# Linear and binary search

Summary
We consider the standard problem of searching an array for a value or values. That is, we consider algorithms that, given a target value, a collection of values, and a notion of equality, find the position of the target value. We explore two common searching algorithms, linear search and binary search. We also consider ways in which object-oriented design (and its instantiation in the Java programming language) may lead us to think about searching in slightly different ways.

## Introduction

One of the most common tasks computer programs do is * search* - look through a collection of information for something matching particular criteria. Sometimes the search is huge and illformed, as in Google searches. Sometimes the search is small and precise.

There are a host of ways to organize information to make searching faster and easier. We’ll consider some of those mechanisms this semester. For now, we’ll start with one of the simplest phrasings of the search problem: given a value to search for and an array of values to search in, find the index of the value in that array.

The simplest searching algorithm is often called linear search, and involves looking at each element in turn until we find one that matches.

For each index, i, in values
If values[i] is equal to val
return i;
If the value was never found, signal that problem


How long does linear search take? Well, it depends. If the thing we’re searching for appears near the beginning of the array, it takes only a few steps. If it appears at the end of the array, we’ll need to look at almost all of the elements of the array. And, if it doesn’t appear at all, we need to look at every element.

Since, in the worst case, we look at every element, the body of our loop will run O(n) times. Each repetition of the loop requires a constant number of steps (or so we hope - comparison should be fast), so the algorithm takes O(n) time.

## Linear search in Java

Is there anything special about linear search in Java? A few things. Let’s start with the “obvious” implementation.

/**
* Search values for an object equal to value.
*
* @param value
*   A value to search for
* @param values
*   An array of values to search
* @return index
*   The index of a value equal to value
* @pre
*   Elements of values are all comparable to value using equals.
* @post
*   value.equals(values[index])
* @throws Exception
*   If the value is incomparable to an element of values.
* @throws NotFound
*   If no equal value appears.
*/
public static <T> int search(T value, T values[]) throws NotFound, Exception {
for (int i = 0; i < values.length; i++) {
if (value.equals(values[i])) {
return i;
} // if
} // for
throw new NotFound(value + " does not appear");
} // search


Experienced functional and object-oriented programmers might note that an important element seems to be implicit in this procedure. That is, we use the equals method of the value to compare values. However, it turns out that when we’re searching an array, we may want to search using different criteria. For example, if we’re searching for a person, we might want to match by last name, id number, or even some physical characteristics. Hence, rather than just relying on the equals method, we really should be passing in an object that knows how to compare things for equality. (A functional programmer would just pass in a function, but we need objects in Java, even if they are objects that represent functions.) More broadly, we should probably use a predicate.

With predicates, our linear search might look something like the following. (We’ve ellided most of the documentation, since it’s not much different.)

/**
* Search values for a value for which pred holds.
*/
public static <T> int search(Predicate<? super T> pred, T[] values)  throws NotFound, Exception {
for (int i = 0; i < values.length; i++) {
if (pred.test(values[i])) {
return i;
} // if
} // for
throw new NotFound(value + " does not appear");
} // search


What’s the ? super T? That’s a “type wildcard”. We want a predicate that works for values of type T. But we’d also accept any predicate that works for values that are a superclass or interface of T.

Should we really be searching arrays? There are certainly times that it is useful to search arrays. However, if we want to write general code, we should probably search any iterable collection. We will consider how to do that in the lab.

Some folks worry that linear search is biased toward elements that appear at the start of the array and hence prefer to search arrays “randomly”. Here’s a simple version of a random search algorithm.

Repeat
i = a random number in the range 0 .. n-1
If values[i] is equal to val
Return i;
Until we've examined all of the values in the array
If the value was not found, signal an error


Of course, this implementation has two significant problems. First, it can be hard to tell if we’ve examined all of the values in the array. (I’m sure that you can figure out some strategies.) Second, we will often look at some values more than once, which is inefficient. What’s the solution? Instead of randomly generating an index, we start with the list of all possible indices and randomly permute it. We can then just look at the elements in that order. We may return to this approach in the future.

The binary search algorithm is a standard divide-and-conquer algorithm. It requires that the input be sorted. At the highest level, binary search does the following

While elements remain
Look at the middle element
If it matches our target, return the index of the middle element
If it is smaller than our target, throw away the bottom half
If it is larger than our target, throw away the top half
If no elements remain, report failure


But how do we “throw away” elements in an array? Traditionally, we keep track of the lower and upper bounds of the portion of the array of interest. So, initially, the lower bound is 0 and the upper bound is length - 1.

While lb <= ub
mid = midpoint(lb, ub);
If values[mid] equals value
return mid;
Else if values[mid] < value
lb = mid + 1;
Else
ub = mid - 1;
Report Failure


We can also express this algorithm recursively.

binarySearch(value, values, lb, ub)
If (lb > ub)
Report Failure
mid = midpoint(lb, ub);
If values[mid] equals value
return mid;
Else if values[mid] *lt; value
return binarySearch(value, values, mid+1, ub);
Else
return binarySearch(value, values, lb, mid-1);


Why prefer one over the other? I find the recursive version easier to analyze. However, unless your compiler optimizes tail calls, the iterative version is likely to be slightly faster.

As noted above, the recursive version is probably a little easier to analyze.

What’s the running time for input of size n? Well, we do a constant number of tests (no more than three, I believe) and we also have to compute the midpoint of two values. And then we probably have to recurse on about half of the input (maybe exactly half, maybe slightly less.) That gives the recurrence relation

• time(n) = c + time(n/2)
• time(1) = b

How can we solve this relation? Let’s try a few values.

• time(1) = b
• time(2) = c + time(1) = c + b
• time(4) = c + time(2) = c + c + b = 2c + b
• time(8) = c + time(4) = c + 2c + b = 3c + b
• time(16) = c + time(8) = c + 3c + b = 4c + b

Do you see a pattern yet? It looks like time(2k) = k*c + b. (You can prove this via induction, if you’d like.) Since k grows as the log of 2k, it looks like binary search has running in O(log2n).

But wait! We’ve only analyzed powers of 2. Does that matter? Probably not. The upper bound on the running time is clearly increasingly monotonically. So, the number of steps, for any n is bounded by the number of steps for a power of two greater than n. (You can finish this proof, too.)

## Binary search in Java

In implementing binary search, one important question is how we compare individual elements. In linear search, we had to compare only for equality, which allowed us to use the built-in equals method (or a few alternates). In binary search, we need to know if the value we’re looking at should appear before or after a non-equal value. (We also still need to know if it’s equal.) Fortunately, Java provides an interface for just that purpose, java.util.Comparator. You should read the documentation for that class. However, here’s a quick snippet of what it might look like.

public interface Comparator<T> {
/**
* Compare o1 and and o2 for order.  Return a negative number of o1
* comes before o2, zero if the two are equal, and a postive number
* if o1 comes after o2.
*/
public int compare(T o1, To2);
} // Comparator<T>


We’ll leave the implementation of the generalized binary search as an exercise for you.

## Citations

This reading is based on a similar reading from the fall 2014 section of CSC 207.

Portions of that reading were based on materials prepared for Grinnell College’s CSC 151 by Janet Davis, Samuel A. Rebelsky, John David Stone, Henry Walker, and Jerod Weinman.