Sequential Search Algorithm

SEQUENTIAL SEARCH; WORST, AVERAGE, AND BEST CASE SCENARIOS; AND THE GREATEST COMMON DIVISOR

Ayman Hajja, PhD

SEARCH PROBLEM (SOLUTION)

Given a list of *unsorted* items, what is the best algorithm to find a particular element?

Let’s say I ask you to find element 4

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Set i to 1.!Repeat this loop:!! If A[i] = x, then exit the loop.!! Set i to i + 1.!Return i.

SEARCH PROBLEM (ANALYSIS)

How efficient/fast is this (sequential search) algorithm?

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Simple (naive) answer is:

It depend on the given array (and element you’re trying to find)

Any suggestions…


The answer is: we calculate how efficient our algorithm is in three different scenarios:

Best case scenario

Worst case scenario

Average case scenario


Answer: When the element we’re trying to find is first in the list.

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Example: Find element with value 12

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Only one comparison

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Program Ends

Best Case Scenario. When would that be?


Worst Case Scenario. When would that be?

Answer: When element we’re trying to find is last in the list.

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N time, where N is the size of array

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Average Case Scenario. When would that be?

Answer: When element we’re trying to find is the middle of the list.

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N/2 time, where N is the size of array

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Program Ends


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MAKE NO MISTAKE

AVERAGE *is not (always) equal to* (BEST + WORST)/2

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Average Case Scenario (Revisited). When would that be?

We said earlier: When element we’re trying to find is the middle of the list.

What we should have said: When element we’re trying to find is in its average position, after (perhaps virtually) trying many input configurations


Let’s define the problem again, find element (could be any number between 1 and 9) in a *randomly generated* list of 9 elements containing numbers from 1 to 9

RECALL: When element we’re trying to find is in its average posit ion, after (perhaps vir tually) trying many input configurations

If we’re trying to find element 2 for example, where would its average position be; in this particular problem definition?


If we’re randomly generate 90 lists, we would get

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? 10 times will land here [0 0 0 0 0 0 0 0 0]? 10 times will land here [1 1 1 1 1 1 1 1 1]? 10 times will land here [2 2 2 2 2 2 2 2 2]? 10 times will land here [3 3 3 3 3 3 3 3 3]? 10 times will land here [4 4 4 4 4 4 4 4 4]? 10 times will land here [5 5 5 5 5 5 5 5 5]? 10 times will land here [6 6 6 6 6 6 6 6 6]? 10 times will land here [7 7 7 7 7 7 7 7 7]? 10 times will land here [8 8 8 8 8 8 8 8 8]


Let’s define a different problem: Find element (could be any number between 1 and 9) in a *randomly generated* list of 9 elements containing numbers from 1 to 100

How many comparisons would we need to perform in the average case?

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Think about it…


Another way of looking at it:

The average (running) time of the sequential search algorithm is the average running time to find a particular element; when running the algorithm for many random input configurations


Same goes for Sorting

The average (running) time of any sorting algorithm is the average time it takes to sort a list; when running the algorithm for many random input/list configurations


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When do you think it would be the best case scenario to sort a list in ascending order?

a) When the (input) list given is already sorted in descending order

b) When the (input) list given is already sorted in ascending order

c) When the (input) list is not sorted in either of the two orders


When do you think it would be the worst case scenario to sort a list in ascending order?

a) When the (input) list given is already sorted in descending order

b) When the (input) list given is already sorted in ascending order

c) When the (input) list is not sorted in either of the two orders

d) Hmm. this is hard; it depends maybe?


Same goes for Greatest Common Divisor

The average (running) cost of the any Greatest Common Divisor algorithm is the average time it takes to of comparisons to find a particular element; when running the algorithm for many random input configurations

BACK TO GCD

The *very slow* algorithm: generate all divisors and find the greatest common one

GCD(m, n) will take m+n seconds,

*Assuming each mod will take one second

BACK TO GCD

I’ll call this: GCD Min Algorithm

The *very slow, but better* algorithm: generate all divisors and find the greatest common one, starting with minimum(m, n)

GCD(m, n) will take 2*min(m, n) seconds


Note here that we are generating divisors for m and n independently; for simplicity

BACK TO GCD

I’ll call this: GCD Half-once Algorithm

The *very slow, but better* algorithm: generate all divisors and find the greatest common one, starting with minimum(m, n)

GCD(m, n) will take 2*min(m, n)/2 = min(m, n) seconds


Again, we are generating divisors for m and n independently; for simplicity

BACK TO GCD

For two very large number, min(m, n) is not that great…

Can we do better?

EUCLID’S ELEMENTS

Many consider Euclid of Alexandria a non-mathematician

13 books were written more than 2000 years ago

Books used to be small, so they were merged into one (only ~100 page) book now

Mostly geometry, but few volumes of the 13 books (or sections) were about number theory (topics such as GCD)

EUCLID’S ALGORITHM IS SIMPLE, YET PROFOUND

It makes the bold statement that factorization is *HARD*

EUCLID’S ALGORITHM

Pseudocode function gcd(a, b)! while b ≠ 0! t := b! b := a mod b! a := t! return a!

Example on board

GEOMETRIC REPRESENTATION OF EUCLID’S ALGORITHM

SEARCH PROBLEM WHEN OUR ELEMENTS ARE SORTED

Given a list of *sorted* items, what is the best algorithm to find a particular element?

Let’s say I ask you to find element 12

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Think about for next class

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Sequential Search Algorithm

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