The basic List interface resembles an array, in that one can access and update elements by specifying their position. The underlying implementation, however, may not be via an array (for example, linked lists), in which case accessing the Nth element may not be as fast as accessing a corresponding array element.

Most list interfaces also support some mechanism of automatic size expansion to accommodate new additions, reducing the risk of array-bounds overflow. However, attempting to access an element (by position) that is not there is always an error: retrieving item 100 of a list that contains items 0 through 37 can never end well.

Java List<T> and ArrayList<T>

Java supports the ArrayList class, and the List interface. We'll only touch on interfaces here.

To declare an ArrayList of strings, use

    import java.util.*;    // or java.util.ArrayList
    ArrayList<String> L = new ArrayList<String>()

The constructor here is the right side; there is no parameter. The <String> is the "type parameter". ArrayList is a generic class, as are most data-structure classes.

The list L created above is initially empty. We use L.add(str) to add entries:


The current size of the list is available as L.size(); this would return 3 for the list above. The elements are at positions 0, 1 and 2.

We can now set and retrieve elements with L.set(i,val) and L.get(i), corresponding to A[i]=val and A[i], for an array A. In the above, L.get(0) returns "apple".

The capacity of the list represents the underlying memory allocation, which, for an ArrayList, is an array. It is 10, by default, though we could also have passed in a numeric parameter to the constructor: new ArrayList<String>(100).

Things get interesting when you keep using .add to insert one more element than there is room for. A new internal array is allocated, and the contents of the old array are copied over. The .add() operation then continues as if nothing unusual had happened.

It is possible to use .add() to add an entire array's worth of elements. We can also use .add(pos, val) to add the value to position pos, rather than to the end of the list. In this case, the value that was at position pos is moved to position pos+1, and so on; nothing is overwritten. Other useful ArrayList options:

Demo: arraylisttest

As a general rule, ArrayLists are best when you're not sure of the list length at the start of the program. Arrays work best when you do know the array size upfront.


Chapter 3 contains Bailey's Vector class, which represents a List of Objects implemented via an array. The main feature is that the vector can grow.

We'll start with the Vector interface on page 46. 

Bailey's examples:

    Vector used for Wordlist

    Vector used for L-systems

There is an issue with Vectors of Objects: we don't really want Objects. Look closely at the code on page 47; it contains a cast to (String):

    targetWord = (String) list.get(index);

What this does is takes the Object list.get(index), and, because it is really a string, allows its use as a string at compile-time.

In Java we will usually use generics, eg Vector<T>.

In the word-frequency example on page 49 (actually starting on p 48) there is

    wordInfo = (Association) vocab.get(i);


    vocabWord = (String) wordInfo.getKey();

Adding to the middle of a vector: you need to move from right to left. See the picture on Bailey p 52.

Vector Growth Demo

C# version

We will use the command-line version. The List<T> class is documented at

The demo program is listgrowth.cs.

We first create the list.

    List<String> s = new List<String>();

We can examine s.Count to get the current length of the list, and s.Capacity to get the current length of the internal array in object s. Initially, both are 0.

If we add an item: s.Add("apple"), then s.Count = 1 and s.Capacity = 4.
If we add three more items, both s.Count and s.Capacity are 4.
If we then execute s.Add("banana"), s.Count is now 5 and s.Capacity is now 8.
If we then execute s.Add("cherry"), s.Capacity becomes 16.

Java version

The demo program is in arraylisttest. Create an empyt arraylisttest object on the Object Bench. We can then call:


  1. Use the inspector to examine the addOne("apple"). Inspect theList. What is the size of elementData? Where is elementData, by the way? Where did it come from?
  2. Same after addOne("apple")
  3. Same after addSome(9,"banana"). Why 9?
  4. Same after addOne("cherimoya")

Keep going until you can guess the expansion pattern for elementData.

�3.5: analysis of costs of expansion

These costs are all linear; that is, proportional to N.

