Describing and Implementing Algorithms

Transcription

1 Describing and Implementing Algorithms ECE2036 Lecture 1 ECE2036 Describing and Implementing Algorithms Spring / 19

2 What is an Algorithm? According to Wikipedia: An algorithm is a sequence of instructions, often used for calculation and data processing. It is formally a type of effective method in which a list of well-defined instructions for completing a task that will, when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. ECE2036 Describing and Implementing Algorithms Spring / 19

3 Graphical Representation of a Simple Algorithm Stop, Lamp Works Yes Does Lamp Work? No No Lamp Plugged In? Yes Bulb Burned Out? Yes No Plug in Lamp Replace Bulb Buy New Lamp Question: Is this algorithm guaranteed to terminate? Start ECE2036 Describing and Implementing Algorithms Spring / 19

4 Revised Simple Algorithm Yes Stop, Lamp Works Does Lamp Work? No No Lamp Plugged In? Plug in Lamp Yes Yes Bulb Burned Out? Replace Bulb No No Stop, Can t Fix Enough Money? Yes Buy New Lamp Guaranteed to terminate assuming less than infinite money. Start ECE2036 Describing and Implementing Algorithms Spring / 19

5 Euclid s Algorithm Algorithm E Euclid s algorithm. Given two positive integers m and n, find their greatest common divisor (GCD), which is the largest positive integer which evenly divides both m and n. E1 Divide m by n and let r be the remainder. (We will have 0 r < n.) E2 If r = 0 the algorithm terminates and n is the answer. E3 Set m n, n r and return to step E1. Algorithm description by Donald E. Knuth, Art of Computer Programming. Question: Is Euclid s algorithm guaranteed to terminate? ECE2036 Describing and Implementing Algorithms Spring / 19

6 Algorithm Primitives SET variable1 newvalue1, variable2 newvalue2 Set one or more variables to a new value Example: E1 SET m 0, n k 3, x ((k + 2) j)/(y + 2) WHILE Condition Perform all substeps as long as Condition is true. Example: E2 WHILE k > 0 AND J = 3 E2.1 SET k k 1 E2.2 SET x = 0 IF Condition Perform all substeps only if the Condition is true Example: E3 IF k = 10 E3.1 SET j j 1 E3.2 SET k k 1 ELSE Perform all substeps only if the Condtion on the previous IF statement is false. Example: E4 IF k = 10 E4.1 SET j j 1 E4.2 SET k k 1 E5 ELSE E5.1 SET x 10 E5.2 SET y k + 50 RETURN Terminate the algorithm and return the result. Example: E6 RETURN Ans ECE2036 Describing and Implementing Algorithms Spring / 19

7 Euclid s Algorithm (revised) Algorithm E Euclid s algorithm. Given two positive integers m and n, find their greatest common divisor (GCD), which is the largest positive integer which evenly divides both m and n. E1 SET Ans 0. E2 WHILE Ans = 0 E2.1 SET r m MODULO n E2.2 IF r = 0 E2.2.1 SET Ans n E2.3 ELSE E2.3.1 SET m n, n r. E3 RETURN Ans ECE2036 Describing and Implementing Algorithms Spring / 19

8 Euclid s Algorithm, C Code / / Implementation of Euclids a l g o r i t h m / / George F. Riley, Georgia Tech, Summer 2008 #include <iostream > using namespace std ; #include " cs1372. h " i n t Euclid ( i n t m, i n t n ) { i n t Ans = 0; cout << " I n i t i a l m " << m << " n " << n << endl ; while ( Ans == 0) { / / Loop u n t i l a l g o r i t h m terminates i n t r = m % n ; / / Remainder when m d i v i d e d by n i f ( r == 0) / / See i f algorithm has terminated { / / Algorithm has terminated, n is the answer Ans = n ; else { / / Interchange m = n ; n = r ; cout << "New m " << m << " n " << n << endl ; PrintLn ( "GCD is ", Ans ) ; return Ans ; i n t main ( ) { Euclid (119, 544); ECE2036 Describing and Implementing Algorithms Spring / 19

9 Euclid s Algorithm, Java Code class e u c l i d { public s t a t i c i n t Euclid ( i n t m, i n t n ) { i n t Ans = 0; while ( Ans == 0) { i n t r = m % n ; i f ( r == 0) { Ans = n ; else { m = n ; n = r ; System. out. p r i n t l n ( "GCD i s " + Ans ) ; return Ans ; public s t a t i c void main ( String [ ] args ) { Euclid (544, 119); ECE2036 Describing and Implementing Algorithms Spring / 19

10 Euclid s Algorithm, TCL Code #! / usr / bin / t c l s h proc Euclid { m n { set Ans 0 while { $Ans == 0 { set r [ expr $m % $n ] i f { $r == 0 { set Ans $n else { set m $n set n $r puts "GCD i s $Ans " return $Ans E u clid ECE2036 Describing and Implementing Algorithms Spring / 19

