CS 615: Design & Analysis of

Algorithms

Chapter 5: SearchingBrassard & Bratley Chap.:

Chapter 9Page 226-228 & 291-327

April 20, 2023 CS 615 Design & Analysis of Algorithms 2

Course Content

1. Introduction, Algorithmic Notation and Flowcharts (Brassard & Bratleyü, Chap. 3)

2. Efficiency of Algorithms (Brassard & Bratley, Chap. 2)

3. Basic Data Structures (Brassard & Bratley, Chap. 5)4. Sorting (Weiss, Chap. 7)

5.5. SearchingSearching ( (Brassard & BratleyBrassard & Bratley,, Chap Chap. 9). 9)6. Graph Algorithms (Weiss, Chap. 9)7. Randomized Algorithms (Weiss, Chap.: 10)8. String Searching (Sedgewick, Chap. 19)9. NP Completeness (Sedgewick, Chap. 40)

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Lecture Content

Binary SearchTraversing TreesDepth-First SearchBreadth-First SearchBacktrackingBranch-and-BoundThe Minimax Principle

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Binary SearchUsed to look up

a word in a dictionaryor a name in telephone directory

Aim: Find a value in a sorted arrayReturn the location of the value in array

Algorithm:Compare the middle element of the arrayif it is equal to the number

Return the index

if the value is less than the middle elementapply the same algorithm to the first part of the array

if the value is greater than the middle elementapply the same algorithm to the second half of the array

execute the operation untilthe value is found

return the index

or there are no more elements to searchreturn not found

April 20, 2023 CS 615 Design & Analysis of Algorithms 5

PseudoCode for Binary Searchfunction binsearch(T[1..n], n)

if n=0 or x>T[n] then return n+1else return binrec(T[1..n],x)

{Binary search for x in subarray T[i..j] with promise that T[i-1]<x<=T[j]}function binrec(T[i..j],x)

if i=j then return ik=(i+j)/2if x=T[k] then return kif x<T[k] then return binrec(T[i..k],x)

else return binrec(T[k+1..j],x)

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Example: Binary SearchSearch for 12:

-5 -2 0 8 8 9 12 13 26 31

i=1 k=5 j=10

i=6 k=8 j=10

i=6 k=7 j=8

1 2 3 4 5 6 7 8 9 10

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Traversing TreesVisiting each node of the tree onceExplore tree from left to right:

PreorderVisit the node itself firstThen all nodes in the left sub-treeFinally all nodes in the right sub-tree

InorderVisit all nodes in the left sub-treeThen the node itselfFinally all nodes in the right sub-tree

PostorderVisit all nodes in the left sub-treeThen all nodes in the right sub-treeFinally the node itself

There are three symmetric techniques

to explore the tree from right to left

A

B

D

H I

E

J K

C

F

L M

G

N O

A B D H I E J K C F L M G N O

Preorder Traversing

H D I B J E K A L F M C N G O

Inorder Traversing

H I D J K E B L M F N O G C A

Postorder Traversing

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Depth-First Search

Go deep in one direction as much as possible

Before looking for another possibility

If the structure isA graph

associate a spanning tree to the graphA tree

a preorder searching

Visited nodes must be markedUse a variable in the strucuture to mark if the node is visited

1

2

5 6

3 4

7 8

1

2

3

6

5

4

7

8

1 2 5 6 3 4 7 8

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Breadth-First Search

Visit all neighboursBefore looking for the child nodes

If the structure isA graph

associate a spanning tree to the graph

A treeVisit all child nodesBefore searching their sub-child nodes

Visited nodes must be markedUse a variable in the strucuture to mark if the node is visited

1

2

5 6

3 4

7 8

1

2

5 6

3 4

7 8

1 2 3 4 5 6 7 8

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Backtracking

When the graph is too large or infinitive

Depth and breadth-first techniques are infeasible It is better to return to a node

That may provide a chance to find the searched node

If the node searched forIs found out that cannot exist in the branchReturn back to previous stepAnd continue the search

1

2

9 14

3 4

7 8

11 15 12 23

Search for 7

Backtrack

Backtrack

Backtrack

Backtrack

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Branch-and-BoundA technique used for

exploring an implicit directed graph

Used to look for optimal solution of a problem

At each nodeCalculate a bound to provide a solutionIf the bound shows that

any such solution is worse than the solution found so far

No need to go on exploring this graphPrune the branchContinue to search

can be used with bothdepth-first or breadth-first search

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The Minimax PrincipleSometimes it is impossible to complete a search

Due to large number of nodesExample: some games like chessThe only solution is

to be content with a partial solution

Minimax:Is a heuristicUsed to find a move

possibly better than all other moves

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Minimax Example

10

-3 10 -8

-7 5 -3 10 -20 0 -5 10 -15 20 1 6 -8 14 -30 0 -8 -9

5 -3 10 10 20 1 14 0 -8

Player: Rule

A max

B

A

B

min

max

min

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End of Chapter 5Searching