Complexity of Python Operations
In this lecture we will learn the complexity classes of various operations on
Python data types. Then we wil learn how to combine these complexity classes to
compute the complexity class of all the code in a function, and therefore the
complexity class of the function. This is called "static" analysis, because we
do not need to run any code to perform it (contrasted with Dynamic or Emperical
Analysis, when we do run code and take measurements of its execution).
------------------------------------------------------------------------------
Python Complexity Classes
In ICS-46 we will write low-level implementations of all of Python's data types
and see/understand WHY these complexity classes apply. For now we just need to
try to absorb (not memorize) this information, with some -but minimal-
justification.
Binding a value to any name (copying a refernce) is O(1). Simple operators on
integers (whose values are small: e.g., under 12 digits) like + or == are also
O(1). You should assume small integers in problems unless explicitly told
otherwise.
In all these examples, N = len(data-type). The operations are organized by
increasing complexity class
Lists:
Complexity
Operation | Example | Class | Notes
--------------+--------------+---------------+-------------------------------
Index | l[i] | O(1) |
Store | l[i] = 0 | O(1) |
Length | len(l) | O(1) |
Append | l.append(5) | O(1) | mostly: ICS-46 covers details
Pop | l.pop() | O(1) | same as l.pop(-1), popping at end
Clear | l.clear() | O(1) | similar to l = []
Slice | l[a:b] | O(b-a) | l[1:5]:O(l)/l[:]:O(len(l)-0)=O(N)
Extend | l.extend(...)| O(len(...)) | depends only on len of extension
Construction | list(...) | O(len(...)) | depends on length of ... iterable
check ==, != | l1 == l2 | O(N) |
Insert | l[a:b] = ... | O(N) |
Delete | del l[i] | O(N) | depends on i; O(N) in worst case
Containment | x in/not in l| O(N) | linearly searches list
Copy | l.copy() | O(N) | Same as l[:] which is O(N)
Remove | l.remove(...)| O(N) |
Pop | l.pop(i) | O(N) | O(N-i): l.pop(0):O(N) (see above)
Extreme value | min(l)/max(l)| O(N) | linearly searches list for value
Reverse | l.reverse() | O(N) |
Iteration | for v in l: | O(N) | Worst: no return/break in loop
Sort | l.sort() | O(N Log N) | key/reverse mostly doesn't change
Multiply | k*l | O(k N) | 5*l is O(N): len(l)*l is O(N**2)
Tuples support all operations that do not mutate the data structure (and they
have the same complexity classes).
Sets:
Complexity
Operation | Example | Class | Notes
--------------+--------------+---------------+-------------------------------
Length | len(s) | O(1) |
Add | s.add(5) | O(1) |
Containment | x in/not in s| O(1) | compare to list/tuple - O(N)
Remove | s.remove(..) | O(1) | compare to list/tuple - O(N)
Discard | s.discard(..)| O(1) |
Pop | s.pop() | O(1) | popped value "randomly" selected
Clear | s.clear() | O(1) | similar to s = set()
Construction | set(...) | O(len(...)) | depends on length of ... iterable
check ==, != | s != t | O(len(s)) | same as len(t); False in O(1) if
the lengths are different
<=/< | s <= t | O(len(s)) | issubset
>=/> | s >= t | O(len(t)) | issuperset s <= t == t >= s
Union | s | t | O(len(s)+len(t))
Intersection | s & t | O(len(s)+len(t))
Difference | s - t | O(len(s)+len(t))
Symmetric Diff| s ^ t | O(len(s)+len(t))
Iteration | for v in s: | O(N) | Worst: no return/break in loop
Copy | s.copy() | O(N) |
Sets have many more operations that are O(1) compared with lists and tuples.
Not needing to keep values in a specific order in a set (which lists/tuples
require an order) allows for faster implementations of set operations.
Frozen sets support all operations that do not mutate the data structure (and
they have the same complexity classes).
