Searching Algorithms

Consider the following simple number game between two parties, A and B. Initially, A chooses an integer between 1 and 100. B then tries to guess this number. In this guessing process, B makes a guess and then A responds by saying either that the guess is correct, or that the guess is too high, or that the guess is too low. This process continues until B makes a correct guess. (In playing the game, B might have some incentive to try to determine the correct number quickly, but the game itself does not require this.)

1. Working in partners, play this game a couple times, so you are sure how it works.

2. Write a procedure that simulates A's role in this game, with the person running the program filling B's role. The procedure first should pick an integer randomly between 1 and 100. Then B (the user) will enter a guess (at the keyboard), and the procedure should respond to the guess. This second part will continue until the correct number is guessed. (All of this can be done with a simple do expression.)

   For this part, you may want to review the use of Scheme's random procedure from the previous labs on Random Number Generators and Input and Output.

3. Now, define a separate variable for A's number, and write the following logical operations which A might undertake:
   • (reset-number) should make A pick a new number.
   • (make-guess number) should return A's response: "too high", "too low", and "correct".

1. B might guess numbers randomly until choosing the correct one.

2. B might guess numbers in sequence 1, 2, 3, ..., 100 (or 100, 99, 98, ..., 2, 1), stopping once the correct integer is found.

3. B might guess a number in the middle (e.g., 50 or 51). On the basis of the answer (``correct'', ``too low'', ``too high''), B could then infer that some numbers need not be tried at all, and B could reduce the range of numbers possible for [1..100] to [1..50] or [51..100] (depending upon A's response). As a second guess, B could try the middle of this new interval, and again the interval of possible values could be cut in half on the basis of A's response. This process of guessing a middle value and cutting the interval of possible candidates in half could continue until the number is found.

4. Write a simple procedure random-B to simulate approach a above. Use the reset-number and make-guess procedures when working with A.

5. For approach b, write out (in English) what information should be kept from one iteration of the loop to the next. Explain what inferences might be made from this kept information, and describe (again, in English) how this information should be updated when the next feedback from A is obtained.

6. Write procedure sequential-B which implement's approach b above. In writing your code, consider the information you described in the previous step. You may want to use local variables or parameters to store that relevant information.

7. Repeat step 5 for approach c.

8. Write procedure binary-B which implements approach c above.

9. Suppose information was stored in a vector. How might you adapt sequential-B to determine whether a given value was stored in the vector? In your approach, would it matter if the data in the vector were sorted? Why or why not?

10. How would you adapt binary-B to determine whether a sorted vector contains a given value?

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http://www.math.grin.edu/~walker/courses/151/lab-merge-sort.html