Chapter 2: Design of Algorithms Representing algorithms How to express algorithms the best? Possible solutions: Natural language Programming language Pseudo-Code + Known by everyone (that speaks it) -- Hides the structure of the algorithm -- Allows ambiguity in expressions + Language that computer understands (e.g. Fortran, C, Pascal) + Very exact and well structured -- Very restrictive particularly in early design phases

Representing Algorithms Pseudo-Code: golden medium Bases on a natural language (e.g. English) Uses “programming-language”-like notation (e.g. statements, conditions etc.) Examples Set x to the value 45 Return Go to step 100 if condition is satisfied Types of pseudo-code instructions Sequential operations Conditional operations Iterative operations

Representing Algorithms Sequential operations Input/Output: Communication with the outside Input: Get the value for “variable1” “variable2” … Example: Get the value of x y z Output Print the value for “variable1” “variable2” … Example: Print the value of x y z Print the message ‘message content’ Example: Print the message ‘An error has been detected’ Computation: Set the value of “variable” to “arithmetic operation” Example: Set the value of Area to p*r2

Representing Algorithms Conditional operations: If “condition” is true then Set of algorithmic operations Else Example: If denominator = 0 then Print the message ‘Operation cannot be done’ Else Set fraction to numerator/denominator

Representing Algorithms Iterative operations Repeat Step i to j until “condition” becomes true Step i: … Step j: Repeat until “condition” becomes true Set of operations End of loop While “condition” remains true do End of the loop

Representing Algorithms Examples of iterative operations Printing squares of 1 to n: Get the value of n While n > 0 do Print n2 Set n to n-1 The same algorithm with a repeat loop: If n > 0 then Repeat until n < 1 becomes true Difference between while and repeat: While: pre-check (continuation) condition then perform work (# of times 0, 1, 2, …) Repeat: perform work then post-check (termination) condition (# of times 1, 2, …)

Representing Algorithms Are the three types of operations (sequential, conditional, iterative) sufficient to represent ANY possible valid algorithm?  Y E S ! Even the most complex algorithms used in e.g. international switching systems can be represented using said operations Compare: We only need few letters (26 in English) to write the most marvelous novels.

Algorithmic Problem Solving Sequential Search Algorithm Problem Statement: Given a name N and a telephone book including names N1 to N1000 and phone numbers T1 to T1000 Task: Find the phone number T of the name N First solution: 1. Get N, N1,…,N1000, T1, …T1000 2. If N = N1 then Print the value T1 … 1001. If N = N1000 then Print the value T1000 1002. Stop

Sequential Search Algorithm Assessment of the first solution +: Algorithm prints all phone numbers of the given name N --: Too long listing (1001 lines) --: Slow: it checks always all entries even after the phone number has been already found --: No error message if the name is not in the list

Sequential Search Algorithm Second solution: 1. Get N, N1, …, N1000, T1, …, T1000 2. Set i to 1 and mark the N as not yet found 3. Repeat 4 to 7 until N is found 4. If N = Ni then 5. Print the Phone number is Ti 6. Mark N as found Else 7. Add 1 to i 8. Stop

Sequential Search Algorithm Assessment of the second solution +: Very short listing (only 8 lines) +: Quick if N is in the list, it does not check all entries in all cases --: Big problem: Endless loop if N is not in the list Final solution: 1. Get N, N1, …, N1000, T1, …, T1000 2. Set i to 1 and mark the N as not yet found 3. Repeat 4 to 7 until either N is found or i > 1000 4. If N = Ni then 5. Print the Phone number is Ti 6. Mark N as found Else 7. Add 1 to i 8. If N is not marked as found then Print message ‘Name is not in the directory’ 9. Stop

Finding the Maximum Given a list of n numbers Task: find the largest number in the list Solution: 1. Get the value n 2. Get the values A1, A2, …, An 3. Set the value of maximum to A1 4. Set the value of location to 1 5. Set the value of i to 2 6. While i <= n do 7. If Ai > maximum then 8. Set maximum to Ai 9. Set location to I 10. Add 1 to I End of the loop 11. Print maximum and location 12. Stop

Pattern Matching Given a text T (sequence of letters) and a pattern (e.g. word) p Find the occurrence of p in T Example: Text: let me know whether or not to go Pattern: no Output: Match starting at position 9 Pattern: no Output: Match starting at position 24 Example: Genome research Genome: … T C G A T T G T C C C A G T G C A A A C T G C A T … Probe: A A A (a match)

Pattern Matching Solution: 1. Get values n and m the size of the text and the pattern 2. Get T1,…,Tn and P1,…,Pm 3. Set k to 1 4. Repeat until k > n-m+1 (until we treated the entire text) 5. Try to match the pattern P1…Pm at position k 6. If there was a match then 7. Print the value of k the starting location of the match 8. Add 1 to k (slides to the right) End of loop 9. Stop

Pattern Matching Step 5 ??? 5. Try to match the pattern P1…Pm at position k Situation: Text: T1 T2 … Tk Tk+1…Tk+m-1 Tk+m …Tn Pattern: P1 P2 … Pm Algorithm for Step 5 Set i to 1 and mismatch to no Repeat until (i > m) or mismatch = yes If Pi not equal Tk+i-1 then Set mismatch to yes Else Set i to i+1 End of loop