Dictionaries: dict and defaultdict
Complexity
Operation | Example | Class | Notes
--------------+--------------+---------------+-------------------------------
Index | d[k] | O(1) |
Store | d[k] = v | O(1) |
Length | len(d) | O(1) |
Delete | del d[k] | O(1) |
get/setdefault| d.get(k) | O(1) |
Pop | d.pop(k) | O(1) | popped key "randomly" selected
Pop item | d.popitem() | O(1) | popped item "randomly" selected
Clear | d.clear() | O(1) | similar to s = {} or = dict()
View | d.keys() | O(1) | same for d.values()
Construction | dict(...) | O(len(...)) | depends # (key,value) 2-tuples
Iteration | for k in d: | O(N) | all forms: keys, values, items
| Worst: no return/break in loop
So, most dict operations are O(1).
defaultdicts support all operations that dicts support, with the same
complexity classes (because it inherits all those operations); this assumes that
calling the constructor when a values isn't found in the defaultdict is O(1) -
which is true for int(), list(), set(), ... (the things we commonly use)
Note that for i in range(...) is O(len(...)); so for i in range(1,10) is O(1).
If len(alist) is N, then
for i in range(len(alist)):
is O(N) because it loops N times. Of course even
for i in range (len(alist)//2):
is O(N) because it loops N/2 times, and dropping the constant 1/2 makes
it O(N): the work doubles when the list length doubles. By this reasoning,
for i in range (len(alist)//1000000):
is O(N) because it loops N/1000000 times, and dropping the constant 1000000
makes it O(N): the work doubles when the list length doubles.
Finally, when comparing two lists for equality, the complexity class above
shows as O(N), but in reality we would need to multiply this complexity class by
O==(...) where O==(...) is the complexity class for checking whether two values
in the list are ==. If they are ints, O==(...) would be O(1); if they are
strings, O==(...) in the worst case it would be O(len(string)). This issue
applies any time an == check is done. We mostly will assume == checking on
values in lists is O(1): e.g., checking ints and small/fixed-length strings.
------------------------------------------------------------------------------
Composing Complexity Classes: Sequential and Nested Statements
In this section we will learn how to combine complexity class information about
simple operations into complexity class information about complex operations
(composed from simple operations). The goal is to be able to analyze all the
statements in a functon/method to determine the complexity class of executing
the function/method. As with computing complexity classes themselves, these
rules are simple and easy to apply once you understand how to use them.
------------------------------------------------------------------------------
Law of Addition for big-O notation
O(f(n)) + O(g(n)) is O( f(n) + g(n) )
That is, we when adding complexity classes we bring the two complexity classes
inside the O(...). Ultimately, O( f(n) + g(n) ) results in the bigger of the two
complexity class (because we alwasy drop the lower-complexity added term). So,
O(N) + O(Log N) = O(N + Log N) = O(N)
because N is the faster growing term: lim (N->infinity) Log N/N = 0.
This rule helps us understand how to compute the complexity class of doing any
SEQUENCE of operations: executing a statement that is O(f(n)) followed by
executing a statement that is O(g(n)). Executing both statements SEQUENTIALLY
is O(f(n)) + O(g(n)) which is O( f(n) + g(n) ) by the rule above.
For example, if some function call f(...) is O(N) and another function call
g(...) is O(N Log N), then doing the sequence
f(...)
g(...)
is O(N) + O(N Log N) = O(N + N Log N) = O(N Log N). Of course, executing the
sequence (calling f twice)
f(...)
f(...)
is O(N) + O(N) which is O(N + N) which is O(2N) which is O(N).
Note that an if statment sequentially evaluates test and then one of the blocks.
if test: assume complexity class of computing test is O(T)
block 1 assume complexity class of executing block 1 is O(B1)
else:
block 2 assume complexity class of executing block 2 is O(B2)
The complexity class for the if is O(T) + max(O(B1),O(B2)). The test is always
evaluated, and one of the blocks is always executed afterward (so, a sequence
of evaulating a test followed by executing a block). In the worst case, the if
will execute the block with the largest complexity class. So, given
if test: complexity class is O(N)
block 1 complexity class is O(N**2)
else:
block 2 complexity class is O(N)
The complexity class for the if is O(N) + max (O(N**2),O(N))) = O(N) + O(N**2)
= O(N + N**2) = O(N**2).
