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CS 420 – Design of Algorithms. Basic Concepts. Design of Algorithms. We need mechanism to describe/define algorithms Independent of the language implementation of the algorithm Pseudo-code. Algorithms.

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design of algorithms
Design of Algorithms
  • We need mechanism to describe/define algorithms
  • Independent of the language implementation of the algorithm
  • Pseudo-code
algorithms
Algorithms
  • Algorithm – “any well defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output” Cormen, et a.
algorithms4
Algorithms
  • Algorithm – “is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. “
    • from Wikipedia.org
      • http://en.wikipedia.org/wiki/Algorithm
algorithms5
Algorithms
  • Human Genome Project
  • Security/encryption for e-commerce
  • Spacecraft navigation
  • Pulsar searches
  • Search Engines
algorithms6
Algorithms
  • Search engines
  • Search algorithms
    • linear search-
      • read-compare-read-…
      • run-time – linear function of n (n=size of database)
      • suppose the DB has 40,000,000 records
      • then 40,000,000 read-compare cycles
      • at 1000 read-compare cycles per second = 40,000 seconds = 667 minutes ~ 11 hours
algorithms7
Algorithms
  • Search Google for “house”
    • 730,000,000 hits in 0.1 seconds
algorithms8
Algorithms
  • Binary tree search algorithm
  • The keyword is indexed in a set of binary indexes – is keyword in left or right half of database?

Database

aaa-mon

moo-zxy

aaa-jaa

jaa-mon

moo-tev

tew-zxy

algorithms9
Algorithms
  • Binary search algorithm
  • So, to search a 40,000,000 record database
  • for a single term –
  • T(40,000,000) = log2(40,000,000)
  • = 26 read compare cycles
  • at 1000 read/compare cycles/sec = 0.026 seconds
algorithms10
Algorithms
  • Binary Search Algorithm
  • So, what about 730,000,000 records
  • Search for a single keyword –
  • 30 read/compare cycles
  • or about 0.03 seconds
pseudo code
Pseudo-code
  • Like English – easily readable
  • Clear and consistent
  • Rough correspondence to language implementation
  • Should give a clear understanding of what the algorithm does
using pseudo code
Using Pseudo-code
  • Use indentation to indicate block structure. Blocks of code at the same level of indentation.
    • Do not use “extra” statements like begin-end
  • Looping constructs and conditionals are similar to Pascal (while, for, repeat, if-then-else). In for loops the loop counter is persistent
using pseudo code13
Using Pseudo-code
  • Use a consistent symbol to indicate comments. Anything on line after this symbol is a comment, not code
  • Multiple assignment is allowed
  • Variables are local to a procedure unless explicitly declared as global
  • Array elements are specified by the array name followed by indices in square brackets… A[i]
pseudo code14
Pseudo-code
  • .. indicates a range of values A[1..4] means elements 1,2,3,and 4 of array A
  • Compound data can be represented as objects with attributes or fields. Reference these attributes array references. For example a variable that is the length of the array A is length[A]
pseudo code15
Pseudo-code
  • An array reference is a pointer
  • Parameters are passed by value
    • assignments to parameters within a procedure are local to the procedure
  • Boolean operators short-circuit
  • Be consistent
    • don’t use read one place and input another unless they have functionally different meaning
insertion sort algorithm
Insertion-Sort Algorithm

INSERTION-SORT(A)

for j = 2 to length[A]

do key = A[j]

C* Insert A[j] into the sorted sequence A[1..j-1]

i=j-1

while i > 0 and A[i]> key

do A[i+1] = A[i]

i=i-1

A[i+1]=key

analysis of algorithms
Analysis of Algorithms
  • Analysis may be concerned with any resources
    • memory
    • bandwidth
    • runtime
  • Need a model for describing runtime performance of an algorith
  • RAM – Random Access Machine
slide18
RAM
  • There are other models but for now…
  • Assume that all instructions are sequential
  • All data is accessible in one step
  • Analyze performance (run-time) in terms of inputs
    • meaning of inputs varies – size of an array, number of bits, vertices and edges, etc.
  • Machine independent
  • Language independent
slide19
RAM
  • Need to base analysis on cost of instruction execution
  • assign costs (run-time) to each instruction
insertion sort21
INSERTION-SORT
  • Run-time = sum of products of costs (instruction runtimes) and execution occurrences
  • T(n)= c1n + c2(n-1) + c4(n-1) +

c5nj=2tj +c6nj=2(tj-1) + c7nj=2(tj-1)

+c8(n-1)

insertion sort22
INSERTION-SORT
  • Best case vs Worst Case
  • Best case
    • Input array already sort
  • Worst case
    • Input array sorted in reverse order
insertion sort23
INSERTION-SORT
  • For sake of discussion…
  • assume that all c=2
  • then, for best case
    • T(n) = 10n-8
    • n=1000, T(n) = 9992
  • for worst case …
    • T(n) = 3n2+7n-8
    • n=1000, T(n) = 3006992
insertion sort performance
Insertion-sort Performance

