|
Ref: Carrano ch.9
elapsed time ref: exampledepot.com
Counting Numbers Of Operations
operations, comparisons, and assignments take time
search loop example:
for (int i = 0; i < n; i++)
if (a[i] == x)
break;
n operations: at most, n, average n/2
n comparisons: at most, n, average n/2
measuring elapsed time in milliseconds:
long start = System.currentTimeMillis();
do something here...
long end = System.currentTimeMillis();
long elapsedTimeMillis = end - start;
Scaling Data Structure Size
the effect on efficiency: #of operations, etc.
in "search loop" example:
doubling n doubles the time
Big Oh
a way to express how an algorithm scales
in "search loop" example: "O(n)"
to get big-oh:
write formula for worst case #of operations
simplify the formula,
so that for values of n
higher than some number,
the simplified formula is
>= the original formula
times some constant
the answer is not unique!
so keep it tight
in the search loop:
#of operations = 2 * n
for n >= 1, O(2n) works
so does O(3n)
for n >= 2, O(n) works
so does O(n2)
but the tightest is O(n)
Big Omega
opposite of big-oh: lower bound
to get big-omega:
write formula for best case #of operations
simplify the formula,
so that for values of n
higher than some number,
the simplified formula is
<= the original formula
times some constant
again, the answer is not unique
|
Big Theta
when big-oh and big-omega overlap:
choose formula that satisfies both
unsuccessful search loop is big-theta(n)
(no solution for successful search loop)
Operations That Are O(1)
retrieval from an array
adding to end of an array-based list
adding to front of a linked list
retrieving the head of a linked list
retrieving the tail of a doubly-linked list
retrieving the 3rd node's value in a linked list
retrieving from a perfect hash table (no collisions)
measuring elapsed time: independent of "n"
Operations That Are O(log n)
note that O(log n), O(lg n),and O(ln n) are the
same: different by a constant multiplier
binary search or bisection
rebalance of a red-black tree
Operations That Are O(n)
adding to middle of an array-based list
traversing a list
retrieving an indexed value from a linked list
radix sort
Operations That Are O(n log n)
mergesort, quicksort, heapsort
Operations That Are O(n2)
insertion sort, selection sort
Operations That Are O(2n)
recursive Fibonacci solution
Confirming Big Oh Determinations
counting operations with private int counters
measuring elapsed time
e.g. for O(n2):
try for various values of n
elapsed time / n2 should be roughly constant
sample test results for O(n2) operation:
| n | n2 | elapsed
millis | elapsed
n2 | norm |
|---|
| 1000 | 106 | 1324 | 1.324 | 1.00 |
| 2000 | 4 x 106 | 5436 | 1.359 | 1.03 |
| 4000 | 1.6 x 107 | 20784 | 1.299 | 0.98 |
| 8000 | 6.4 x 107 | 86120 | 1.345 | 1.02 |
"norm" is the col 4 value divided by the first line's column 4 value
|