Datastructures and algorithms are essential to computer science, which is the study of data, its representation in memory, and its transformation from one form to another. In programming, we use datastructures to store and organize data, and we use algorithms to manipulate the data in those structures. The more you understand about datastructures and algorithms, the more efficient your Java programs will be.

This article launches a three-part series introducing datastructures and algorithms. In Part 1, you'll learn what a datastructure is and how datastructures are classified. You'll also learn what an algorithm is, how algorithms are represented, and how to use time and space complexity functions to compare similar algorithms.

## What is a datastructure?

Datastructures are based on abstract data types (ADT), which Wikipedia defines as follows:

An ADT doesn't care about the memory representation of its values or how its operations are implemented. It's like a Java interface, which is a data type that's disconnected from any implementation. In contrast, a *datastructure* is a concrete implementation of one or more ADTs, similar to how Java classes implement interfaces.

Examples of ADTs include Employee, Vehicle, Array, and List. Consider the List ADT (also known as the Sequence ADT), which describes an ordered collection of elements that share a common type. Each element in this collection has its own position and duplicate elements are allowed. Basic operations supported by the List ADT include:

- Creating a new and empty list
- Appending a value to the end of the list
- Inserting a value within the list
- Deleting a value from the list
- Iterating over the list
- Destroying the list

Datastructures that can implement the List ADT include fixed-size and dynamically sized one-dimensional arrays and singly-linked lists. (You'll be introduced to arrays in Part 2, and linked lists in Part 3.)

## Classifying datastructures

There are many kinds of datastructures, ranging from single variables to arrays or linked lists of objects containing multiple fields. All datastructures can be classified as primitives or aggregates, and some are classified as containers.

## Primitives versus aggregates

The simplest kind of datastructure stores single data items; for example, a variable that stores a Boolean value or a variable that stores an integer. I refer to such datastructures as *primitives*.

Many datastructures are capable of storing multiple data items. For example, an array can store multiple data items in its various slots, and an object can store multiple data items via its fields. I refer to these datastructures as *aggregates*.

All of the datastructures we'll look at in this series are aggregates.

## Containers

Anything in which data items are stored and retrieved could be considered a datastructure. Examples include the datastructures derived from the previously mentioned Employee, Vehicle, Array, and List ADTs.

Many datastructures are designed to describe various entities. Instances of an `Employee`

class are datastructures that exist to describe various employees, for instance. In contrast, some datastructures exist as generic storage vessels for other datastructures. For example, an array can store primitive values or object references. I refer to this latter category of datastructures as *containers*.

As well as being aggregates, all of the datastructures we'll look at in this series are containers.

## Design patterns and datastructures

It's become fairly common practice to use design patterns to introduce university students to datastructures. A Brown University paper surveys several design patterns that are useful for designing high-quality datastructures. Among other things, the paper demonstrates that the Adapter pattern is useful in the design of stacks and queues. The demonstration code is shown in Listing 1.

#### Listing 1. DequeStack.java

```
public class DequeStack implements Stack
{
Deque D; // holds the elements of the stack
public DequeStack()
{
D = new MyDeque();
}
@Override
public int size()
{
return D.size();
}
@Override
public boolean isEmpty()
{
return D.isEmpty();
}
@Override
public void push(Object obj)
{
D.insertLast(obj);
}
@Override
public Object top() throws StackEmptyException
{
try
{
return D.lastElement();
}
catch(DequeEmptyException err)
{
throw new StackEmptyException();
}
}
@Override
public Object pop() throws StackEmptyException
{
try
{
return D.removeLast();
}
catch(DequeEmptyException err)
{
throw new StackEmptyException();
}
}
}
```

Listing 1 excerpts the Brown University paper's `DequeStack`

class, which demonstrates the Adapter pattern. Note that `Stack`

and `Deque`

are interfaces that describe Stack and Deque ADTs. `MyDeque`

is a class that implements `Deque`

.

`DequeStack`

adapts `MyDeque`

so that it can implement `Stack`

. All of `DequeStack`

's method are one-line calls to the `Deque`

interface's methods. However, there is a small wrinkle in which `Deque`

exceptions are converted into `Stack`

exceptions.

## What is an algorithm?

Historically used as a tool for mathematical computation, algorithms are deeply connected with computer science, and with datastructures in particular. An *algorithm* is a sequence of instructions that accomplishes a task in a finite period of time. Qualities of an algorithm are as follows:

- Receives zero or more inputs
- Produces at least one output
- Consists of clear and unambiguous instructions
- Terminates after a finite number of steps
- Is basic enough that a person can carry it out using a pencil and paper

Note that while programs may be algorithmic in nature, many programs do not terminate without external intervention.

