1 algorithms. algorithms introduction recipe for baking a cake. 2 sticks butter 2 cups flour 1 cup...
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Algorithms
Algorithms Introduction Recipe for baking a cake….• 2 sticks butter• 2 cups flour• 1 cup sugar• 4 eggs• 1 cup milk• 1 tsp baking powder• Cocoa powder (1/2 pound)Mix the sugar, baking powder and flour, mix in beaten
eggs, melted butter and bake at 325F for 40 mins.
Cooking exampleSalad: 25m prep, 0m cooking
Chicken noodle: 10m prep, 40 min cooking
Rice pudding:15 mins prep, 20m cooking
In what order should Martha make the dishes?
• Martha can work on preparing one dish at a time, however once something is cooking, she can prepare another dish.
• How quickly can she get all the dishes ready?
• She starts at 5pm, and her guests will arrive at 6pm….
First try
5:00pm 5:25pm5:35pm
5:50pm
6:15pm 6:10pm
(25,0) (10,40) (15,20)
Prep time Cook time
Second try
5:00pm 5:10pm5:25pm 5:50pm
5:50pm 5:45pm
(10,40) (15,20)(25,0)
First work on dishes with shortest preparation time?
This rule may not work all the time
Suppose the required times are: Bulgur (5,10) Lentils (10, 60) Lamb (15, 75)Shortest prep time order: start at 5pm, and
finish lamb at 6:45pmLongest cooking time first: food ready at
6:30pm.
What if she had to make several dishes?
• For 3 dishes, there are only 6 possible orders. SCR,SRC,RSC,RCS,CSR,CRS.
• The number of possible orderings of 10 dishes is 3,628,800.
• For 15 dishes the number of possible orderings is 1,307,674,368,000!
• This leads to a combinatorial explosion.
Key Idea• Order dishes in longest cooking time order.• Chicken noodle soup goes first (40 mins of cook
time), next is the Rice pudding (20 mins of cook time), followed by the Salad (0 mins of cook time).
• This is the best ordering. In other words, no other order can take less time.
• This does not work if there are very few stovetops (now the problem becomes really difficult).
What if we had a small number of burners?
• Problem becomes very difficult if we have 2, 3, 4 burners..
• Problem can be solved optimally if we only have one burner (Johnson, 1954)
Figure 8-1
Informal definition of an algorithm used in a computer
Informal definition
Finding the largest integer among five integers
Defining actions in FindLargest algorithm
FindLargest refined
Generalization of FindLargest
Three constructs
Algorithm representation
• Flowchart – A flowchart is a pictorial representation of
an algorithm.
• Pseudocode – Pseudocode is an Englishlike
representation of an algorithm.
Flowcharts for three constructs
Pseudocode for three constructs
Write an algorithm to find the largest of a set of numbers.
FindLargestInput: A list of positive integers
1. Set Largest to 02. while (more integers)
2.1 if (the integer is greater than Largest) then 2.1.1 Set largest to the value of the
integer End ifEnd while
3. Return LargestEnd
Find largestFind largest
Find Largest
FIND-LARGEST (A, n) ⊳ largestlargest ← 0for i ← 1 to n
if A[i] > largestlargest ← A[i]
return largest
“pseudocode”
The problem of sorting
Input: sequence a1, a2, …, an of numbers.
Example:Input: 8 2 4 9 3 6
Output: 2 3 4 6 8 9
Output: a'1, a'2, …, a'n such that
a'1 <= a'2 <=… <= a'n
Insertion sortINSERTION-SORT (A, n) ⊳ A[1 . . n]
for j ← 2 to ndo key ← A[ j]
i ← j – 1while i > 0 and A[i] > key
do A[i+1] ← A[i]i ← i – 1
A[i+1] = key
“pseudocode”
i j
keysorted
A:1 n
Example of insertion sort8 2 4 9 3 6
Example of insertion sort8 2 4 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
2 3 4 8 9 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
2 3 4 8 9 6
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
2 3 4 8 9 6
2 3 4 6 8 9 done
https://www.youtube.com/watch?v=ROalU379l3U&list=PL58zywNQ04Laefu_tC8oMwbH4M929HPnG&index=13
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . n/2
] and A[ n/2+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
Merging two sorted arrays
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https://www.youtube.com/watch?v=XaqR3G_NVoo&index=9&list=PL58zywNQ04Laefu_tC8oMwbH4M929HPnG
Searching
• Searching– The process of finding the location of a
target among a list of objects.
– Sequential search– Binary search
Figure 8-19
Search concept
Figure 8-20: Part I
Example of a sequential search
Figure 8-20: Part II
Example of a sequential search
Figure 8-21
Example of a binary search
References• www.cs.umd.edu/~samir/DSTTalk.ppt• http://courses.csail.mit.edu/6.046/spring04/
lectures/l1.ppt• http://www.csie.ntnu.edu.tw/~violet/cs92/
ch08.PPT
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