* Best case is a linear function of n

so what are we really interested in
So, what are we really interested in?
  • the big picture
  • the trend in run-time performance as the problem grows
  • not concerned about small differences in algorithms
  • what happens to the algorithm as the problem gets explosively large
  • the order of growth
abstractions and assumptions
Abstractions and assumptions
  • The cost coefficients will not vary that much… and will not contribute significantly to the growth of run-time performance
    • so we can set them to a constant
    • … and we can ignore them
    • remember the earlier example –
      • c1 = c2 = … = 2
abstractions and assumptions27
Abstractions and assumptions
  • In a polynomial run-time function the order of growth is controlled by the higher order term
    • T(n) = 3n2+7n-8
    • so we can ignore (discard) the lower order terms
    • T(n) = 3n2
abstractions and assumptions28
Abstractions and assumptions
  • It turns out that with sufficiently large n the coefficient of the high order term is not that important in characterizing the order of growth of a run-time function
    • So, from that perspective the run-time function of the Insertion-Sort algorithm (worst-case) is -
    • T(n) = n2
abstractions and assumptions29
Abstractions and assumptions
  • Are these abstraction assumptions correct?
  • for small problems – no
  • but for sufficiently large problem
  • they do a pretty good job of characterizing the run-time function of algorithms
design of algoritms
Design of Algoritms
  • Incremental approach to algorithm design
    • Design for a very small case
    • expand the complexity of the problem and algorithm
  • Divide and Conquer
    • Start with a large (full)problem
    • Divide it into smaller problems
    • Solve smaller problems
    • Combine results from smaller problems
another look at sort algorithms
Another look at Sort algorithms
  • Suppose:
    • you have an array evenly divisible by two
    • in each half (left and right) values are already sorted in order
    • but not in order across the whole array
    • task: sort the array so that it is in order across the entire array
merge sorted subarrays
Merge Sorted subarrays
  • Split the array into two subarrays
  • Add a marker to each subarrays to indicate the end
  • Set index to first value of each subarray
  • Compare indexed (pointed to) value of each subarray
  • If either indexed value is an end-marker: move all remaining values (except the end-mark from the other subarray to the output array; Stop
  • Move the smallest of the two values to the output array (sorted); increment the index to that subarray
  • Go to step 4
merge a p q r
Merge(A, p, q, r)
  • Where A is the array containing values to be sorted, each half is already sorted from smallest to largest
  • p = is the starting point index for the array A
  • q = is the end point index for the left side of array A (end of first half… sort of)
  • r = end index for array A
  • So, sort values from p to r from two halves of array A where q marks where to split the array into subarray
merge a p q r34
Merge(A, p, q, r)
  • n1 = q – p + 1
  • n2 = r – q
  • c* create subarrays L[1..n1+1] and R[1..n2+1]
  • for i = 1 to n1
  • do L[i] = A[p+i-1]
  • for j = 1 to n2
  • do R[j] = A[q+j]
  • L[n1+1] = 
  • R[n2+1] = 
  • i = 1
  • j = 1
  • for k = p to r
  • do if L[i] <= R[j]
  • then A[k] = L[i]
  • i = i + 1
  • else A[k] = R[j]
  • j = j + 1
merge sort a p r
MERGE_SORT(A,p,r)
  • if p < r
  • then q = (p+r)/2
  • MERGE_SORT(A, p, q)
  • MERGE_SORT(A, q+1, r)
  • MERGE(A, p, q, r)
asymptotic notation
Asymptotic Notation
  • Big  (theta)
  • (g(n)) = {f(n) : there exists two constants c1 and c2, n0 such that

0<=c1g(n)<=f(n)<=c2g(n) for all n >=n0}

asymptotic notation38
Asymptotic Notation
  • Big O (oh)
  • O(g(n)) = {f(n) : there positive constants c and n0 such that 0<=f(n)<=cg(n) for all n >=n0}
asymptotic notation39
Asymptotic Notation
  • Big  (Omega)
  • (g(n)) = {f(n) : there positive constants c and n0 such that 0<=cg(n)<=f(n) for all n >=n0}
asymptotic notation40
Asymptotic Notation
  • Little o (oh)
  • o(g(n)) = {f(n) : there positive constants c>0 and n0>0 such that 0<=f(n)<cg(n) for all n >=n0}
asymptotic notation41
Asymptotic Notation
  • Little  (omega)
  • (g(n)) = {f(n) : there positive constants c>0 there exists a constant n0 such that

0<=cg(n)<f(n) for all n >=n0 }

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