If the test had complexity class O(N**3), then the complexity class for the if
is O(N**3) + max (O(N**2),O(N))) = O(N**3) + O(N**2) = O(N**3 + N**2) = O(N**3).
------------------------------------------------------------------------------
Law of Multiplcation for big-O notation
O(f(n)) * O(g(n)) is O( f(n) * g(n) )
If we repeat an O(f(N)) process O(N) times, the resulting complexity class is
O(N)*O(f(N)) = O( N*f(N) ). An example of this is, if some function call f(...)
is O(N**2), then executing that call N times (in the following loop)
for i in range(N):
f(...)
is O(N)*O(N**2) = O(N*N**2) = O(N**3)
This rule helps us understand how to compute the complexity class of doing some
statement INSIDE A BLOCK controlled by a statement that is REPEATING it. We
multiply the complexity class of the number of repetitions by the complexity
class of the statement (sequence; using the summing rule) being repeated.
Compound statements can be analyzed by composing the complexity classes of
their constituent statements. For sequential statements (including if tests and
their block bodies) the complexity classes are added; for statements repeated
in a loop the complexity classes are multiplied.
Let's use the data and tools discussed above to analyze (determine their
complexity classes) three different functions that each compute the same
result: whether or not a list contains only unique values (no duplicates). We
will assume in all three examples that len(alist) is N and that we can compare
the list elements in O(1): e.g., they are small ints or strs.
1) Algorithm 1: A list is unique if each value in the list does not occur in any
later indexes: alist[i+1:] is a list slice containing all values after the one
at index i.
def is_unique1 (alist : [int]) -> bool:
for i in range(len(alist)): O(N) - for every index
if alist[i] in alist[i+1:]: O(N) - index+add+slice+in: O(1)+O(1)+O(N)+O(N) = O(N)
return False O(1) - never executed in worst case; ignore
return True O(1)
The complexity class for executing the entire function is O(N) * O(N) + O(1)
= O(N**2). So we know from the previous lecture that if we double the length of
alist, this function takes 4 times as long to execute.
-----
Many students want to write this as O(N) * ( O(N) + O(1) ) + O(1) because the
if statement's complexity is O(N) + O(1): complexity of test + complexity of
block when test is True. But in the worst case, the return is NEVER EXECUTED
(the loop keeps executing) so it should not appear in the formula. Although,
even if it appears in this formula, the formula still computes the same
complexity class (because O(N) + O(1) is still O(N)): O(N**2).
So, in the worst case, we never return False and keep executing the loop, so
this O(1) does not appear in the formula. Also, in the worst case the list
slice is aliset[1:] which is O(N-1) = O(N), although when i is len(alist) the
slice contains 0 values: is empty. The average list slice taken in the if has
N/2 values, which is still O(N).
-----
2) Algorithm 2: A list is unique if when we sort its values, no ADJACENT values
are equal. If there were duplicate values, sorting the list would put these
duplicate values right next to each other (adjacent). Here we copy the list so
as to not mutate (change the order of) the parameter's list by sorting it
(functions generally shouldn't mutate their arguments unless that is the purpose
of the function): it turns out that copying the list does not increase the
complexity class of the method, because the O(N) used for copying is not the
largest added term.
def is_unique2 (alist : [int]) -> bool:
copy = list(alist) O(N)
copy.sort() O(N Log N) - for fast Python sorting
for i in range(len(alist)-1): O(N) - really N-1, but that is O(N); len and - are both O(1)
if copy[i] == copy[i+1]: O(1): +, 2 [i], and == on ints: all O(1)
return False O(1) - never executed in worst case
return True O(1)
The complexity class for executing the entire function is given by the sum
O(N) + O(N Log N) + O(N)*O(1) + O(1) = O(N + N Log N + O(N*1) + 1) =
O(N + N Log N + N + 1) = O(N Log N + 2N + 1) = O(N Log N). So the complexity
class for this algorithm/function is lower than the first algorithm, the
is_unique1 function. For large N unique2 will eventually run faster. Because
we don't know the constants, we don't know which is faster for small N.