Now suppose we want to insert N items into a list initially of length 0, perhaps searching the list each time in order to insert in alphabetical order. Each item's required position is more-or-less random, and so takes on average size()/2 moves. That is, to insert the 1st element takes 0/2 moves, the 2nd takes 1/2, the 3rd takes 2/2, the 4th takes 3/2, the 5th takes 4/2, etc. Adding all these up gives us a total "average" number of moves of

    1/2 + 2/2 + 3/2 + 4/2 + 5/2 + ... + (N-1)/2 = 1/2*(1 + 2+ 3 + ... + (N-1))
=  1/2*(N(N-1)/2) = N2/4-N/4

Now, for large N this is approximately N2/4, that is, proportional to N2, or quadratic. Using the big-O notation, later, this is equivalent to saying the number of moves is O(N2).

Search and insert costs

These operations both take, on average, currsize/2 steps, or N/2 if we follow the convention that N represents the current size. For search, we have to check on average half the list; the steps we are counting are the comparisons. For insert, the operations are the assignments things[i+1]=things[i].

Both these can be described as O(N).


We would really like to be able to declare containers with a fixed type, where the type is supplied as parameter:

    List<String> s = new List<String>();

If we use Bailey's Vector class, we would have pretty much the same performance, but getting strings out of the Vector would always require a cast:

    String s = (String) vect.get(3);


A stack is a data structure that supports push() and pop() operations. A stack looks like a list except there is no direct way to access anything but the topmost element; you cannot even do that except by also deleting that element from the stack. The basic operations are
The Stack class, with specific methods push(), pop(), and isEmpty(), is sort of the canonical example of an Abstract Data Type, that is, a class where the focus is on representing a "thing" (as is usually the case). A stack can be implemented as an array, but we have no access to the top element except through pop(), and no access to the middle elements at all. Alternatively, we could change the implementation to that of, say, "linked list", and the class users would be unaffected. Note that push() and pop() do not simply perform individual field updates.

Finding something to do with the stack is harder; why would you need that very specific last-in, first-out (LIFO) access? There are lots of examples from system design and programming-language design, but they tend not to be trivial. One straightforward example is to confirm that a line consisting of ()[]{} has all the braces in balance. The algorithm is as follows:
    if you encounter an opening symbol, (, [, or {, push it.
    if you encounter a closing symbol, ), ], or }, pop what is on the stack and verify the two correspond.
    when you get to the end of the input, verify that the stack is empty.

Note that generally popping something off an empty stack is an error, so that you should check with isEmpty().

Implementing a stack

Here's a stack of strings:

class Stack {
   private List<string> L;
   public void push(string s) {L.Add(s)}
   public string pop() {string s = L[L.Count - 1]; L.remove[L.Count-1]; return s;}
   public boolean is_empty() {return L.Count == 0;}

push(e) corresponds to L.add(e),
is_empty() corresponds to L.size() == 0.

pop() corresponds to {e=L.get(size()-1; L.delete(size()-1); return e;}

Deleting from a list

If all we do is add, then the growth strategy of doubling the internal space when necessary makes perfect sense.

But what happens if we will regularly grow lists to large size, and then delete most of the entries? A list grown to have internal capacity 1024 will retain that forever, even if we shrink down to just a few elements.

One approach is to re-allocate to a smaller elements[] whenever L.Count < L.Capacity/2, or something like that.

Morin in §2.6 (p 49) introduces what he calls a RootishArrayStack, which is an array-based list with an efficient delete operation. Here are the key facts (big-O notation is officially introduced in the next section):

The idea is to keep a list of arrays (an array of pointers to arrays). These sub-arrays have size 1, 2, 3, 4, etc respectively. For 10 elements, the RootishArrayStack would have four arrays, and thus a capacity of 1+2+3+4.

If the RootishArrayStack has N elements in n arrays, then N ≃ n²/2; this follows because 1+2+...+n ≃ n²/2. When the RootishArrayStack needs to expand, it will add an array of size n+1 to the pool; this is about √(2n). Thus, growth is "slower" than for C# Lists or Java ArrayLists. However, when a new allocation is made for growth, the old space is not discarded.

