analysis_of_algorithms.md

Analysis of Algorithms

This appendix is an edited excerpt from Think Complexity, by Allen B. Downey, also published by O’Reilly Media (2012). When you are done with this book, you might want to move on to that one.

Analysis of algorithms is a branch of computer science that studies the performance of algorithms, especially their run time and space requirements. See https://en.wikipedia.org/wiki/Analysis_of_algorithms.

The practical goal of algorithm analysis is to predict the performance of different algorithms in order to guide design decisions.

During the 2008 United States Presidential Campaign, candidate Barack Obama was asked to perform an impromptu analysis when he visited Google. Chief executive Eric Schmidt jokingly asked him for “the most efficient way to sort a million 32-bit integers.” Obama had apparently been tipped off, because he quickly replied, “I think the bubble sort would be the wrong way to go.” See https://www.youtube.com/watch?v=k4RRi_ntQc8.

This is true: bubble sort is conceptually simple but slow for large datasets. The answer Schmidt was probably looking for is “radix sort” (https://en.wikipedia.org/wiki/Radix_sort).

But if you get a question like this in an interview, I think a better answer is, “The fastest way to sort a million integers is to use whatever sort function is provided by the language I’m using. Its performance is good enough for the vast majority of applications, but if it turned out that my application was too slow, I would use a profiler to see where the time was being spent. If it looked like a faster sort algorithm would have a significant effect on performance, then I would look around for a good implementation of radix sort.”

The goal of algorithm analysis is to make meaningful comparisons between algorithms, but there are some problems:

• The relative performance of the algorithms might depend on characteristics of the hardware, so one algorithm might be faster on Machine A, another on Machine B. The general solution to this problem is to specify a machine model and analyze the number of steps, or operations, an algorithm requires under a given model.

• Relative performance might depend on the details of the dataset. For example, some sorting algorithms run faster if the data are already partially sorted; other algorithms run slower in this case. A common way to avoid this problem is to analyze the worst case scenario. It is sometimes useful to analyze average case performance, but that’s usually harder, and it might not be obvious what set of cases to average over.

• Relative performance also depends on the size of the problem. A sorting algorithm that is fast for small lists might be slow for long lists. The usual solution to this problem is to express run time (or number of operations) as a function of problem size, and group functions into categories depending on how quickly they grow as problem size increases.

The good thing about this kind of comparison is that it lends itself to simple classification of algorithms. For example, if I know that the run time of Algorithm A tends to be proportional to the size of the input, n, and Algorithm B tends to be proportional to n^2, then I expect A to be faster than B, at least for large values of n.

This kind of analysis comes with some caveats, but we’ll get to that later.

Order of growth

Suppose you have analyzed two algorithms and expressed their run times in terms of the size of the input: Algorithm A takes 100n+1 steps to solve a problem with size n; Algorithm B takes n^2 + n + 1 steps.

The following table shows the run time of these algorithms for different problem sizes:

Input size	Run time of Algorithm A	Run time of Algorithm B
10	1 001	111
100	10 001	10 101
1 000	100 001	1 001 001
10 000	1 000 001	100 010 001

At n=10, Algorithm A looks pretty bad; it takes almost 10 times longer than Algorithm B. But for n=100 they are about the same, and for larger values A is much better.

The fundamental reason is that for large values of n, any function that contains an n^2 term will grow faster than a function whose leading term is n. The leading term is the term with the highest exponent.

For Algorithm A, the leading term has a large coefficient, 100, which is why B does better than A for small n. But regardless of the coefficients, there will always be some value of n where a n^2 > b n, for any values of a and b.

The same argument applies to the non-leading terms. Even if the run time of Algorithm A were n+1000000, it would still be better than Algorithm B for sufficiently large n.

In general, we expect an algorithm with a smaller leading term to be a better algorithm for large problems, but for smaller problems, there may be a crossover point where another algorithm is better. The location of the crossover point depends on the details of the algorithms, the inputs, and the hardware, so it is usually ignored for purposes of algorithmic analysis. But that doesn’t mean you can forget about it.

If two algorithms have the same leading order term, it is hard to say which is better; again, the answer depends on the details. So for algorithmic analysis, functions with the same leading term are considered equivalent, even if they have different coefficients.

An order of growth is a set of functions whose growth behavior is considered equivalent. For example, 2n, 100n and n+1 belong to the same order of growth, which is written O(n) in Big-Oh notation and often called linear because every function in the set grows linearly with n.

All functions with the leading term n^2 belong to O(n^2); they are called quadratic.

