Lecture 8: Notes

Some topics from this lecture are also covered in the Introduction to Algorithms textbook:

common time complexities

Here is a table reviewing the time complexities that we will commonly see. I’ve also listed recurrences and sums that typically lead to these complexities.

It may not be obvious how to solve a given recurrence to find the associated complexity. In the last lecture we saw how a recursion tree can sometimes point us to the solution for a recurrence, though that will not work for all of the cases here. You may study general methods for solving recurrences in a more advanced mathematics or computer science course. But for now, try to become familiar with these common cases.

notation

name

typical recurrences

typical sums

O(1)

constant

T(n) = k


O(log n)

logarithmic

T(n) = T(n / 2) + k


O(n)

linear

T(n) = T(n - 1) + k
T(n) = T(n / 2) + O(n)
T(n) =
2 ⋅ T(n / 2) + k

O(1 + 1 + 1 + … + n)
O(1 + 2 + 4 + 8 + … + n)

O(n log n)

log linear

T(n) = 2 ⋅ T(n / 2) + O(n)


O(n2)

quadratic

T(n) = T(n - 1) + O(n)

O(1 + 2 + 3 + … + n)

O(n3)

cubic

T(n) = T(n - 1) + O(n2)

O(12 + 22 + 33 + … + n2)

O(nk)

polynomial

T(n) = T(n - 1) + O(nk - 1)


O(kn)

exponential

T(n) = kT(n - 1) + O(n)


O(n!)

factorial

T(n) = nT(n - 1) + O(n)


function pointers

A function pointer is a value that refers to a function.

You can declare a function pointer type like this:

type
  intFun = function(a, b: integer): integer

Here is a function whose signature matches this function pointer type:

function add(a, b: integer): integer;
begin
  add := a + b;
end;

Use the @ (“address-of”) operator to get a pointer to a named function in your program:

var
  p: intFun;
begin
  p := @add;

You can call a function through a function pointer:

  writeln(p(3, 4));  // writes 7

using function pointers

We can often generalize a function by giving it a function pointer argument that indicates how some operation is to be carried out.

For example, consider the insertion sort function we saw in the last lecture. Here it is again, with minor changes so that it operates on an array of strings rather than of integers:

procedure insertionSort(var a: array of string);
var 
  i, j: integer;
  v: string;
begin
  for i := 1 to length(a) - 1 do
    begin
      v := a[i];
      j := i;
      while (j > 0) and (v < a[j - 1]) do
        begin
          a[j] := a[j - 1];
          j := j - 1;
        end;
      a[j] := v;
    end;
end;

As written, this procedure always compares strings using Pascal’s built-in comparison operator ‘<’, which is case-sensitive. So if we sort the array

    ('apple', 'cherry', 'watermelon', 'Pear', 'Banana')

we will get

    ('Banana', 'Pear', 'apple', 'cherry', 'watermelon')

The words beginning with a capital letter come first, since capital letters precede lowercase letters in the ASCII encoding.

We might want to sort strings differently, e.g. case-insensitively, or ignoring diacritical marks, or in other ways. We can generalize our insertion sort procedure to work with any string ordering. To do this, we define a function pointer type:

type
  ordering = function(s, t: string): boolean;  // returns true if s < t

And now we modify insertionSort so that any caller can pass a function that implements the ordering they wish to use. Here is the modified implementation:

{$mode delphi}

uses sysutils;

type
  ordering = function(s, t: string): boolean;  // returns true if s < t

procedure insertionSort(var a: array of string; less: ordering);
var 
  i, j: integer;
  v: string;
begin
  for i := 1 to length(a) - 1 do
    begin
      v := a[i];
      j := i;
      while (j > 0) and less(v, a[j - 1]) do
        begin
          a[j] := a[j - 1];
          j := j - 1;
        end;
      a[j] := v;
    end;
end;

function less(s, t: string): boolean;
begin
  less := s < t;
end;

function lessIgnoreCase(s, t: string): boolean;
begin
  lessIgnoreCase := compareText(s, t) < 0;
end;

function reverseIgnoreCase(s, t: string): boolean;
begin
  reverseIgnoreCase := compareText(t, s) < 0;
end;

var
  a: array[1..5] of string = ('apple', 'cherry', 'watermelon', 'Pear', 'Banana');

begin
  insertionSort(a, @less);
  insertionSort(a, @lessIgnoreCase);
  insertionSort(a, @reverseIgnoreCase);
end.

queues

In the last lecture we saw stacks, which are an abstract data type: we can implement a stack in various ways, such as using an array or using a linked list.