The real advantage of the RootishArrayStack is for deletions. If the list shrinks so that the last sub-array is now empty, that sub-array and that sub-array only is discarded. This is a relatively efficient operation.

Big-O notation and Bailey Chapter 5: Analysis

We will need to be able to talk about runtime costs. To this end, the big-O and (to a lesser extent) little-o notations are useful. If N is the size of the data structure, and f(N) is a growth function (like f(N) = log(N) or f(N) = N or f(N) = N2), then we say that a cost is O(f(N)) provided the number of steps is bounded by k×f(N), for a constant k, as N grows large. We say that a cost is o(f(N)) if cost(N)/f(N) → 0 as N grows large. (Alternatively, cost(N) is O(f(N)) if cost(N)/f(N) is bounded by constant k as N grows large.)

See Figure 5.1 on page 83.

Here are a few examples for an array-based Vector as in Bailey of length N:

provisos and notes
Inserting at the end of a Vector O(1)                        if no expansion is necessary
Inserting in the middle of a Vector O(N) N/2 moves on average
Searching a Vector O(N) N/2 comparisons on average if found

Inserting and searching are both linear.

Adding an element to a SetVector takes O(n) comparisons, because we have to make sure it isn't already there.

Now suppose we want to insert N items into a Vector initially of length 0, perhaps searching the list each time in order to insert in alphabetical order. Each item's required position is more-or-less random, and so takes on average size()/2 moves. That is, to insert the 1st element takes 0/2 moves, the 2nd takes 1/2, the 3rd takes 2/2, the 4th takes 3/2, the 5th takes 4/2, etc. Adding all these up gives us a total "average" number of moves of

    1/2 + 2/2 + 3/2 + 4/2 + 5/2 + ... + (N-1)/2 = 1/2*(1 + 2+ 3 + ... + (N-1))
=  1/2*(N(N-1)/2) = N2/4-N/4

Now, for large N this is approximately N2/4, that is, O(N2), or quadratic.

Building a list up by inserting each element at the front (or inserting each element at random) is O(n2). (This is the last example on Bailey page 87.)

Taking the union or intersection of two sets is O(n2) (Why? Is there a faster way?)

Finding if a number is prime by checking every k < sqrt(n) is O(n1/2).

How hard is it to find the minimum of an array of length N? O(N)

How hard is it to find the median of an array of length N? Somewhat surprisingly, this can also be done in O(N) time. See sorting.html#median.

See sorting.html#binsearch for an analysis of binary search

A function is said to be polynomial if it is O(nk) for some fixed k; quadratic growth is a special case.

So far we've been looking mainly at running time. We can also consider space needs. As an example, see the Table of Factors example on Bailey page 88. Let us construct a table of all the k<=n and a list of all the factors (prime or not) of k, and ask how much space is needed. This turns out to be n log n. The argument here is a bit mathematical; see Bailey. If the table length is n, then factor f can appear no more than n/f times (once every fth line).

The running time to construct the table varies with how clever the algorithm is, it can be

Now suppose we want to search a large string for a specific character. How long should this take? Bailey has an example on p 90. The answer depends on whether we're concerned with the worst case or the average case (we are almost never interested in the best case). If the average case, then the answer typically depends on the probability distribution of the data.

Linked Lists

The standard "other" way of implementing a list is to build it out of cells, where each cell contains a pointer to the next item. See
Linked lists are very efficient in terms of time to allocate and de-allocate space. Insertion is O(1). Finding an element is O(n), however, even if the list is sorted; there is no fast binary search.

Each linked-list block contains two pointers, one for data and one for the link. That's a 2× space overhead. For array-based lists, that would correspond to having each list have a Capacity that was double its Count. That's not necessarily bad, but the point is that linked lists have limited space efficiency. (They may be quite efficient in terms of allocation time, though; each block allocated amounts to one list cell, and if many linked lists are growing and shrinking then the allocator can in effect just trade cells back and forth. With array-based lists, however, if two lists have just deleted blocks of size 256 and a third list now needs a block of size 512, the deleted blocks cannot be recycled into the new block unless they just happen to be adjacent.

