Loop Invariant Review

1
CS100A Sections Dec 1-2 1998

• Loop Invariant Review
• C Review and Example
• Recursion

2

• Loop Invariant Review
• A loop invariant is a true-false statement (about
  the vari-ables used in a loop) that is true
  before and after each iteration.
• Given an invariant P, we can develop a loop with
  initialization in four steps
• (1) Create initialization that makes P true.
• (2) Determine the condition B so that from !B
  and P you can see that the desired result is
  true. (Or, equivalently, determine the condition
  B such that you know the desired result is not
  true.)
• (3) Figure out what statement(s) to put in the
  loop body to make progress toward termination of
  the loop.
• (4) Figure out what else the loop body should do
  to en-sure that after execution of the loop body
  invariant P is still true.
• init
• // invariant P S must make progress
• while ( B ) toward termination
• S and keep P true

3
Example of a loop developed from an
invariant Problem. Sorted array b0..n-1 may
contain duplicates. Rearrange b0..n-1 and store
in k so that b0..k-1 con-tains the
non-duplicates of b0..n-1. Invariant P (1)
Initialization Make first section empty item
using k 0 make second section contain 1 item
using i 0 (2) Stop when last section is empty,
i.e. when in. So, continue as long as i !
n. (3) Make progress using i i1. (4) Before
incrementing i if bi is not a duplicate, place
it in section b0..k-1. k 0 i 0 while ( i
! n) if (k 0 bk-1 ! bi
) bk bi k k1 i i1
looked at, but not changed
4

• (A version of selection sort). Write an
  algorithm that, given
• a lt b1, sorts ma..b. Use the invariant P
  given below. Do not write a loop within the body
  of the main loop instead, write commands in the
  body of the loop as high-level statements, in
  English.
• P
• j b1
• // Invariant P (see the problem description)
• while ( a lt j )
• j j-1
• Store in h the index of the maximum value
• of ma..j
• Swap mh and mj

a
j
b
m
lt sorted, gt
5

• Each element bm (where 1ltmlt12) of array
  b0..12 contains the number of days in month m.
  (January is month 1, a 0 is in b0, etc.) Given
  a date (day, month) in two variables day and
  month, store in n the day of the year on which
  (day, month) occurs. For example, if January has
  31 days, then (5,2), i.e. day 5 of February,
  occurs on day 36 of the year, and if February has
  28 days, then (10,3) occurs on day 68 of the
  year.
• Your algorithm will need a loop plus some other
  statements. For the loop, use the loop invariant
  P
• P 1 lt m lt month and
• n is the total number of days in months 1..m-1
• m 1 n 0
• // Invariant P (see the problem description)
• while (m ! month)
• n n bm m m1
• n n day

6

• Pointers and References in C
• By default, C methods are call_by_value for
  integer args
• void f (int x, int y)
• int tmp
• tmp x
• xy
• ytmp
• void main (void)
• int j10
• int k20
• f(j,k)
• No change to j and k as a result of the call to
  f.

7

• Pointers and References in C
• What if you WANT f to modify j and k
• j
• k
• j is a pointer to j j must be a variable
• type of j int
• type of j pointer to int, int
• So if you see
• int p1 p1 j
• int p2 k
• then p1 is 10 (follow the pointer to the value)
• and p2 is 20.

8

• Pointers and References in C
• Rewrite f so that it modifies the arguments
• void f (int x, int y)
• int tmp
• tmp x
• xy
• ytmp
• void main (void)
• int j10
• int k20
• f(j, k)
• Now j and k change as a result of the call to f.

9
C Example
10
Calls to One

• (a) One (a, c, b, d)
• (b) One ((ab), c, b, d)
• (c) One (a, c, b, d)
• (d) One (a, d, b, d)
• (e) One (b, b, b, d)
• (a) Illegal. First and second arguments must be
  pointers to integer variables, not integer
  values.
• (b) Illegal. The operator can only be applied
  to a variable, not an expression.
• (c)
• 20 2 1 10.0
• 1 2 20 8.0
• (d) Second parameter must be a pointer to an
  integer. d is a pointer to a double.
• (e)
• 20 2 20 10.0
• 1 20 7 8.0

11
Recursion Example

• Write a function prod that computes the product
  of integers between lo and hi inclusive. (Assume
  that lo lt hi.)
• Header
• public static int product (int lo, int hi)
• Example
• lo 3 hi 5 3 x 4 x 5 60
• Algorithm
• base case if lo hi, return lo
• otherwise (go from lo to hi or vice versa)
• return ( lo product
  (lo1, hi))
• Code
• //computes the product of integers between lo and
  hi
• //inclusive. (Assume that lo lt hi.)
• public static int prod (int lo, int hi)
• if (lo hi)
• return lo