Notice that the complexity class for sorting is dominant in this code: it does
most of the work. If we double the length of alist, this function takes a bit
more than twice the amount of time. In N Log N: N doubles and Log N gets a tiny
bit bigger (i.e., Log 2N = 1 + Log N; e.g., Log 2000 = 1 + Log 1000 = 11, so
compared to 1000 Log 1000, doubling N is 2000 Log 2000, which is just 2.2 times
bigger, or 10% bigger than just doubling).
Looked at another way if T(N) = c*(N Log N), then T(2N) = c*(2N Log 2N) =
c*2N(Log N + 1) = c*2N Log N + c*2N = 2*T(N) + c*2N. Or, computing the doubling signature
T(2N) c*2(N Log N) + c*2N 2
----- ~ ------------------- = 2 + -------
T(N) c*(N Log N) Log N
So, the ratio is 2 + a bit (and that bit gets smaller -very slowly- as N
increases): for N >= 10**3 it is <= 2.2; for N >= 10**6 it is <= 2.1; for N >=
10**9 it it < 2.07. So, it is a bit worse than doubling each time, but much
better than O(N**2) which is quadrupling each time.
We could also simplify
copy = list(alist) O(N)
copy.sort() O(N Log N) - for fast Python sorting
to just
copy = sorted(alist) O(N Log N) - for fast Python sorting
Which won't change the complexity analysis becaus O(N + N Log N) = O (N Log N).
Finally, Algorithm 2 works only if all the values in the list are comparable:
it would fail if the list contained both integers and strings.
3) Algorithm 3: A list is unique if when we turn it into a set, its length is
unchanged: if duplicate values were added to the set, its length would be
smaller than the length of the list by exactly the number of duplicates in the
list added to the set.
def is_unique3 (alist : [int]) -> bool:
aset = set(alist) O(N): construct set from alist values
return len(aset) == len(alist) O(1): 2 len (each O(1)) and == ints O(1)
The complexity class for executing the entire function is O(N) + O(1) =
O(N + 1) = O(N). So the complexity class for this algortihm/function is lower
than both the first and second algorithms/functions. If we double the length of
alist, this function takes just twice the amount of time. We could write the
body of this function more simply as: return len(set(alist)) == len(alist),
where evaluating set(alist) takes O(N) and then computing the two len's and
comparing them for equality are all O(1).
Unlike Algorithm 2, it can work for lists containing both integers and strings.
But, Algorithm 3 works only if all the values in the list are immutable (a
requirement for storing values in a set). So, it would not work for a list of
lists.
So the bottom line here is that there might be many algorithms/functions to
solve some problem. If the function bodies are small, we can analyze them
statically (looking at the code, not needing to run it) to determine their
complexity classes. For large problem sizes, the algorithm/function with the
smallest complexity class will ultimately be best, running in the least amount
of time. But, for small problem sizes, complexity classes don't determine which
is best: for small problem we need to take into account the CONSTANTS and lower
order terms that we ignored when computing complexity classes). We can run the
functions (dynamic analysis, aka empirical analysis) to test which is fastest
on small problem sizes.
------------------------------------------------------------------------------
Using a Class (implementable 3 ways) Example:
We will now look at the solution of a few problems (combining operations on a
priority queue: pq) and how the complexity class of the result is affected by
three different classes/implementations of priority queues.
In a priority queue, we can add values to and remove values from the data
structure. A correctly working priority queue always removes the maximum value
remaining in the priority queue (the one with the highest priority). Think of a
line/queue outside of a Hollywood nightclub, such that whenever space opens up
inside, the most famous person in line gets to go in (the "highest priority"
person), no matter how long less famous people have been standing in line
(contrast this with first come/first serve, which is a regular -non priority-
queue; in a regaular queue, whoever is first in the line -has been standing in
line longest- is admitted next).