Many code sequences qualify as algorithms. One example is a code sequence that prints a report. More famously, Euclid's algorithm is used to calculate the mathematical greatest common divisor. A case could even be made that a datastructure's basic operations (such as *store value in array slot*) are algorithms. In this series, for the most part, I'll focus on higher-level algorithms used to process datastructures, such as the Binary Search and Matrix Multiplication algorithms.

## Representing algorithms

How do you represent an algorithm? Writing code before fully understanding its underlying algorithm can lead to bugs, so what's a better alternative? Two options are flowcharts and pseudocode.

## Using flowcharts

A *flowchart* is a visual representation of an algorithm's control flow. This representation illustrates statements that need to be executed, decisions that need to be made, logic flow (for iteration and other purposes), and terminals that indicate start and end points. Figure 1 reveals the various symbols that flowcharts use to visualize algorithms.

Consider an algorithm that initializes a counter to 0, reads characters until a newline (`\n`

) character is seen, increments the counter for each digit character that's been read, and prints the counter's value after the newline character has been read. The flowchart in Figure 2 illustrates this algorithm's control flow.

A flowchart's simplicity and its ability to present an algorithm's control flow visually (so that it's is easy to follow) are its major advantages. Flowcharts also have several disadvantages, however:

- It's easy to introduce errors or inaccuracies into highly-detailed flowcharts because of the tedium associated with drawing them.
- It takes time to position, label, and connect a flowchart's symbols, even using tools to speed up this process. This delay might slow your understanding of an algorithm.
- Flowcharts belong to the structured programming era and aren't as useful in an object-oriented context. In contrast, the Unified Modeling Language (UML) is more appropriate for creating object-oriented visual representations.

## Using pseudocode

An alternative to flowcharts is *pseudocode*, which is a textual representation of an algorithm that approximates the final source code. Pseudocode is useful for quickly writing down an algorithm's representation. Because syntax is not a concern, there are no hard-and-fast rules for writing pseudocode.

You should strive for consistency when writing pseudocode. Being consistent will make it much easier to translate the pseudocode into actual source code. For example, consider the following pseudocode representation of the previous counter-oriented flowchart:

```
DECLARE CHARACTER ch = ''
DECLARE INTEGER count = 0
DO
READ ch
IF ch GE '0' AND ch LE '9' THEN
count = count + 1
END IF
UNTIL ch EQ '\n'
PRINT count
END
```

The pseudocode first presents a couple of `DECLARE`

statements that introduce variables `ch`

and `count`

, initialized to default values. It then presents a `DO`

loop that executes `UNTIL`

`ch`

contains `\n`

(the newline character), at which point the loop ends and a `PRINT`

statement outputs `count`

's value.

For each loop iteration, `READ`

causes a character to be read from the keyboard (or perhaps a file--in this case it doesn't matter what constitutes the underlying input source) and assigned to `ch`

. If this character is a digit (one of `0`

through `9`

), `count`

is incremented by `1`

.

## Choosing the right algorithm

The datastructures and algorithms you use critically affect two factors in your applications:

- Memory usage (for datastructures).
- CPU time (for algorithms that interact with those datastructures).

It follows that you should be especially mindful of the algorithms and datastructures you use for applications that will process lots of data. These include applications used for big data and the Internet of Things.

## Measuring algorithm efficiency

Some algorithms perform better than others. For example, the Binary Search algorithm is almost always more efficient than the Linear Search algorithm--something you'll see for yourself in Part 2. You want to choose the most efficient algorithm for your application's needs, but that choice might not be as obvious as you would think.

For example, what does it mean, from an efficiency perspective, for the Selection Sort algorithm (also introduced in Part 2) to take 0.4 seconds to sort 10,000 integers on a given machine? That benchmark is only valid for the machine on which the algorithm's implementation runs, for the implementation itself, and for the size of the input data.

A computer scientist measures an algorithm's efficiency in terms of its time complexity and space complexity by using *complexity functions* to abstract implementation and runtime environment details. Complexity functions reveal the variance in an algorithm's time and space requirements based on the amount of input data:

- A
*time-complexity function*measures an algorithm's*time complexity*--meaning how long an algorithm takes to complete. - A
*space-complexity function*measures an algorithm's*space complexity*--meaning the amount of memory overhead required by the algorithm to perform its task.

