  a net.
• A bipartite graph is a graph that can be
  partitioned in to two sets X, and Y so that each
  edge has one end in X, and the other end in Y.

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Algorithm Strategies

• Divide-and-conquer.
• Recursively divide the problem into distinct
  sub-problems, and construct the global solution
  from sub-solutions.
• Merge sort.
• Dynamic programming
• Often utilized for optimization problems.
• Recursively divide the problem into sub-problems
  (possibly overlapping), and construct the global
  solution from sub-solutions.
• Sub-problems are not independent.
• Solutions maintained in a table, hence the term
  programming.
• Longest common sequence.
• Greedy algorithms
• Often utilized for optimization problems.
• No division into sub-problems.
• Decision making is applied in a locally optimal
  manner.
• Kruskal minimum spanning tree.

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P and NP Problems

• The class P Solution is given by an algorithm
  whose time complexity can be specified by a
  polynomial p(nk).
• The class NP Set of problems whose solution can
  be verified by a polynomial time algorithm.

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NP Complete and NP hard

• NP Complete
• Belongs to class NP.
• Can be reduced to another NP complete problem by
  a polynomial time algorithms.
• NP hard
• Satisfies property 2, but not property 1.

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Strategies for solving hard problems optimally

• Brute force
• Enumerate all solution instances.
• Branch-and-Bound
• Smart(er) strategy for enumerating solution
  instances.
• Integer linear programming
• Problem specified by linear mathematical
  equations, and then a solver (computer tool) is
  utilized for generating the solution.

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Other strategies for solving hard problems

• Approximation algorithms
• Instead of solving the original problem P, solve
  a problem P whose solution is within certain
  bounds of the optimum.
• P typically solvable in polynomial time.
• Heuristic strategies
• Deterministic algorithms
• Non-deterministic algorithms

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Deterministic algorithms

• Every invocation of the algorithm with identical
  inputs, generates the same solution (hence,
  deterministic).
• Fast, but inherently greedy in nature.

Local minima
Cost
Successive solutions
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Non-deterministic algorithms

• Also known as probabilistic or stochastic
  algorithms.
• Every invocation of the algorithm with identical
  inputs generates a different solution.
• Slower than non-deterministic, but demonstrates
  non-greedy behavior.

Hill-climbing behavior
Cost
Successive solutions
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Classic non-deterministic algorithms

• Simulated annealing
• Maintains a temperature variable that is reduced
  from high value to a low value.
• Number of solutions explored at each temperature
  by modification of existing solution.
• Solution that decreases cost is always accepted.
• Accept solutions that increase cost at high
  temperatures with greater probability.
• At low temperatures accept solutions that
  increase cost with very low probability.

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Classic non-deterministic algorithms

• Genetic algorithms
• Solution represented by a set of chromosones or
  a generation.
• Chromosone an array of 0s, and 1s.
• Successive generations are obtained by
• Duplication of the best solution (survival of the
  fittest).
• Mutation Random generation of a chromosone.
• Crossover Slicing, and inter-changing parts of
  two chromosones.
• Solution(s) generated after many generations.

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• Read chapter 1.