Queues are another abstract data type. The interface to a queue looks like this:

type queue = …
procedure init(var q: queue);
procedure enqueue(var q: queue; i: integer);
function dequeue(var q: queue): integer;
function isEmpty(q: queue): boolean;

The enqueue function adds a value to a queue, and dequeue removes a value from queue. Unlike stacks, queues are a first in first out data structure: the first value added to a queue will be the first one to be removed. For example:

var
  q: queue;
begin
  init(q);
  enqueue(q, 4);
  enqueue(q, 77);
  enqueue(q, 12);
  writeln(dequeue(q));  // writes 4
  writeln(dequeue(q));  // writes 77

A queue, then, is like a set of values waiting in line. We say that the value that will next be removed is at the head of the queue, and the value that was last added is at the tail.

queue implementation using a linked list

We can easily implement a queue using a linked list. We must keep pointers to the first and last nodes of the list, which represent the head and tail of the queue. A complete implementation appears below.

With this implementation, the enqueue and dequeue operations run in constant time, i.e. O(1).

type
  node = record
    i: integer;
    next: ^node;
  end;
  
  queue = record
    head: ^node;
    tail: ^node;
  end;

procedure init(var q: queue);
begin
  q.head := nil;
  q.tail := nil;
end;

procedure enqueue(var q: queue; i: integer);
var
  n: ^node;
begin
  new(n);
  n^.i := i;
  n^.next := nil;
  
  if q.head = nil then
    begin
      q.head := n;
      q.tail := n;
    end
  else
    begin
      q.tail^.next := n;
      q.tail := n;
    end;
end;

function dequeue(var q: queue): integer;
var
  p: ^node;
begin
  dequeue := q.head^.i;
  p := q.head;
  q.head := q.head^.next;
  dispose(p);
  if q.head = nil then q.tail := nil;
end;

function isEmpty(q: queue): boolean;
begin
  isEmpty := (q.head = nil);
end;

queue implementation using an array

Another way to implement a queue is using a dynamic array. In addition to the array itself (a), we keep integer variables head and tail that indicate the array positions of the first and next-to-be-added elements. For example, if head = 3 and tail = 6 then there are currently three elements in the queue, with values a[3], a[4] and a[5].

As we add elements to the queue, they may wrap past the end of the array back to the beginning. For example, if length(a) = 8, head = 6 and tail = 2 then there are four elements in the queue: a[6], a[7], a[0] and a[1].

If the tail reaches the head, then the array is full and we need to expand it. Expanding the existing array would be awkward if the queue currently wraps past the end of the array, so we instead allocate a new array, doubling the size of the existing array. We then copy all elements from the old array to the new array.

Suppose that we insert N elements into an initially empty queue. The time to insert each element when the array is not expanded is O(1). We can assume that expanding the array to size S (including copying the elements) takes time O(S). Because we double the array size each time, the total array expansion time will be proportional to

1 + 2 + 4 + … + N = O(N)

So we can insert N elements in O(N) time. Then the average time to insert each element will be O(1) (just as when we implemented a stack using an array). The dequeue operation never changes the array size and always runs in O(1).