Both complexity functions are based on the size of input (*n*), which somehow reflects the amount of input data. Consider the following pseudocode for array printing:

```
DECLARE INTEGER i, x[] = [ 10, 15, -1, 32 ]
FOR i = 0 TO LENGTH(x) - 1
PRINT x[i]
NEXT i
END
```

## Time complexity and time-complexity functions

You can express the time complexity of this algorithm by specifying the time-complexity function `t(`

, where *n*) = a*n*+b`a`

(a constant multiplier) represents the amount of time to complete one loop iteration, and `b`

represents the algorithm's setup time. In this example, the time complexity is linear.

The `t(`

function assumes that time complexity is measured in terms of a chronological value (such as seconds). Because you'll want to abstract machine details, you'll often express time complexity as the number of *n*) = a*n*+b*steps* to complete.

How we define "step" can vary from one algorithm to another. In this case, if you identified the single print instruction as the program's step, you would rewrite the time-complexity function in terms of the printing step:

`t(`

; forn) =nnarray elements,nsteps are needed to print the array.

It's important to take care when defining an algorithm's steps, so that the definition is meaningful and correlates with the algorithm's input size. For example, it makes sense to define printing as the *step* for the array-printing algorithm, because printing dominates the runtime and depends on the input size (number of array elements to print).

It's also possible to define steps in terms of *comparisons and exchanges*. In a sorting algorithm, for instance, you might define steps in terms of *comparisons* if comparisons dominate the runtime or *exchanges* if exchanges dominate the runtime.

It's fairly easy to choose a time-complexity function for the array-printing example, but it can be more difficult to find this function for more complicated algorithms. Use the following rules-of-thumb to simplify this task:

- Algorithms with single loops are typically
*linear*--their time-complexity functions are specified in terms of*n*. - Algorithms with two nested loops are typically
*quadratic*--their time-complexity functions are specified in terms of*n*^{2}. - Algorithms with a triply-nested loop are typically
*cubic*--their time-complexity functions are specified in terms of*n*^{3}. - This pattern continues with quadruply and higher nested loops.

These rules-of-thumb work best when a loop executes *n* times (where *n* is the size of the input data). This isn't always the case, however, as demonstrated by the Selection Sort algorithm represented in pseudocode below:

```
DECLARE INTEGER i, min, pass
DECLARE INTEGER x[] = [ ... ]
FOR pass = 0 TO LENGTH(x) - 2
min = pass
FOR i = pass + 1 TO LENGTH(x) - 1
IF x[i] LT x[min] THEN
min = i
END IF
NEXT i
IF min NE pass THEN
EXCHANGE x[min], x[pass]
END IF
NEXT pass
END
```

Because this algorithm consists of two nested loops, you might think that its performance is quadratic. That's only partially correct, however, because the algorithm's performance depends on whether you choose comparisons or exchanges as the algorithm's step:

- If you choose an exchange as one step (because you think that exchanges dominate the runtime) you end up with a linear time-complexity function because
*n*-1 exchanges are required to sort*n*data items. This function is specified as`t(`

.*n*) =*n*-1 - If you choose a comparison as one step ((because you think that comparison dominate the runtime) you end up with
`t(`

, which shortens to*n*) = (*n*-1)+(*n*-2)+...+1`t(`

. Comparisons occur in the inner loop, which executes*n*) =*n*^{2}/2-*n*/2*n*-1 times for the first outer loop iteration,*n*-2 for the second, and so on down to once for the final outer loop iteration.

## Space complexity and space-complexity functions

An algorithm's space complexity indicates the amount of extra memory needed to accomplish its task. For printing an array, a constant amount of extra memory (for code storage, stack space to store the return address when `PRINT`

is called, and space for variable `i`

's value) is needed no matter how large the array.

You can express the array-printing algorithm's space complexity via space-complexity function `s(`

, where *n*) = c`c`

signifies how much constant additional space is required. This value represents overhead only; it doesn't include space for the data being processed. In this case, it doesn't include the array.

Space complexity is expressed in terms of machine-independent *memory cells* instead of machine-dependent bytes. A memory cell holds some kind of data. For the array-printing algorithm, `i`

's memory cell stores an integer value.

## Comparing algorithms

You use time complexity and space complexity functions to compare the algorithm to others of a similar nature (one sorting algorithm to another sorting algorithm, for example). In order to ensure a fair comparison, you must use the same definition for *step* and *memory cell* in each algorithm.