11 Euclid s Algorithm, Assembly Code # Program to execute euclid s a l g o r i t h m # George F. Riley, F a l l 2008 LDI R0, 0 # Set R0 t o zero LDM R1,m # Get value f o r m LDM R2, n # Get value f o r n Loop : MOD R3, R1, R2 # Remainder of m / n BEQ R3, R0, Done # Done, s t o r e r e s u l t LD R1, R2 # Set m to n LD R2, R3 # Set n to r BEQ R0, R0, Loop # Return to step E1 Done : STM R2, gcd # Store r e s u l t HLT ECE2036 Describing and Implementing Algorithms Spring / 19

12 Defining and Accessing Arrays A variable name followed by brackets [] indicates an Array. Example: Given an array d[], compute... The number of elements in the array is not specified. Usually a separate variable would indicate the length. Example: Given an array d[] containing Length elements... Individual variables in an array are identified by an index, which is specified in one of two ways: SET d[i] 1 SET d i 1 IMPORTANT! The index of the first array element is always ZERO. SET d[0] 10 Set the first element of d to 10 If variable Length indicates the number of elements, then the last element is d[length 1] Arrays can be multi dimensional. Example: Given an array d2[][], compute... Multi dimensional arrays have more than one index. SET d[i, j] 1 SET d ij 1 ECE2036 Describing and Implementing Algorithms Spring / 19

13 Averaging Algorithm Algorithm A Compute the average of an array of numbers. Given an array d[] of length N, compute the average value of all elements of d[]. A1 SET SUM 0, i 0 A2 WHILE i < N A2.1 SET Sum Sum + d[i] A2.2 SET i i + 1 A3 RETURN Sum / N ECE2036 Describing and Implementing Algorithms Spring / 19

14 Algorithmic Complexity For algorithm A, can we predict how long we could expect the algorithm to take to execute? This does not mean in terms of seconds or time, but rather in terms of steps or operations We can think of each step in the algorithm as an operation In our simple algorithm, it is easy to see that the number of operations is C 0 + C 1 N, where C 0 and C 1 are constants we will compute next, and N is the number of elements in array d[] C 0 is the number of operations needed to initialize and exit the algorithm, three in this case (two initializations and one return). C 1 is harder to determine. We could claim it is as small as three or as large as eight. Regardless of the values of C 0 and C 1, we can say definitively that the computational complexity of the averaging algorithm is proportional to N. If the running time of A for N = 1000 is t seconds The running time of A for N = will be approximately 10t ECE2036 Describing and Implementing Algorithms Spring / 19

15 Averaging Algorithm, C Code / / Compute a r i t h m e t i c average of an array of values / / George F. Riley, Georgia Tech, Summer 2008 double Average ( i n t d [ ], i n t n ) { / / d [ ] i s array of values, n i s l e n gth of the array i n t Sum = 0; i n t i = 0; while ( i < n ) { Sum = Sum + d [ i ] ; i ++; return ( double )Sum / n ; ECE2036 Describing and Implementing Algorithms Spring / 19

16 Averaging Algorithm, Assembly Code # Program t o compute average o f an a r r a y o f numbers LDI, R0,0 # Index i n t o d LDM, R1, Length # Get value of l ength LDI, R2,0 # I n i t a l i z e Sum to zero Loop : LDM, R4, R0, d # Read next element ADD, R2, R2, R4 # Add to sum STM, R2,Sum # Store r e s u l t i n Sum ADDI, R0, R0,1 # Increment index BLT, R0, R1, Loop # Repeat while index < length DIV, R0, R2, R1 # Sum / n STM, R0, Avg # Store r e s u l t i n Avg HLT # Done ECE2036 Describing and Implementing Algorithms Spring / 19

17 Bubble Sort Algorithm Algorithm B Sort an array of numbers in ascending order. Given an array d[] of length N, re-order the elements in ascending order, such that d[i 1] <= d[i] for all 1 <= i < n. B1 SET i 1 B2 WHILE i < N B2.1 SET k = i B2.2 WHILE k > 0 AND d[k] < d[k 1] Need to swap B2.2.1 SET tmp d[k], d[k] d[k 1], d[k 1] tmp B2.2.2 SET k k 1 B2.3 SET i i + 1 B3 RETURN ECE2036 Describing and Implementing Algorithms Spring / 19

18 Bubble Sort Computational Complexity What is the computational complexity of the bubble sort algorithm? Clearly, the loop at step B2 executes exactly N 1 times, so the complexity of that loop is proportional to N. However, we have a nested loop at B2.2, and it is harder to say exactly how many times this loop executes. We can say that, on average, d[k] < d[k 1] will be true one half of the time. We can also say that k > 0 is true i times, since k was initialized to i at step B2.1. We can say the average value of i is N/2. Therefore, the complexity of the bubble sort algorithm is proportional to N 2. ECE2036 Describing and Implementing Algorithms Spring / 19

19 Bubble Sort Algorithm Animation You can see an animation of the bubble sort algorithm. ECE2036 Describing and Implementing Algorithms Spring / 19