Here is some code from the demo file

class TLinkedList<T> {
    private T data;
    private TLinkedList<T> next;
    public TLinkedList(T d, TLinkedList<T> n) {data=d; next=n;}
    public T first() {return data;}
    public TLinkedList<T> rest() {return next;}

The interface is peculiar here; ignore that for now.

A program that uses this might be:

static void main(String[] args) {
    TLinkedList<String> slist = new TLinkedList<String>("apple", null);
    slist = new TLinkedList<String>("banana", slist);
    slist = new TLinkedList<String>("cherry", slist);
    slist = new TLinkedList<String>("daikon", slist);
    slist = new TLinkedList<String>("eggplant", slist);
    slist = new TLinkedList<String>("fig", slist);

    TLinkedList<String> p = slist;
    while (p!= null) {
        p =;

This is not exactly what we want: too many internals are exposed. A more contained implementation would be as follows:

class TLinkedList<T> {
    class Cell<T> {
        private T data;
        private Cell<T> next;
        public Cell(T d, Cell<T> n) {data=d; next=n;}
        public T first() {return data;}
        public Cell<T> rest() {return next;}
    private Cell<T> head = null;
    public void AddToFront(T element) {head = new Cell<T>(element, head);}
    public bool is_empty() {return head == null;}
    public T First() {return head.first();}
    public void DelFromFront() {head =;}

A slightly more complete Cell subclass is the following (changes in bold)
    public class Cell<T> {
       private T data_;
       private Cell<T> next_;
       public Cell(T s, Cell<T> n) {data_ = s; next_ = n;}
       public T data() {return data_;}
       public Cell<T> next() {return next_;}
       public void setData(T s) {data_ = s;}
       public void setNext(Cell<T> c)   {next_ = c;}

Implementing a stack using linkedlist

push(e) corresponds to AddToFront(e), is_empty() corresponds to head == null.

pop() corresponds to ...

Implementing a set

In section 3.7 Bailey uses vectors/Mylists to implement an abstract Set. Note the more limited set of operations; there is no get() and no set().

add() now works very differently: add(E e) is basically if (!contains(e) ) add(e), where the second add(e) is Vector.add(e).

On the face of it, to form the union of two sets A and B of size N, we need N2 equality comparisons: each element of A has to be compared with each element of B to determine if it is already there. This cost is sometimes said to be O(N2) if we don't care if it's N2, or N2/2, or 3N2.

Later we'll make this faster with hashing. Brief summary: choose a relatively large M, maybe quite a bit larger than N. Define h(obj) = hashCode(obj) % M. Now choose a big array ht (for hash table) of size M, initially all nulls. For each a in A, do something with ht[hash(a)] to mark the table. Then, for each b in B, if ht[hash(b)] is still null, put it in; it's not a duplicate! If ht[hash(b)] is there already, then we have to check "the long way", but in general we save a great deal.

Is there an intersect option?

Java LinkedList

Java has a LinkedList<T> class. It works like ArrayList, except it uses linked lists. That makes lookup of an arbitrary element O(N), but insertion (once you've found the postion) is now O(1).

Using List<T> to implement a Matrix class in C#

Suppose we want to construct a two-dimensional object, Matrix. Values in the Matrix will have type double. The class should have the following operations:

How should we proceed?

Here's a simpler problem: how should we implement a Vector<T> class, where vector objects have a fixed length, and are initialized to 0? C# does take care of that latter, but List<T>'s do not automatically have the right length. Also, ideally we'd like to "hide" the add() operation, that can make a List<T> grow longer than we'd like.

class Vector { public Vector(int l) {...}
     public int getlength() {...}
     public double get(int i) {...}
     public void set(int i, double val) {...}

Now let's return to the Matrix class. As in Vector, we will pre-allocate space for all the elements. Here is some simple code to implement a matrix class with TList objects (not yet converted to Java).

