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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

- 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.

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

Algorithms

- Human Genome Project
- Security/encryption for e-commerce
- Spacecraft navigation
- Pulsar searches
- Search Engines

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

Algorithms

- Search Google for “house”
- 730,000,000 hits in 0.1 seconds

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

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

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

- 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

- 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-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-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-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(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 may be concerned with any resources
- memory
- bandwidth
- runtime
- Need a model for describing runtime performance of an algorith
- RAM – Random Access Machine

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

RAM

- Need to base analysis on cost of instruction execution
- assign costs (run-time) to each instruction

INSERTION-SORT

- Run-time = sum of products of costs (instruction runtimes) and execution occurrences
- T(n)= c1n + c2(n-1) + c4(n-1) +

c5nj=2tj +c6nj=2(tj-1) + c7nj=2(tj-1)

+c8(n-1)

INSERTION-SORT

- Best case vs Worst Case
- Best case
- Input array already sort
- Worst case
- Input array sorted in reverse order

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

* Best case is a linear function of n

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

- 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 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 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 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

- 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

- 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

- 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)

- 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, 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)

- 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

- 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 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 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 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 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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