Here is the complete implementation:

type
  queue = record
    a: array of integer;
    head: integer;  // index of the first item on the queue
    tail: integer;  // index of the next item to be added
  end;

procedure init(var q: queue);
begin
  setLength(q.a, 2);
  q.head := 0;
  q.tail := 0;
end;

function next(const q: queue; i: integer): integer;
begin
  next := (i + 1) mod length(q.a);
end;

procedure enqueue(var q: queue; i: integer);
var
  b: array of integer;
  size, count, j: integer;
begin
  if next(q, q.tail) = q.head then  // queue is full
    begin
      size := length(q.a);
      count := size - 1;
      
      setLength(b, size * 2);
      for j := 0 to count - 1 do
        b[j] := q.a[(q.head + j) mod size];
      q.a := b;
      q.head := 0;
      q.tail := count;
    end;
    
  q.a[q.tail] := i;
  q.tail := next(q, q.tail);
end;

function dequeue(var q: queue): integer;
begin
  dequeue := q.a[q.head];
  q.head := next(q, q.head);
end;

function isEmpty(const q: queue): boolean;
begin
  isEmpty := (q.head = q.tail);
end;

quicksort

In the last lecture we saw several sorting algorithms (bubble sort, insertion sort, merge sort). Quicksort is another sorting algorithm. It is efficient in practice and is very commonly used.

Quicksort sorts values in an array in place. In this discussion we will assume that array indices begin at 0, like in dynamic arrays in Pascal. Given an array a[0 .. n – 1] of values to sort, Quicksort divides the array into two non-empty partitions a[0 .. p] and a[p + 1 .. n – 1] for some integer p. It does this in such a way that every element in the first partition (a[0 .. p]) is less than or equal to every element in the second partition (a[p + 1 .. n – 1]). After partitioning, Quicksort recursively calls itself on each of the two partitions.

The high-level structure of Quicksort is similar to merge sort: both algorithms divide the array in two and call themselves recursively on both halves. With Quicksort, however, there is no more work to do after these recursive calls. Sorting both partitions completes the work of sorting the array that contains them. Another difference is that merge sort always divides the array into two equal-sized pieces, but in Quicksort the two partitions are generally not the same size, and sometimes one may be much larger than the other.

There are two partitioning algorithms commonly used in Quicksort. In the class we will discuss and use the Hoare partitioning method, which works as follows. The array to be partitioned must have at least two elements. We first choose some element in the array to use as the pivot. The pivot has some value v. All elements in the first partition will have values less than or equal to v, and elements in the right partition will have values greater than or equal to v. In our initial implementation, we will choose a value at the midpoint of the array to use as the pivot, specifically a[(n – 1) div 2].

We define integer variables i and j representing array indexes. Initially i is positioned at the left end of the array and j is positioned at the right. We move i rightward, looking for a value that is greater than v (i.e. it cannot be in the first partition). And we move j leftward, looking for a value that is less than v. Once we have found these values, we swap a[i] and a[j]. After the swap, a[i] and a[j] now hold acceptable values for the left and right partitions, respectively. Now we continue the process, moving i to the right and j to the left. Eventually i and j meet. The point where they meet is the division point between the partitions, i.e. is the index p mentioned above.

Note that the pivot element itself may move to a different position during the partitioning process! There is nothing wrong with that. The pivot really just serves as an arbitrary value to use for separating the partitions.

A complete implementation of Quicksort appears below.

procedure swap(var i, j: integer);
var
  k: integer;
begin
  k := i;
  i := j;
  j := k;
end;

// Partition an array of at least two elements.
// Return an index p < high(a) such that
//   every element of a[0 .. p] <= every element of a[p + 1 .. high(a)]
function partition(var a: array of integer): integer;
var
  v, i, j: integer;
begin
  v := a[high(a) div 2];    // pivot value
  i := -1;
  j := high(a) + 1;
  
  while true do
    begin
      repeat
        i := i + 1;
      until a[i] >= v;
      repeat
        j := j - 1;
      until a[j] <= v;
      
      if i >= j then break;
      swap(a[i], a[j]);
    end;
    
    partition := j;
end;

procedure quicksort(var a: array of integer);
var
  p: integer;
begin
  if length(a) < 2 then exit;
  p := partition(a);
  quicksort(a[0 .. p]);
  quicksort(a[p + 1 .. high(a)]);
end;

partition function correctness

The partition function is simple, but it may not be immediately obvious that it is correct. We make three claims:

  1. All array accesses that happen as partition runs are within the array bounds.

  2. The returned value p is in the range 0 <= p < high(a) (and so both partitions contain at least one element).

  3. All elements in the left partition are less than or equal to all elements in the right partition.

Here is an argument in support of these claims. On each iteration of the while loop, at the point where we test ‘if i >= j’, one of three things is true:

  1. i < j: the values will be swapped, and the loop will continue

  2. i = j: the indices are equal, and the loop will exit

  3. j < i: the indices have crossed, and the loop will exit

The condition (a) will hold for some number of iterations. During those iterations all array accesses will be within bounds, since we have 0 <= i < j <= high(a). Furthermore, the following will always be true at the beginning of the while loop:

This invariant is vacuously true at the beginning of the first iteration, and is preserved by the loop body. Immediately after the repeat loops, a modified invariant with the stricter conditions 0 <= k < i and j < k <= high(a) is true.

Eventually we will reach condition (b) or (c). If we reach (b), then certainly 0 <= i = j <= high(a), so i = j is within bounds and all array accesses have stayed within bounds as well. And in fact we must have j < high(a). That’s because if we reach (b) on the first iteration, we must have reached the pivot element, whose index is always less than high(a) due to the way we computed it. That’s because the array has at least two elements, so high(a) >= 1, so high(a) div 2 < high(a). And if we reach (b) on any other iteration, we have decremented j at least twice, so j < high(a) in that case as well. In any case the loop will exit, and the value i = j is a valid partition point.

Or we may reach condition (c), in which case the pointers have crossed. This cannot happen on the first iteration, where both pointers will stop at the pivot. So let i0 and j0 be the values of i and j at the end of the previous iteration. We know that 0 <= i0 < j0 <= high(a). Now, at the moment where we test ‘if i >= j’, we have j < i. It must be that i <= j0, since a[j0] >= v so i will stop there (if not before) as if advances. Furthermore we must have j = i – 1, since a[i – 1] <= v, so j cannot have moved past that point. Also, i > i0, so j = i – 1 >= i0 >= 0. This shows that both i and j remain within bounds. The loop will exit. At this point, for any k <= j we have k < i, so by the invariant a[k] <= v. And for any k > j the invariant gives us a[k] >= v. The partition point p = j is valid.

performance of Quicksort

To analyze the performance of Quicksort, we may start by observing that partition(a) runs in time O(n), where n = length(a). This is because the number of swaps must be no more than n / 2, and because i and j stay within bounds so their values will increment or decrement at most n times.

In the best case, Quicksort divides the input array into two equal-sized partitions at each step. Then we have

T(n) = 2 T(n / 2) + O(n)

This is the same recurrence we saw when we analyzed merge sort in the last lecture. Its solution is

T(n) = O(n log n)

In the worst case, at each step one partition has a single element and the other partition has all remaining elements. Then

T(n) = T(n – 1) + O(n)

This yields

T(n) = O(n2)

So, then, how will Quicksort typically perform?

This depends critically on the choice of pivot elements. Suppose that in the implementation above we had chosen a[0] as the pivot element at each step. Then if the input array was already sorted, the pivot element would always be the smallest in the array, and Quicksort would run in the worst-case time of O(n2).

Choosing the element at the midpoint will generally be more robust, and should perform well almost all the time. Still, it is possible to construct input arrays for which Quicksort will perform poorly even with this pivot choice. To be safer, we can choose a random pivot element whose index is from 0 to high(a) - 1. (The random element must not be the last, since then partition could return p = high(a) !) If we do so, then Quicksort’s expected performance will be O(n log n) for any input array. We will not prove this fact in this class. Note, however, that it is somewhat similar to our (also here unproven) claim that if elements are inserted into a binary tree in a random order, then the expected height of the tree will be O(log n), where n is the number of elements in the tree.

greatest common denominator

Let us review a bit of number theory. We say that an integer d divides an integer a, written d | a, if a = kd for some integer k. Note that if d | a, then d | ka for any integer k. Furthermore, if d | a and d | b then d | (a + b).