The following table shows some of the orders of growth that appear most commonly in algorithmic analysis, in increasing order of badness.

Order of growth	Name
O(1)	constant
O(\log_b n)	logarithmic (for any b)
O(n)	linear
O(n \log_b n)	linearithmic
O(n^2)	quadratic
O(n^3)	cubic
O(c^n)	exponential (for any c)

For the logarithmic terms, the base of the logarithm doesn’t matter; changing bases is the equivalent of multiplying by a constant, which doesn’t change the order of growth. Similarly, all exponential functions belong to the same order of growth regardless of the base of the exponent. Exponential functions grow very quickly, so exponential algorithms are only useful for small problems.

Read the Wikipedia page on Big-Oh notation at https://en.wikipedia.org/wiki/Big_O_notation and answer the following questions:

1. What is the order of growth of n^3 + n^2? What about 1000000 n^3 + n^2? What about n^3 + 1000000 n^2?

2. What is the order of growth of (n^2 + n) \cdot (n + 1)? Before you start multiplying, remember that you only need the leading term.

3. If f is in O(g), for some unspecified function g, what can we say about af+b?

4. If f_1 and f_2 are in O(g), what can we say about f_1 + f_2?

5. If f_1 is in O(g) and f_2 is in O(h), what can we say about f_1 + f_2?

6. If f_1 is in O(g) and f_2 is O(h), what can we say about f_1 \cdot f_2?

Programmers who care about performance often find this kind of analysis hard to swallow. They have a point: sometimes the coefficients and the non-leading terms make a real difference. Sometimes the details of the hardware, the programming language, and the characteristics of the input make a big difference. And for small problems asymptotic behavior is irrelevant.

But if you keep those caveats in mind, algorithmic analysis is a useful tool. At least for large problems, the “better” algorithm is usually better, and sometimes it is much better. The difference between two algorithms with the same order of growth is usually a constant factor, but the difference between a good algorithm and a bad algorithm is unbounded!

Analysis of basic Python operations

In Python, most arithmetic operations are constant time; multiplication usually takes longer than addition and subtraction, and division takes even longer, but these run times don’t depend on the magnitude of the operands. Very large integers are an exception; in that case the run time increases with the number of digits.

Indexing operations—reading or writing elements in a sequence or dictionary—are also constant time, regardless of the size of the data structure.

A for loop that traverses a sequence or dictionary is usually linear, as long as all of the operations in the body of the loop are constant time. For example, adding up the elements of a list is linear:

total = 0
for x in t:
    total += x

The built-in function sum is also linear because it does the same thing, but it tends to be faster because it is a more efficient implementation; in the language of algorithmic analysis, it has a smaller leading coefficient.

As a rule of thumb, if the body of a loop is in O(n^a) then the whole loop is in O(n^{a+1}). The exception is if you can show that the loop exits after a constant number of iterations. If a loop runs k times regardless of n, then the loop is in O(n^a), even for large k.

Multiplying by k doesn’t change the order of growth, but neither does dividing. So if the body of a loop is in O(n^a) and it runs n/k times, the loop is in O(n^{a+1}), even for large k.

Most string and tuple operations are linear, except indexing and len, which are constant time. The built-in functions min and max are linear. The run-time of a slice operation is proportional to the length of the output, but independent of the size of the input.

String concatenation is linear; the run time depends on the sum of the lengths of the operands.

All string methods are linear, but if the lengths of the strings are bounded by a constant—for example, operations on single characters—they are considered constant time. The string method join is linear; the run time depends on the total length of the strings.

Most list methods are linear, but there are some exceptions:

• Adding an element to the end of a list is constant time on average; when it runs out of room it occasionally gets copied to a bigger location, but the total time for n operations is O(n), so the average time for each operation is O(1).

• Removing an element from the end of a list is constant time.

• Sorting is O(n \log n).

Most dictionary operations and methods are constant time, but there are some exceptions:

• The run time of update is proportional to the size of the dictionary passed as a parameter, not the dictionary being updated.

• keys, values and items are constant time because they return iterators. But if you loop through the iterators, the loop will be linear.

The performance of dictionaries is one of the minor miracles of computer science. We will see how they work in Section Hashtables.

Read the Wikipedia page on sorting algorithms at https://en.wikipedia.org/wiki/Sorting_algorithm and answer the following questions:

1. What is a “comparison sort?” What is the best worst-case order of growth for a comparison sort? What is the best worst-case order of growth for any sort algorithm?

