1. Analysis & Design of Algorithms (CSCE 321)
Prof. Amr Goneid
Department of Computer Science, AUC
Part 4. Brute Force Algorithms
Prof. Amr Goneid, AUC

2. Brute Force Algorithms
Prof. Amr Goneid, AUC

3. Brute Force Algorithms
Elementary Sorting Algorithms
Selection Sort
Bubble Sort
Insertion Sort
Brute Force Sequential Search
Brute Force String Matching
Closest Pair Problem
Exhaustive Search
Traveling Salesman Problem
0/1 Knapsack Problem
Assignment problem
Prof. Amr Goneid, AUC

4. 1. Brute Force Algorithms
Brute Force is a straightforward approach to solving a problem, usually directly based on the problem’s statement and definitions of the concepts involved
In many cases, Brute Force does not provide you a very efficient solution
Brute Force may be enough for moderate size problems with current computers….
Prof. Amr Goneid, AUC

5. 2. Elementary Sorting Algorithms
Selection Sort
Bubble Sort
Insertion Sort
Prof. Amr Goneid, AUC

6. Sorting The Sorting Problem:
Input: A sequence of keys {a1 , a2 , …, an}
output: A permutation (re-ordering) of the input,
{a’1 , a’2 , …, a’n} such that a’1 ≤ a’2 ≤ …≤ a’n
Sorting is the most fundamental algorithmic problem in computer science
Sorting can efficiently solve many problems
Different sorting algorithms have been developed, each of which rests on a particular idea.
Prof. Amr Goneid, AUC

7. Some Applications of Sorting
Efficient Searching
Ordering a set so as to permit efficient binary search.
Uniqueness Testing
Test if the elements of a given collection of items are all distinct.
Deleting Duplicates
Remove all but one copy of any repeated elements in a collection.
Prof. Amr Goneid, AUC

8. Some Applications of Sorting
Closest Pair
Given a set of n numbers, find the pair of numbers that have the smallest difference between them
Median/Selection
Find the kth largest item in set S. Can be used to find the median of a collection.
Frequency Counting
Find the most frequently occurring element in S, i.e., the mode.
Prioritizing Events
Sorting the items according to the deadline date.
Prof. Amr Goneid, AUC

9. Types of Sorts In-Place Sort:
An in-place sort of an array occurs within the array and uses no external storage or other arrays
In-place sorts are more efficient in space utilization
Stable Sorts:
A sort is stable if it preserves the ordering of elements with the same key.
i.e. If elements e1 and e2 have the same key, and e1 appears earlier than e2 before sorting, then e1 is located before e2 after sorting.
Prof. Amr Goneid, AUC

10. Sorting What will count towards T(n):
Comparisons of array elements
Swaps or shifts of array elements
We compare sorting methods on the bases of:
The time complexity of the algorithm
If the algorithm is In-Place and Stable or not
Prof. Amr Goneid, AUC

11. Sorting We examine here 3 elementary sorting algorithms:
Selection Sort
Bubble Sort
Insertion Sort
These have a complexity of O(n2)
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12. (a) Selection Sort
The general idea of the selection sort is that for each slot, find the element that belongs there
Assume elements to be in locations 0..n-1
Let (i) be the start of a sub-array of at least 2 elements, i.e. i = 0 .. n-2
for each i = 0 .. n-2
Find smallest element in sub-array a[i..n-1]
Swap that element with that at the start of the
sub-array (i.e. with a[i]).
Prof. Amr Goneid, AUC

13. How it works 4 3 1 6 2 5 1 3 4 6 2 5 1 2 4 6 3 5 1 2 3 6 4 5 1 2 3 4 6 5 1 2 3 4 5 6
Prof. Amr Goneid, AUC

14. Demos http://www.cosc.canterbury.ac.nz/people/mukundan/dsal/SSort.html
Prof. Amr Goneid, AUC

15. Selection Sort Algorithm
First, let us compute the cost of finding the location of the minimum element in a subarray a[s..e ]
minimum (a,s,e)
{ m = s ;
for j = s+1 to e
if (a[j]  a[m]) m = j ;
return m; }
Hence, T(s,e) = e – s
For an array of size (n) it will cost T(n) = n-1 comparisons
Prof. Amr Goneid, AUC

16. Selection Sort Algorithm
SelectSort (a[0..n-1 ]) {
for i = 0 to n-2
m = minimum (a , i , n-1) ;
swap (a[i] , a[m]); }
Prof. Amr Goneid, AUC