Even when you have chosen identical step and memory cell definitions, however, comparing algorithms can still prove tricky. Because complexities are often nonlinear, an algorithm's input size can greatly affect the comparison result. As an example, consider two time-complexity functions:

`t`

_{1}(*n*) = 10*n*^{2}+15*n*`t`

_{2}(*n*) = 150*n*+5

When *n* equals 1, `t`

yields 25 steps, whereas _{1}`t`

yields 155 steps. In this case, _{2}`t`

is clearly better. This pattern continues until _{1}*n* equals 14, at which point `t`

yields 2170 steps and _{1}`t`

yields 2105 steps. In this case, _{2}`t`

is the much better choice for this and successor values of _{2}*n*.

## Using Big Oh to represent upper bounds

Computer scientists commonly compare algorithms as *n* tends to infinity; this is known as *asymptotic analysis*. Complexity functions serve as the upper bound of the algorithm's asymptotic behavior (as *n* approaches infinity), and a notation called *Big Oh* is used to represent these upper bounds. Here's the formal definition for Big Oh:

Note: *n*, *f(n)*, *g(n)*, *c*, and *n _{0}* must be positive.

*f(n)* represents the algorithm's computing time. When we say that this function is "O(*g(n)*)," we mean that (in terms of steps) it takes no longer than a *constant* multiplied by *g(n)* for this function to execute. For example, here are the Big Oh notations for the previous time-complexity functions:

`t`

_{1}(n) = O(n^{2})

`t`

_{2}(n) = O(n)

You read the first equation as "t_{1} is order *n*^{2}" or "t_{1} is quadratic," and read the second equation as "t_{2} is order *n*" or "t_{2} is linear." The result is that you can think of time complexities as being linear, quadratic, and so on; because this is how the algorithms respond as *n* greatly increases.

To prove that t_{1}(*n*) is O(*n*^{2}), all you need to do is find any two constants *c* and *n _{0}* where the relation holds. For example, choosing 25 for

*c*and 1 for

*n*is sufficient because every value of 10

_{0}*n*

^{2}+15

*n*is less than or equal to 25

*n*for

^{2}*n*>= 1. You can similarly prove that t

_{2}(

*n*) is O(

*n*).

## Comparing algorithms with Big Oh

Suppose the Selection Sort algorithm is followed by the Array Printing algorithm. Because each algorithm offers its own time-complexity function, what is the overall time-complexity function for both algorithms? The answer is governed by the following rule:

If:fand_{1}(n) = O(g(n))fthen_{2}(n) = O(h(n))

- (A)
f= max(O(_{1}(n)+f_{2}(n)g(n)), O(h(n)))- (B)
f= O(_{1}(n)*f_{2}(n)g(n)*h(n)).

Part A covers cases where algorithms follow each other sequentially. For the Selection Sort algorithm followed by the Array Printing algorithm, the overall time-complexity function is the maximum of each algorithm's time-complexity function, which happens to be O(*n*^{2}) (assuming that comparisons are the dominant steps).

Part B covers cases where one algorithm nests inside another. For example, suppose the Array Printing algorithm is called after Selection Sort performs an exchange. Assuming that the sort's time-complexity function is O(*n*^{2}) (comparisons are dominant), the overall time complexity changes to O(*n*^{3}).

How do you choose an efficient algorithm that meets your application's needs? Start by obtaining the Big Oh-bounded time-complexity functions for the candidate algorithms being considered, then deciding the range of *n* values that will be input to these functions (and, hence, the algorithms).

Because it helps to see the impact of various *n* values in a tabular format, I've constructed a table that correlates the number of steps with common Big Oh-bounded time-complexity functions and various *n* values. This table is presented in Figure 3.

#### Figure 3. Correlating step counts with common Big Oh-bounded time-complexity functions and various *n* values. (Click to enlarge.)

The Big Oh-bounded time-complexity functions are sorted from the most efficient function (constant) at the top to the least efficient function (exponential) at the bottom. As you move down the table, notice the functions becoming less efficient (with more steps to complete) for *n* values starting at 16.

It would be great if all algorithms were *O(1)* because they would all be equally efficient. Because this doesn't happen in the real world, you need to carefully choose the most efficient algorithm based on Big Oh-bounded time-complexity functions and the desired range of *n* values.

Keep in mind that more efficient algorithms may be harder to code than less efficient ones. If the range of *n* input values doesn't result in too many steps, you may find that it's better to use a less efficient algorithm with a smaller input range than a more efficient algorithm with a larger input range. You'll see an example of this in Part 2.

## Conclusion to Part 1

Part 1 has introduced you to the fundamentals of datastructures and algorithms. Part 2 moves from the theoretical to the practical, taking you on a tour of Java arrays and their algorithms.