2. What is the order of growth of bubble sort, and why does Barack Obama think it is “the wrong way to go?”

3. What is the order of growth of radix sort? What preconditions do we need to use it?

4. What is a stable sort and why might it matter in practice?

5. What is the worst sorting algorithm (that has a name)?

6. What sort algorithm does the C library use? What sort algorithm does Python use? Are these algorithms stable? You might have to Google around to find these answers.

7. Many of the non-comparison sorts are linear, so why does does Python use an O(n \log n) comparison sort?

Analysis of search algorithms

A search is an algorithm that takes a collection and a target item and determines whether the target is in the collection, often returning the index of the target.

The simplest search algorithm is a “linear search”, which traverses the items of the collection in order, stopping if it finds the target. In the worst case it has to traverse the entire collection, so the run time is linear.

The in operator for sequences uses a linear search; so do string methods like find and count.

If the elements of the sequence are in order, you can use a bisection search, which is O(\log n). Bisection search is similar to the algorithm you might use to look a word up in a dictionary (a paper dictionary, not the data structure). Instead of starting at the beginning and checking each item in order, you start with the item in the middle and check whether the word you are looking for comes before or after. If it comes before, then you search the first half of the sequence. Otherwise you search the second half. Either way, you cut the number of remaining items in half.

If the sequence has 1,000,000 items, it will take about 20 steps to find the word or conclude that it’s not there. So that’s about 50,000 times faster than a linear search.

Bisection search can be much faster than linear search, but it requires the sequence to be in order, which might require extra work.

There is another data structure, called a hashtable that is even faster—it can do a search in constant time—and it doesn’t require the items to be sorted. Python dictionaries are implemented using hashtables, which is why most dictionary operations, including the in operator, are constant time.

Hashtables

To explain how hashtables work and why their performance is so good, I start with a simple implementation of a map and gradually improve it until it’s a hashtable.

I use Python to demonstrate these implementations, but in real life you wouldn’t write code like this in Python; you would just use a dictionary! So for the rest of this chapter, you have to imagine that dictionaries don’t exist and you want to implement a data structure that maps from keys to values. The operations you have to implement are:

• add(k, v):
  Add a new item that maps from key k to value v. With a Python dictionary, d, this operation is written d[k] = v.

• get(k):
  Look up and return the value that corresponds to key k. With a Python dictionary, d, this operation is written d[k] or d.get(k).

For now, I assume that each key only appears once. The simplest implementation of this interface uses a list of tuples, where each tuple is a key-value pair.

class LinearMap:

    def __init__(self):
        self.items = []

    def add(self, k, v):
        self.items.append((k, v))

    def get(self, k):
        for key, val in self.items:
            if key == k:
                return val
        raise KeyError

add appends a key-value tuple to the list of items, which takes constant time.

get uses a for loop to search the list: if it finds the target key it returns the corresponding value; otherwise it raises a KeyError. So get is linear.

An alternative is to keep the list sorted by key. Then get could use a bisection search, which is O(\log n). But inserting a new item in the middle of a list is linear, so this might not be the best option. There are other data structures that can implement add and get in log time, but that’s still not as good as constant time, so let’s move on.

One way to improve LinearMap is to break the list of key-value pairs into smaller lists. Here’s an implementation called BetterMap, which is a list of 100 LinearMaps. As we’ll see in a second, the order of growth for get is still linear, but BetterMap is a step on the path toward hashtables:

class BetterMap:

    def __init__(self, n=100):
        self.maps = []
        for i in range(n):
            self.maps.append(LinearMap())

    def find_map(self, k):
        index = hash(k) % len(self.maps)
        return self.maps[index]

    def add(self, k, v):
        m = self.find_map(k)
        m.add(k, v)

    def get(self, k):
        m = self.find_map(k)
        return m.get(k)

__init__ makes a list of n LinearMaps.

find_map is used by add and get to figure out which map to put the new item in, or which map to search.

find_map uses the built-in function hash, which takes almost any Python object and returns an integer. A limitation of this implementation is that it only works with hashable keys. Mutable types like lists and dictionaries are unhashable.

Hashable objects that are considered equivalent return the same hash value, but the converse is not necessarily true: two objects with different values can return the same hash value.

find_map uses the modulus operator to wrap the hash values into the range from 0 to len(self.maps), so the result is a legal index into the list. Of course, this means that many different hash values will wrap onto the same index. But if the hash function spreads things out pretty evenly (which is what hash functions are designed to do), then we expect n/100 items per LinearMap.

