Lecture 1
Analyze the efficiency of algorithms
CSE354-Algorithms and Data Structure © Dr. Ahmed Ayoub, 2019
Learning Objectives
By the end of this lecture you will be able to:
 Define the Model of computation
 Select what to Analyze
 Perform Running-Time calculations
 Solve a simple Example
 Recognize the General Rules
 State the limitations of worst-case
analysis
Model of computation
 To analyze algorithms in a formal framework.
 Model: a computer with instructions executed
sequentially.
 Standard simple instructions: addition,
multiplication, comparison, and assignment.
 Takes exactly one time unit to do anything
(simple.)
 Has fixed-size integers.
 Has infinite memory.
Model of computation
 This model clearly has some weaknesses:
 In real life, not all operations take exactly the
same time.
 By assuming infinite memory, we ignore the fact
that the cost of a memory access can increase
What to Analyze?
 The most important resource to analyze is
generally the running time.
 Factors affect the running time of a program:
 compiler and computer used (cannot deal with)
 algorithm used and
 the input size to the algorithm.
What to Analyze?
 We define two functions:
 Tavg(N), the average running time
 Tworst(N), worst-case running time
 N, input of size used by an algorithm
 Clearly, Tavg(N) ≤ Tworst(N). If there is more than one
input.
 These functions may have more than one
argument!
Example
Given integers: A1, A2, …, AN find the maximum value of
 Small inputs: no need to design a clever algorithm.
 Large inputs, algorithm 4 is clearly the best choice
Running-Time calculations
 Generally, there are several algorithmic ideas, and
we would like to eliminate the bad ones early
 An analysis is usually required.
 The ability to do an analysis usually provides insight
into designing efficient algorithms.
 The analysis also generally pinpoints the
bottlenecks, which are worth coding carefully.
Running-Time calculations
 To simplify the analysis, we adopt the convention
that there are no particular units of time.
 Thus, we throw away leading constants.
 Also, throw away loworder terms,
 So what we are essentially doing is computing a
Big-Oh running time.
 The answer provided guarantees that the program
will terminate within a certain time period, but
never later.
A simple Example
Program fragment to calculate:
 The declarations count for no time.
 Lines 1 and 4 count for one unit each.
 Line 3 counts for 4 units per time
executed N times, for a total of 4N.
 Line 2 has the hidden costs of initializing
i, testing i ≤ N, and incrementing i.
 The total cost of all these is 1 to
initialize, N+1 for all the tests, and N for
all the increments, which is 2N + 2.
 We ignore the costs of calling the
function and returning,
 for a total of 6N + 4.
 Thus, we say that this function is O(N).
The General Rules




Rule 1—FOR loops: O(N)
Rule 2—Nested loops: O(N2)
Rule 3—Consecutive Statements: O(N)
Rule 4—If/Else: O(N)
The limitations of worst-case analysis
 Sometimes the analysis is shown empirically to be
an overestimate:
 the analysis needs to be tightened; or
 the average running time is significantly less
than the worst-case running time
 For many complicated algorithms the worst-case
bound is achievable by some bad input
 Unfortunately, an average-case analysis is
extremely complex, and
 a worst-case bound is the best analytical result
known!
Conclusion
 Define the Model of computation
 Select what to Analyze
 Perform Running-Time calculations
 Solve a simple Example
 Recognize the General Rules
 State the limitations of worst-case
analysis