Algorithm Analysis

1
Algorithm Analysis

• Algorithm
• An algorithm is a clearly specified set of simple
  instructions to be followed to solve a problem .
• Running time analysis
• Once an algorithm is decided to be correct for a
  problem , an important step is to determine how
  much recourses it requires
• time
• space

Our concern is from now on is not only to things
right , but to them efficiently !
2
Algorithm Analysis

• In our analysis our main concern will be the
  running time , as a funcition of the input size .
  
• We will find the big Oh for the algorithm we will
  analyze (O(f(n))
• There are two aspects usually considered
• .Worst case
• .Average case
• Generally the required quantity required is the
  worst case time , unless otherwise specified .
  One reason for this is to provide a bound for all
  input , including very bad ones .

3
Algorithm Analysis

• Maximum subsequence sum problem
• Given (possibly negative ) integers A1, A2,, An
  find the maximum value of Ak ( k form I to j )
  ( for convenience , the maximum subsequence sum
  is 0 if all integers are negative )
• Example
• For input 2 , 11, -4 , 13, -5, -2 the answer
  is 20 ( A2 through A4).

What is the most efficient solution?
4
Algorithm Analysis-Maximum subsequence sum
problem first solution

• Public static in maxSubSum1(inta)
• int maxSum 0
• for (int I 0 I lta.length I)
• for ( int j I j lt a.length j )
• for ( int k I k lt j k )
• thisSum a k
• if (thisSum gtmaxSum)
• maxSum thisSum
• return maxSum

What is the running time ?
5
Algorithm Analysis-Maximum subsequence sum
problem first solution

• The first loop is preformed n times ( the size of
  the input)
• The second loop is preformed different amount of
  times each time , but since we are looking for an
  upper bound we take the worst case and consider
  it to be O(n)
• The third loop is O(n) for the same reason .
• The running time is thus O(1NNN) O(n3)
• A more accurate analysis will bring us to the
  following computation ((n3)(3N2)(2N))/6
• This leaves us with our original analysis.

6
Algorithm Analysis-Maximum subsequence sum
problem second solution

• Public static int maxSubSum2(inta)
• int maxSum 0
• for ( int I 0 Ilt a. length I )
• int thisSum 0
• for ( int j I j lt a.length j )
• thisSum aj
• if ( thisSum gt maxSum)
• maxSum thisSum
• return maxsum

Running time is O(n2)
7
Algorithm Analysis-Maximum subsequence sum
problem third solution

• Public static int maxSum4(inta)
• int maxSum 0 , thisSum 0
• for (int j 0 j lta.length j)
• thisSum a j
• if ( thisSum gt maxsum )
• maxSum thisSum
• else if ( thisSum lt 0)
• thisSum 0
• return maxSum

8
Algorithm Analysis-Maximum subsequence sum
problem third solution

• As in many clever algorithms the running time is
  obvious , but the correctness is not .
• The running time is linear O (n)
• convince yourself about its correctness .

We have seen that there are many ways to solve a
problem , and only a few ways to solve it
efficiently !!
9
Algorithm Analysis
Evaluate the number of operations in the
following code
10
Algorithm Analysis

• Solution
• for-loop executed n times O(n)
• inner for-loop executed i timesO(i)
• while-loop executed 1 time O(1)