Since the run time of LinearMap.get is proportional to the number of items, we expect BetterMap to be about 100 times faster than LinearMap. The order of growth is still linear, but the leading coefficient is smaller. That’s nice, but still not as good as a hashtable.

Here (finally) is the crucial idea that makes hashtables fast: if you can keep the maximum length of the LinearMaps bounded, LinearMap.get is constant time. All you have to do is keep track of the number of items and when the number of items per LinearMap exceeds a threshold, resize the hashtable by adding more LinearMaps.

Here is an implementation of a hashtable:

class HashMap:

    def __init__(self):
        self.maps = BetterMap(2)
        self.num = 0

    def get(self, k):
        return self.maps.get(k)

    def add(self, k, v):
        if self.num == len(self.maps.maps):
            self.resize()

        self.maps.add(k, v)
        self.num += 1

    def resize(self):
        new_maps = BetterMap(self.num * 2)

        for m in self.maps.maps:
            for k, v in m.items:
                new_maps.add(k, v)

        self.maps = new_maps

Each HashMap contains a BetterMap; __init__ starts with just 2 LinearMaps and initializes num, which keeps track of the number of items.

get just dispatches to BetterMap. The real work happens in add, which checks the number of items and the size of the BetterMap: if they are equal, the average number of items per LinearMap is 1, so it calls resize.

resize make a new BetterMap, twice as big as the previous one, and then “rehashes” the items from the old map to the new.

Rehashing is necessary because changing the number of LinearMaps changes the denominator of the modulus operator in find_map. That means that some objects that used to hash into the same LinearMap will get split up (which is what we wanted, right?).

Rehashing is linear, so resize is linear, which might seem bad, since I promised that add would be constant time. But remember that we don’t have to resize every time, so add is usually constant time and only occasionally linear. The total amount of work to run add n times is proportional to n, so the average time of each add is constant time!

To see how this works, think about starting with an empty HashTable and adding a sequence of items. We start with 2 LinearMaps, so the first 2 adds are fast (no resizing required). Let’s say that they take one unit of work each. The next add requires a resize, so we have to rehash the first two items (let’s call that 2 more units of work) and then add the third item (one more unit). Adding the next item costs 1 unit, so the total so far is 6 units of work for 4 items.

The next add costs 5 units, but the next three are only one unit each, so the total is 14 units for the first 8 adds.

The next add costs 9 units, but then we can add 7 more before the next resize, so the total is 30 units for the first 16 adds.

After 32 adds, the total cost is 62 units, and I hope you are starting to see a pattern. After n adds, where n is a power of two, the total cost is 2n-2 units, so the average work per add is a little less than 2 units. When n is a power of two, that’s the best case; for other values of n the average work is a little higher, but that’s not important. The important thing is that it is O(1).

Figure below shows how this works graphically. Each block represents a unit of work. The columns show the total work for each add in order from left to right: the first two adds cost 1 units, the third costs 3 units, etc.

Figure B.1: The cost of a hashtable add

The extra work of rehashing appears as a sequence of increasingly tall towers with increasing space between them. Now if you knock over the towers, spreading the cost of resizing over all adds, you can see graphically that the total cost after n adds is 2n - 2.

An important feature of this algorithm is that when we resize the HashTable it grows geometrically; that is, we multiply the size by a constant. If you increase the size arithmetically—adding a fixed number each time—the average time per add is linear.

You can download my implementation of HashMap from http://thinkpython2.com/code/Map.py, but remember that there is no reason to use it; if you want a map, just use a Python dictionary.

Glossary

• analysis of algorithms:
  A way to compare algorithms in terms of their run time and/or space requirements.

• machine model:
  A simplified representation of a computer used to describe algorithms.

• worst case:
  The input that makes a given algorithm run slowest (or require the most space.

• leading term:
  In a polynomial, the term with the highest exponent.

• crossover point:
  The problem size where two algorithms require the same run time or space.

• order of growth:
  A set of functions that all grow in a way considered equivalent for purposes of analysis of algorithms. For example, all functions that grow linearly belong to the same order of growth.

• Big-Oh notation:
  Notation for representing an order of growth; for example, O(n) represents the set of functions that grow linearly.

• linear:
  An algorithm whose run time is proportional to problem size, at least for large problem sizes.

• quadratic:
  An algorithm whose run time is proportional to n^2, where n is a measure of problem size.

• search:
  The problem of locating an element of a collection (like a list or dictionary) or determining that it is not present.

• hashtable:
  A data structure that represents a collection of key-value pairs and performs search in constant time.