17. Analysis of Selection Sort
Exact algorithm
Prof. Amr Goneid, AUC

18. Performance of Selection Sort
The complexity of the selection sort is  (n2), independent of the initial data ordering.
In-Place Sort Yes
Stable Algorithm No
This technique is satisfactory for small jobs, not efficient enough for large amounts of data.
Prof. Amr Goneid, AUC

19. (b) Bubble Sort
The general idea is to compare adjacent elements and swap if necessary
Assume elements to be in locations 0..n-1
Let (i) be the index of the last element in a sub-array of at least 2 elements, i.e. i = n
Prof. Amr Goneid, AUC

20. Bubble Sort Algorithm for each i = n-1…1
Compare adjacent elements aj and aj+1 , j = 0..i-1 and swap them if aj > aj+1
This will bubble the largest element in the sub-array
a[0..i] to location (i).
Prof. Amr Goneid, AUC

21. How it works 4 3 1 2 3 4 1 2 3 1 4 2 3 1 2 4 1 3 2 4 1 2 3 4
Prof. Amr Goneid, AUC

22. Demos http://www.cosc.canterbury.ac.nz/people/mukundan/dsal/BSort.html
Prof. Amr Goneid, AUC

23. Bubble Sort Algorithm BubbleSort (a[ 0..n-1]) { for i = n-1 down to 1
for j = 0 to i-1
if (a[j] > a[j+1] ) swap (a[j] , a[j+1]); }
Prof. Amr Goneid, AUC

24. Analysis of Bubble Sort
Worst case when the array is inversely sorted:
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25. Bubble Sort (a variant)
void bubbleSort (itemType a[ ], int n)
{ int i , j; bool swapped;
for (i = n-1; i >= 1; i--)
{ swapped = false;
for (j = 0; j < i; j++ )
{ if (a[j] > a[j+1] )
{ swap(a[j] , a[j+1]); swapped = true; } }
if (!swapped) return;
Prof. Amr Goneid, AUC

26. Performance of Bubble Sort
The worst case complexity of the bubble sort is when the array is inversely sorted: O(n2).
The variant has a best case when the array is already sorted in ascending order: Ω(n). Outer loop works for only i = n-1 and there will be no swaps.
In-Place Sort Yes
Stable Algorithm Yes
For small jobs, not efficient enough for large amounts of data.
Prof. Amr Goneid, AUC

27. (c) Insertion Sort
The general idea of the insertion sort is that for each element, find the slot where it belongs.
Performs successive scans through the data. When an element is out of sequence (less than its predecessor), it is pulled out and then inserted where it should belong.
Array elements have to be shifted to the right to make space for the insertion.
Prof. Amr Goneid, AUC

28. How it works 2 7 1 5 8 6 3 4 1 2 7 5 8 6 3 4 1 2 5 7 8 6 3 4 1 2 5 6 7 8 3 4 1 2 3 5 6 7 8 4 1 2 3 4 5 6 7 8
Prof. Amr Goneid, AUC

29. Demos http://www.cosc.canterbury.ac.nz/people/mukundan/dsal/ISort.html
Prof. Amr Goneid, AUC

30. Insertion Sort Algorithm
InsertSort (a[0..n-1 ])
{ for i =1 to n-1 {
j = pointer to element ai;
v = copy of ai;
while( j > 0 && aj-1 > v)
{ move data right: aj ← aj-1
move pointer left: j-- }
Insert v at last (j) location: aj ← v; }
Prof. Amr Goneid, AUC

31. Analysis of Insertion Sort
Worst case when the array is inversely sorted, while loop iterates (i) times:
Best case when the array is already sorted in ascending order, while loop works zero times
Prof. Amr Goneid, AUC

32. Performance of Insertion Sort
The worst case complexity of the insertion sort is when the array is inversely sorted: O(n2).
Best case when the array is already sorted in ascending order: Ω(n). While loop will not execute and there will be no data movement.
In-Place Sort Yes
Stable Algorithm Yes
For modest jobs, not efficient enough for large amounts of data.
Prof. Amr Goneid, AUC

33. 3. Brute Force Sequential Search
Find whether a search key is present in an array
Worst case:
Key not found.
T(n) = O(n)
comparisons
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34. 4. Brute Force String Matching
Pattern: a string of m characters to search for
Text: a (longer) string of n characters to search in
Problem: find a substring in the text that matches the pattern
Example: Pattern – ‘NOT’,
text – ‘NOBODY_NOTICED_HIM’
Typical Applications
‘find’ function in the text editor, e.g., MS-Word, Google search
Prof. Amr Goneid, AUC

