Algorithms: The recipe for computation

Transcription

1 Algorithms: The recipe for computation 2B Experimenting with algorithms Principles of Computation, Carnegie Mellon University - CORTINA 1 Fundamentals of Computational Algorithms A variable is used to represent a data value needed for the algorithm. When algorithms are implemented as computer programs, a variable corresponds to a memory cell where data is stored for the computer program. A variable can store only one data value at a time. If an algorithm instructs us to store a data value in a variable (e.g. Set x = 1 ), the previous value associated with this variable is lost. x Principles of Computation, Carnegie Mellon University - CORTINA 2 1

2 Fundamentals of Computational Algorithms A variable can also be used to represent an ordered collection of data values. Such a variable is called a vector. A data value in a vector is named using the variable name and a subscript. A This value is labeled A[0] Principles of Computation, Carnegie Mellon University - CORTINA 3 Fundamentals of Computational Algorithms Example: Let A be a vector of 8 integers. Set A[0] = 1. Set A[0] = 1. Set A[1] = 2. Set i = 0. Set A[2] = 4. Do the following 7 times: Set A[3] = 8. Add 1 to i. Set A[4] = 16. Set A[i] = A[i-1] * 2 Set A[5] = 32. Set A[6] = 64. Set A[7] = Principles of Computation, Carnegie Mellon University - CORTINA 4 2

3 Algorithm Example 1. Input m 2. Set n = While m > 0 do the following: a. Multiply n by 2. b. Subtract 1 from m. 4. Output n. m n Principles of Computation, Carnegie Mellon University - CORTINA 5 Vector Example 3. Set i = 0 4. Set m = A[0] 5. Do the following while i < n-1: a. Add 1 to i b. If A[i] > m, set m = A[i] SELECTION 6. Output m if this is true then we do this otherwise, we don t do anything for this substep Principles of Computation, Carnegie Mellon University - CORTINA 6 3

4 Vector Example 3. Set i = 0 4. Set m = A[0] 5. Do the following while i < n-1: a. Add 1 to i b. If A[i] > m, set m = A[i] 6. Output m LOOP repeat this pair of steps as long as the condition is true NOTE: You cannot exit out of this loop in the middle of its work! Principles of Computation, Carnegie Mellon University - CORTINA 7 Tracing Points 3. Set i = 0 4. Set m = A[0] 5. Do the following while i < n-1: a. Add 1 to i b. If A[i] > m, set m = A[i] 6. Output m In order to trace what an algorithm is doing, take a snapshot immediately after a step is executed, and examine the contents of all variables Principles of Computation, Carnegie Mellon University - CORTINA 8 4

5 Tracing the algorithm n = 5 A = [4, 7, 3, 9, 8] i m Input a vector of values A[0], A[1],..., A[n-1] 3. Set i = 0 4. Set m = A[0] 5. Do the following while i < n-1: a. Add 1 to i b. If A[i] > m, set m = A[i] 6. Output m Principles of Computation, Carnegie Mellon University - CORTINA 9 Another way to look at it 3. Set i = 0 4. Set m = A[0] 5. Do the following while i < n-1 : a. Add 1 to i b. If A[i] > m, set m = A[i] 6. Output m Principles of Computation, Carnegie Mellon University - CORTINA 10 5

6 Another algorithm LOOP exchange A[i] and A[i+1] LOOP Loop: repeat each step in the box the indicated number of times IMPORTANT: You cannot exit the loop in the middle of its steps Principles of Computation, Carnegie Mellon University - CORTINA 11 Another algorithm exchange A[i] and A[i+1] if this is true then we do this otherwise, we don t do anything for this substep Principles of Computation, Carnegie Mellon University - CORTINA 12 6

7 Tracing Points exchange A[i] and A[i+1] In order to trace what an algorithm is doing, take a snapshot immediately after a step is executed, and examine the contents of all variables Principles of Computation, Carnegie Mellon University - CORTINA 13 Tracing the algorithm (1) n = 5 A = [8, 4, 9, 3, 7] i A 0 [8, 4, 9, 3, 7] 2. Input a vector of values A[0], A[1],, A[n-1] exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 14 7