A linear combination of integers a and b is any value of the form ua + vb for some integers u and v. Suppose that x is a linear combination of a and b, and that d | a and d | b. Then x = ua + vb for some u and v, and d | ua and d | vb, so d | x as well.

The division theorem states that for any integers a and b with b > 0, there exist unique integers q and r with 0 <= r < b such that a = bq + r. q is called the quotient and r is the remainder. We have seen these values before: as long as a >= 0, we have q = a div b and r = a mod b. So, rewriting the division theorem in Pascal’s terminology,

a = (a div b) * b + (a mod b)

For integers a and b, the greatest common divisor of a and b, also written as gcd(a, b), is the largest integer d such that d | a and d | b. The greatest common divisor is useful in many situations, such as simplying fractions. For example, suppose that we want to simplify 35/15. gcd(35, 15) = 5, so we can divide both the numerator and denominator by 5, yielding the simplified fraction 7/3.

One way the find the greatest common divisor of two numbers is to factor the numbers into primes, then look for common primes. For example consider finding gcd(252, 90). 252 = 9 * 28 = 22 * 32 * 71. And 90 = 9 * 10 = 21 * 32 * 51. So the common primes are 21 * 32 = 18, and we have gcd(252, 90) = 18.

Euclid’s algorithm

Euclid’s algorithm is a much more efficient way to find the gcd of two numbers than prime factorization. It is based on the fact (proven below) that gcd(a, b) = gcd(b, a mod b). So, for example,

gcd(252, 90) = gcd(90, 72) = gcd(72, 18) = gcd(18, 0) = 18.

It’s easy to write Euclid’s algorithm in Pascal:

function gcd(a: integer; b: integer): integer;
begin
  if b = 0 then
    gcd := a
  else
    gcd := gcd(b, a mod b);
end;

To show that this works, we must prove that gcd(a, b) = gcd(b, a mod b).

Suppose that d | b and d | (a mod b). We know that a is a linear combination of b and (a mod b) by the division theorem above. So d | a.

Now, conversely, suppose that d | a and d | b. We can rewrite the division theorem as

a mod b = a – (a div b) * b

This shows that (a mod b) is a linear combination of a and b. So d | (a mod b).

We have shown that every divisor of b and (a mod b) is also a divisor of a and b. Conversely, every divisor of a and b is also a divisor of b and (a mod b). So the common divisors of a and b are the same as the common divisors of b and (a mod b). It follows immediately that gcd(a, b) = gcd(b, a mod b).

running time of Euclid’s algorithm

What is the running time T(a) of the above function gcd(a, b) , where a > b? First note that if a > b, then a mod b < a / 2. For if a / 2 < b < a, then a div b = 1, so a mod b = a – b < a – a / 2 = a / 2. Or if 0 < b <= a / 2, then a mod b <= b – 1 < a / 2.

Suppose that Euclid’s algorithm run on gcd(a, b) recurses for at least two iterations. Then it computes

gcd(a, b) = gcd(b, a mod b) = gcd(a mod b, ...) = …

In two iterations the first argument has dropped from a to (a mod b) < a / 2. So after every two iterations of the algorithm, the first argument will be reduced by at least a factor of 2. This shows that the algorithm cannot iterate more than 2 log2(a) times. It follows that T(a) = O(log a).

To find the gcd of several numbers, we can invoke Euclid’s algorithm more than once. The gcd function is associative, so we can compute either

gcd(gcd(a, b), c)

or

gcd(a, gcd(b, c))

least common multiple

The least common multiple of integers a and b is the smallest integer m such that a | m and b | m.

We can compute the least common multiple by factoring a and b into prime powers, just like with the greatest common divisor. But there is an easier way. For all a and b,

lcm(a, b) = a b / gcd(a, b)

To compute the lcm of several numbers, we can use the above formula more than once:

lcm(lcm(a, b), c)

or

lcm(a, lcm(b, c))