ECE556: Design Automation of Digital Systems

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ECE556 Design Automation of Digital Systems

• Spring 2007
• Instructor Azadeh Davoodi
• Lecture 2
• January 25, 2007

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Graph Theory

• Graph A mathematical object representing a set
  of points and interconnections between
  them
• Notation G(V, E), where
• V is the vertex set v1, v2, v3, v4, v5, v6
• E is the edge set e1, e2, e3, e4, e5
• An edge has two endpoints, e.g. e1 (v1, v2)

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Terminology

• degree of a vertex
• subgraph of a graph
• complete (sub)graph
• clique
• parallel edges
• bipartite graph

• an edge incident with a vertex
• adjacent vertices

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Terminology

• Path e.g. v1, e1, v2, e3, v3, e4, v4

• cycle a closed path
• connected vertices, connected graph, connected
  components

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Terminology

• directed graph
• directed path
• directed cycle
• strongly connected vertices
• weakly connected vertices
• weighted graphs
• edge weighted and/or vertex weighted

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Data Structures for Storing Graphs

• Different data structures for different
  applications
• Adjacency matrix
• -- a 0 or 1 entry for every vertex pair

v1 v2 v3 v4 v5 v6
v1 v2 v3 v4 v5 v6
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Data Structures for Storing Graphs

• 2) Adjacency list

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Explicit Edges and Vertices
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Graph Traversal

• Can you think of generic ways to visit all the
  vertices and edges of a graph? And why do you
  think it is important to answer this question in
  VLSI-CAD problems?
• Popular graph traversal methods
• Depth First Search (DFS)
• Breadth First Search (BFS)

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Algorithm

• Algorithm
• A step-by-step procedure for solving a problem in
  a finite number of steps
• Properties
• Precision each step precisely stated
• Termination ends in finite time
• Correctness

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Algorithm

• There can be many different algorithms to solve
  the same problem
• Need some way to compare algorithms
• Desired attributes of a good algorithm in VLSI
  CAD
• 1- Optimality or exactness need the best
  quality solution (e.g., minimum cost power,
  area, etc.)
• 2- Efficiency need to find this optimal solution
  fast

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Some Algorithmic Paradigms
Some of these techniques guarantee achieving the
optimal solution, but might not be necessarily
efficient. Some might be efficient but not
necessarily guarantee optimality!

• Exhaustive search
• Branch and Bound
• Divide and conquer
• Greedy algorithm
• Dynamic programming
• Mathematical programming
• Simulated annealing

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Comparison of algorithmsMotivation

• Will first start by comparing run-time of
  algorithms
• Assume two algorithms both generate a solution to
  a problem
• Ignoring the quality of solution for now, lets
  focus on ways for comparing their efficiency

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Comparing Run-Time of Algorithms

• Comparing pure run-time of two algorithms
• Difficult since algorithms may be implemented on
  different machines, use different languages, etc
• Also input-dependent (i.e., which input to use?)
• Notion of computational complexity

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Computational Complexity

• Computational complexity
• an abstract measure of the time and space
  necessary to execute an algorithm as function of
  its input size
• Input size examples
• sort n words of bounded length ?
• size n
• the input is the integer n ?
• size log n
• the input is the graph G(V,E) ?
• size V and E

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Computational Complexity

• Computational complexity
• An abstract measure of the time and space
  necessary to execute an algorithm as function of
  its input size
• Run-Time complexity
• Expressed in elementary computational steps
  (e.g., an addition, multiplication, pointer
  indirection is one step)
• 2) Space Complexity
• Expressed in memory locations (e.g. bits, bytes,
  words)

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Computational ComplexityBig-O Notation

• Definition f(n) O(g(n))
• If cgt0 and n0gt 0 such that 0 f(n) cg(n) for all
  n n0
• Intuition
• f(n) g(n) when we ignore constant multiples
  and small values of n

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Run-Time Complexity of Algorithmsusing Big-O
Notation

• Obtain upper bound on run-time
• Express run time as a function of input size n
• Ignore multiplicative constant
• Ignore lower order terms
• Examples
• 3n26n2.7 is
• O(n2)
• n1.110000000000n is
• O(n1.1)
• n1.1 is
• O(n1.1)

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Effect of Multiplicative Constant
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Growth Rates of some Functions
Polynomial
Exponential
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Computational Complexity

• Relevant growth rates for the time complexity
• polynomial vs. exponential
• linear vs. quadratic
• sublinear

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• Question
• If I know the expressions f1(n) and f2(n)
  corresponding to the actual run-times of two
  algorithms, I can directly compare these run-time
  expressions
• Why do I need the O-notation then?
• Answer
• Many of the times, it is not possible to obtain
  exact f(n) expression for the run-time of an
  algorithm
• It is much easier to think of an algorithm in
  terms of union of sub-algorithms, and then to
  find its timing complexity we can only consider
  those sub-algorithms that have a dominant timing
  complexity and ignore the rest

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Problem of Exponential Function

• Consider 2n, value doubled when n is increased by
  1

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Facts

• It is desirable to develop polynomial-time
  algorithms to optimally solve VLSI CAD problems
• Unfortunately many VLSI CAD problems can not be
  optimally solved in polynomial time
• They belong to a class of problems known as
  NP-Complete which indicate the set of problems
  that are difficult to be solved (can not be
  efficiently solved)