 * Class Matrix is implemented by a TList of rows.

class Matrix {
    // instance variables - replace the example below with your own
    private TList<TList<double> > m;
		// list of lists
    private int height, width;

     * Constructor for objects of class TList
    public Matrix(int h, int w)
        // initialize instance variables
        height = h;
	width = w;
        m = new TList<TList<double>>(height);
	// we must preallocate all the rows
	for (int i = 0; i<height; i++) {
		TList<double> theRow = new TList<double>(width);
		// we must preallocate all the slots (columns) in each row
    public int getwidth() {return width;}
    public int getheight() {return height;}

    // get nth value, with range check
    public double get(int r, int c) {
        if (r<0 || r >= height) {
            Console.WriteLine("Warning: Matrix.get() called with out-of-range row = " + r);
            return 0.0;	
        if (c<0 || c >= width) {
            Console.WriteLine("Warning: Matrix.get() called with out-of-range column = " + c);
            return 0.0;	
        return m.get(r).get(c);
    // set nth value, with range check
    public void set(int r, int c, double val) {
        if (r<0 || r >= height) {
            Console.WriteLine("Warning: Matrix.set() called with out-of-range row = " + r);
        if (c<0 || c >= width) {
            Console.WriteLine("Warning: Matrix.get() called with out-of-range column = " + c);

Things to note:

Linked List Efficiency

What good are linked lists? Inserting in the middle is fast, but finding a point in the middle is slow. So almost everything is O(n).

But inserting at the head is always fast.

Also, linked lists use memory efficiently if you have a great many shorter lists. While the next_ fields require space, there are no "empty" slots as in an array-based stack. And no memory wasted due to list expansion.

These are singly linked lists; a doubly linked list has a pointer prev_ as well as next_, that points to the previous element in the chain.

Stacks and Linked Lists

While the array implementation of a stack is quite fast, the linked list approach is equally straightforward. All we have to do is maintain a pointer to the head:

class stack<T> {
    private Cell<T> head_;
    public bool is_empty() {return (head_ == null);}
    public T pop() {T val =; head_ =; return val;}
    public void push(T val) {head_ = new Cell<T>(val, head_);}

What would we need to do in C++ if we wanted to be sure we deleted a popped cell?

Sorting Linked Lists

How would you sort a linked list? QuickSort is out.


"When in doubt, use a hash table"
- Brian Fitzpatrick, Google engineering manager and former Loyola undergrad

One way to search through a large number of values is to create a hash function hash(T) that returns an integer in the range 0..hmax-1. Then, given a data value d, we calculate h = hash(d) and then put d into "bucket" h. A convenient way to do this is to have an array htable of lists, and add d to the list htable[i]. This particular technique is sometimes called "bucket hashing" or "chain hashing"; see Bailey 15.4.2.

Linked lists are particularly convenient for representing the buckets, as we will have a relatively large number of them, and most will be small.

What shall we use as a hash function? This comes up often, and a great number of standard data structures rely on having something available. Therefore, Java provides every object with a hashCode() method. It returns a 32-bit value.

Demo: what are hashcodes of
On my system, for a two-character string hashCode() returns 31*first_char + second_char, where the values first_char and second_char are the ascii numeric values. So, for string "db", where 'd' is 100 and 'b' is 98, hashCode() returns 3198. See class HashCodes in demo hash.

Example: bucket hashing of
where hash(s) = s.length();

Many classes choose to "tune" the standard hashCode() by providing their own version. Many data structures will simply assume that two objects with different hashcodes are unequal, so it is important when providing an overriding .Equals() method to also provide .hashCode(). In lab 3, I provided equals() and hashCode() for class LinkedList<T>; for lab 2 I did this for StrList.

If you were to create a class with its own .equals(), but no .hashCode(), search might fail with some containers. Given a container of your class, Java might determine that there was no value in the container that had the same hashCode() value as the search target, and give up, even if there was in fact a value in the container that was .equals() to the search target.

Mid-class exercise: call hashCode() on the following strings:
The above can be assigned to an array string[] A; this is done in and (in hash).

1. Do all of you get the same values for s.hashCode()? On linux, for "avocado" I get -622659773 and for "guava" I get  98705182. (Java is a little more standard across different platforms than C#.)