For the problems below, all we need to know is the complexity class of the
"add" and "remove" operations.
add remove
+-------------+-------------+
Implementation 1 | O(1) | O(N) |
+-------------+-------------+
Implementation 2 | O(N) | O(1) |
+-------------+-------------+
Implementation 3 | O(Log N) | O(Log N) |
+-------------+-------------+
Implementation 1 adds the new value into the pq by appending the value at the
rear of a list or the front of a linked list: both are O(1); it removes the
highest priority value by scanning through the list or linked list to find the
highest value, which is O(N), and then removing that value, also O(N) in the
worst case (removing at the front of a list; at the rear of a linked list).
Implementation 2 adds the new value into the pq by scanning the list or linked
list for the right spot to put it and putting it there, which is O(N). Lists
store their highest priority at the rear (linked lists at the front); it
removes the highest priority value from the rear for lists (or the front for
linked lists), which is O(1).
So Implementations 1 and 2 swap the complexity classes in their add/remove
method. Implementation 1 doesn't keep the values in order: so easy to add but
hard to find/remove the maximum (must scan). Implementation 2 keeps the values
in order: so hard to add (need to scan to find where it goes) but easy to
find/remove the maximum (at one end).
Implementation 3, which is discussed in ICS-46, uses a binary heap tree (not a
binary search tree) to implement both operations with "middle" complexity
O(Log N): this complexity class greater than O(1) but less than O(N). Because
Log N grows so slowly, O(Log N) is actually closer to O(1) than O(N) even though
O(1) doesn't grow at all: Log N grows that slowly.
Problem 1: Suppose we wanted to use the priority queue to sort N values: we
add N values in the pq and then remove all N values (first the highest, next
the second highest, ...). Here is the complexity of these combined operations
for each implementation.
Implementation 1: N*O(1) + N*O(N) = O(N) + O(N**2) = O(N**2)
Implementation 2: N*O(N) + N*O(1) = O(N**2) + O(N) = O(N**2)
Implementation 3: N*O(Log N) + N*O(Log N) = O(NLogN) + O(NLogN) = O(NLogN)
Note N*O(...) is the same as O(N)*O(...) which is the same as O(N * ...)
Here, Implementation 3 has the lowest complexity class for the combined
operations. Implementations 1 and 2 each do one operation quickly but the other
slowly: both are done O(N) times. The slowest operation determines the
complexity class, and both are equally slow. The complexity class O(Log N) is
between O(1) and O(N); surprisingly, it is actually "closer" to O(1) than O(N),
even though it does grow -because it grows so slowly; yes, O(1) doesn't grow at
all, but O(Log N) grows very slowly: the known Universe has about 10**90
particles of matter, and Log 10**90 = Log (10**3)**30 = 300, which isn't very
big compared to 10**90 (like 86 orders of magnitude less).
Problem 2: Suppose we wanted to use the priority queue to find the 10 biggest
(of N) values: we would enqueue N values and then dequeue 10 values. Here is
the complexity of these combined operations for each implementation..
Implementation 1: N*O(1) + 10*O(N) = O(N) + O(N) = O(N)
Implementation 2: N*O(N) + 10*O(1) = O(N**2) + O(1) = O(N**2)
Implementation 3: N*O(Log N) + 10*O(Log N) = O(NLogN) + O(LogN) = O(NLogN)
Here, Implementation 1 has the lowest complexity for the combined operations.
That makes sense, as the operation done N times (add) is very simple (add to
the end of a list/the front of a linked list, where each add is O(1)) and the
operation done a constant number of times (10, independent of N) is the
expensive operation (remove, which is O(N)). It even beats the complexity of
Implementation 3. So, as N gets bigger, implementation 1 will eventually become
faster than the other two for the "find the 10 biggest" task.
So, the bottom line here is that sometimes there is NOT a "best all the time"
implementation for a data structure. We need to know what problem we are
solving (the complexity classes of all the operations in various
implementations and how often we must do these operations) to choose the most
efficient implementation for solving the problem.
------------------------------------------------------------------------------
Problems:
TBA