8 Tracing the algorithm (2) n = 5 A = [8, 4, 9, 3, 7] i A 0 [4, 8, 3, 7, 9] 2. Input a vector of values A[0], A[1],, A[n-1] exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 15 Tracing the algorithm (3) n = 5 A = [8, 4, 9, 3, 7] i A 0 [4, 3, 7, 8, 9] 2. Input a vector of values A[0], A[1],, A[n-1] exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 16 8

9 Tracing the algorithm (4) n = 5 A = [8, 4, 9, 3, 7] i A 0 [3, 4, 7, 8, 9] 2. Input a vector of values A[0], A[1],, A[n-1] exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 17 Tracing the algorithm (5) n = 5 A = [8, 4, 9, 3, 7] i A 0 [3, 4, 7, 8, 9] 2. Input a vector of values A[0], A[1],, A[n-1] exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 18 9

10 Analyzing the algorithm Step 3 says to perform the subsequent substeps a total of n times. Does step 3 need to be performed n times in general? Principles of Computation, Carnegie Mellon University - CORTINA 19 The algorithm revised 3. Do the following n 1 times: b. Do the following n 1 times: exchange A[i] and A[i+1] Principles of Computation, Carnegie Mellon University - CORTINA 20 10

11 Analyzing the algorithm Step 3b says to examine n 1 pairs of data values in the vector. Do we need to examine all n 1 pairs in general? Principles of Computation, Carnegie Mellon University - CORTINA 21 The algorithm revised 3. Let j = 1 4. Do the following n 1 times: b. Do the following n j times: exchange A[i] and A[i+1] c. Add 1 to j 5. Output A Principles of Computation, Carnegie Mellon University - CORTINA 22 11

12 Analyzing the algorithm The algorithm continues to execute even if the vector becomes sorted. Can we stop the computation even earlier if the vector becomes sorted? How do we know if the vector is sorted? Principles of Computation, Carnegie Mellon University - CORTINA 23 The final algorithm (Bubble Sort) 3. Let j = 1 4. Do the following n 1 times: b. Let s = 0 c. Do the following n j times: exchange A[i] and A[i+1], and add 1 to s d. Add 1 to j e. If s = 0, exit step 4 and goto step 5 5. Output A Principles of Computation, Carnegie Mellon University - CORTINA 24 12

13 The evils of GOTO Edgsar Dijkstra, Dutch computer scientist who created the shortestpath algorithm used in networking today (see Chap. 4) Received the Turing Award (from ACM) in 1972 for his contributions to Computer Science Go To Statement Considered Harmful Communications of the ACM, March 1968 For a number of years I have been familiar with the observation that the quality of programmers is a decreasing function of the density of go to statements in the programs they produce. More recently I discovered why the use of the go to statement has such disastrous effects, and I became convinced that the go to statement should be abolished from all "higher level" programming languages (i.e. everything except, perhaps, plain machine code) Principles of Computation, Carnegie Mellon University - CORTINA 25 What does this algorithm do? 1. Set d = an integer between 0 and 6, inclusive. 2. Set n = an integer between 28 and 31, inclusive. 3. Draw a grid with 6 rows and 7 columns. 4. Skip the first d cells in the first row of the grid. 5. Set i = While i < n do the following: a. Write the current value of i in the next available cell of the current row of the grid. b. If (i+d) modulo 7 = 0, move to the next row. c. Add 1 to i. Trace this with d = 4 and n = Principles of Computation, Carnegie Mellon University - CORTINA 26 13

14 Tracing the algorithm d = 4 n = 31 Output: 1. Set d = an integer in [0,6]. 2. Set n = an integer in [28,31]. 3. Draw a grid with 6 rows and 7 columns. 4. Skip the first d cells in the first row of the grid. 5. Set i = While i < n do the following: a. Write the current value of i in the next available cell of the current row of the grid. b. If (i+d) modulo 7 = 0, move to the next row. c. Add 1 to i Principles of Computation, Carnegie Mellon University - CORTINA 27 What does this algorithm do? 1. Set X = 0 2. Set Y = 1 3. Set Z = 1 4. Set X = X + 4 / Z 5. While X -X > do the following: a. Set X equal to X b. Add 1 to Y c. Add 2 to Z d. If Y is odd, set X = X + 4 / Z ; otherwise set X = X 4 / Z 6. Output X Principles of Computation, Carnegie Mellon University - CORTINA 28 14