2. Now use hash(s), in the file above, and put the strings into htable. For what htablesize do you get buckets with "collisions": more than one string assigned to it? For what htablesize is this particular table collision-free?

3. Can you think of an orderly way of searching for the answer for #2?

The hash table in hash.cs is not actually an object. What do we have to do to make it one? Perhaps htablesize could be a parameter to the constructor.

Open Hashing

Another way to do hashing is so-called "open" hashing: a data object d is simply put into htable[hash(d)]. If that position is taken, the next position is used. For this to work, we need to be sure that htablesize is quite a bit larger (eg at least double) the number of elements added. Deletions require careful thought. See Bailey 15.4.1.

Traversing a Hash Table

If we want to print out a hash table, or construct an iterator to step through each element in turn, we can simply run linearly through the hashtable array. For bucket hashing, each hashtable[i] represents a linked list to be traversed. For open hashing, we simply skip over the unused elements.

This traversal is in no particular order!

A class based on this is in; note the print() method. This class uses the string type; there is also inthashclass.cs that uses int (yes, I should have made this use a generic type).

Hash Sets and Hash Dictionaries

One way to implement sets of strings (or of a generic type T) is with lists:
class StrSet {

    private StrList sl;
    public StrSet() {sl = new StrList(100); }
    public boolean isMember(String s) {
        for (int i=0; i<sl.size(); i++) {
            if (sl.get(i) == s) return true;
        return false;

    public void add(string s) {
	if (isMember(s)) return;
But there is a problem here: the isMember() and add() methods are O(N). [why?]

Can we do better? Yes, with hashing.

To create a HashSet, we use a hash table as in The code for this is in
class hashset {
    private hashtable ht;
    public hashset(int size) {ht = new hashtable(size);}

    public boolean isMember(String s) {
        return ht.isMember(s);

    public void add(String s) {
	if (isMember(s)) return;

    public void print() {
To run this, it must be linked with


This section is still partly in C#

To create a dictionary, we will use generic type parameters K for the key and V for the values. We will rewrite our hashtable class so that the Cell contains fields for the key (of type K) and value (of type V).

The interface will then be:

The dictionary example is at dictionary.cs. It contains a simple driver program.

Demo: use dictionary.cs and count the word occurrences in a paragraph pasted in from some other source (these notes, or else a paragraph from Bailey). If s is a long string holding the paragraph, use s.Split() to divide it into words. Extra features:

The file dictionary.cs contains a full-fledged implementation of a dictionary type, complete with generics, support for "iterators", and the d[] notation.

This section is still in C#

hashtable enumerator: demos/dictionary.cs

I want this to work:
	foreach (KeyValuePair<string,int> kvp in d) 
Console.WriteLine("{0}: {1}", kvp.Key, kvp.Value);
The hashtable is an array of linked lists; the linked-list cell type is
    public class Cell {
	private K key_;
	private V val_;
	private Cell  next_;
	public Cell(K k, V v, Cell n) {key_ = k; val_ = v; next_ = n;}
	public K getKey() {return key_;}
	public V getVal() {return val_;}
	public Cell   next() {return next_;}
        public void setVal(V v) {val_ = v;}
        public void setNext(Cell c)   {next_ = c;}
To start, I must have class dictionary inherit from System.Collections.Generic.whatever. This works:
	class dictionary : System.Collections.Generic.IEnumerable> {
Then I must implement the IEnumerable method. The exact method signature is as follows; note the return type.
    IEnumerator> IEnumerable>.GetEnumerator() {
	return foonumerator();
What is up with foonumerator()? That's here:

    IEnumerator> foonumerator() {
	for (int i = 0; i(p.getKey(), p.getVal());
		p =;
	yield break;
Why didn't I just define this in IEnumerable, above? Because we also must implement the non-generic form of IEnumerable, due to inheritance constraints. I did that this way:
    System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() { 
	return foonumerator(); 
Otherwise I would have to type everything twice.

I figured this all out by reading the MSDN Dictionary.cs reference code, here.