35. Brute Force String Matching
Brute-force algorithm
Step 1 Align pattern at beginning of text
Step 2 Moving from left to right, compare each character of pattern to the corresponding character in text until
all characters are found to match (successful search); or
a mismatch is detected
Step 3 While pattern is not found and the text is not yet exhausted, realign pattern one position to the right and repeat Step 2
Prof. Amr Goneid, AUC

36. Brute Force String Matching
T[0..n-1]: Text string P[0..m-1]: Pattern string
Show that the worst case number of comparisons is O(mn)
Prof. Amr Goneid, AUC

37. 5. Closest Pair Problem Problem:
Find the two closest points in a set of n points (in the two
dimensional Cartesian plane).
Brute-force algorithm
Compute the distance between
every pair of distinct points
Return the indexes of the points for
which the distance is the smallest.
Prof. Amr Goneid, AUC

38. Closest Pair Problem
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39. Closest Pair Problem
The Hamming distance between two strings of equal length is defined as the number of positions at which the corresponding symbols are different. It is named after Richard Hamming (1915–1998), a prominent American scientist.
Suppose that the points in the problem of closest
pairs are strings and hence the distance between
two of such strings is now measured by the
Hamming Distance. Write an algorithm to find the
closest pair of strings in an array of strings and find
the number of comparisons done by this algorithm.
Prof. Amr Goneid, AUC

40. 6. Exhaustive Search A brute-force approach to combinatorial problem
Generate each and every element of the problem’s domain
Then compare and select the desirable element that satisfies the set constraints
Involve combinatorial objects such as permutations, combinations, and subsets of a given set
The time efficiency is usually bad – usually the complexity grows exponentially with the input size
Three examples
Traveling salesman problem
Knapsack problem
Assignment problem
Prof. Amr Goneid, AUC

41. (a)Traveling Salesman Problem (TSP)
Given n cities with known distances between each pair, find the shortest tour that passes through all the cities exactly once before returning to the starting city
Alternatively: Find shortest Hamiltonian circuit in a weighted connected graph (The circuit problem is named after Sir William Rowan Hamilton)
Example: a b c d 8 2 7 5 3 4
Prof. Amr Goneid, AUC

42. Traveling Salesman Problem (TSP)
Hamiltonian Circuit Cost
a→b→c→d→a = 17 Optimal
a→b→d→c→a = 21
a→c→b→d→a = 20
a→c→d→b→a = 21
a→d→b→c→a = 20
a→d→c→b→a = 17 Optimal
Number of candidate circuits = (n-1)!
Very High complexity O(n!)
Prof. Amr Goneid, AUC

43. (b) 0/1 Knapsack Problem Given n items: weights: w1 w2 … wn
values: v1 v2 … vn
a knapsack of capacity W
Find most valuable subset of the items that fit into the knapsack
Example: Knapsack capacity W=16
item weight value ($)
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44. 0/1 Knapsack Problem Generates 2n possible subsets, T(n) = O(2n)
Subset Total weight Total value
{ } $00
{1} $20
{2} $30
{3} $50
{4} $10
{1,2} $50
{1,3} $70
{1,4} $30
{2,3} $80 Optimal
{2,4} $40
{3,4} $60
{1,2,3} not feasible
{1,2,4} $60
{1,3,4} not feasible
{2,3,4} not feasible
{1,2,3,4} not feasible
Generates 2n possible subsets,
T(n) = O(2n)
Prof. Amr Goneid, AUC

45. (c) Assignment Problem by Exhaustive Search
There are n people who need to be assigned to n jobs, one person per job. The cost of assigning person i to job j is C[i,j]. Find an assignment that minimizes the total cost.
Job 1 Job 2 Job 3 Job 4
Person
Person
Person
Person
Algorithmic Plan:
Generate all legitimate assignments, compute their costs, and select the cheapest one.
How many assignments are there? n!
Pose the problem as one about a cost matrix:
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46. Assignment Problem by Exhaustive Search
Assignment (col.#s) Total Cost
1, 2, 3, =18
1, 2, 4, =30
1, 3, 2, =24
1, 3, 4, =26
1, 4, 2, =33
1, 4, 3, =23
etc.
T(n) = # of assignments = O(n!)
Prof. Amr Goneid, AUC