The Analysis Application
The examples in this document are all part of the Java Application Analysis. Download the source archive Analysis.zip and install it as a Java Application with Existing Sources. See the Using NetBeans document for details. Like others, the Exceptions project has multiple Main Classes intended to illustrate various independent features. The simplest way to run the various classes is to rightclick the file and select Run File either in the file itself or from the listing in the Projects window.Linear Search
For example, consider simple linear search of an integer array for a given integer key. The following class representing a simple random linear search might be something like this:Coding
In order to create a general version which could be reused, we should consult what Java does in the Arrays class: There is no linear search algorithm, but there is a binary search. Here are two relevant static functions:int binarySearch(int[] a, int key) int binarySearch(int[] a, int from, int to, int key)The "toIndex" argument name is misleading since the range is really between fromIndex and toIndex 1, inclusive.
...  from  from+1  ...  to1  to  ... 
java.util.Arrays.binarySearch(A, key); java.util.Arrays.binarySearch(A, 0, A.length, key);The int return value is supposed to be the position at which the key is found. If not found, a negative value is returned. These algorithm specifications suggest a more general way to write linear search, namely using our own package, util, and our own class MyArrays with two versions:
util.MyArrays.linearSearch(int[] a, int key) util.MyArrays.linearSearch(int[] a, int from, int to, int key)We should also consider what to do if the latter of these calls is provided with invalid arguments such as these:
linearSearch(A, 3, 2, 15); linearSearch(A, 3, 2, 15);In these cases, we want to throw an Exception. A few tests of Arrays.binarySearch leads us to add this starter code:
if (from < 0  from > to) { // implies that to >= 0 throw new IllegalArgumentException(); }
util.MyArrays Show
Here, we
apply this search algorithm in the following main class:
util.MyArrays
package util; public class MyArrays { public static int linearSearch(int[] A, int from, int to, int key) { if (from < 0  from > to) { throw new IllegalArgumentException(); } int i; for (i = from; i < to; ++i) { if (key == A[i]) { return i; } } return 1; } public static int linearSearch(int[] A, int key) { return linearSearch(A, 0, A.length, key); } }
Other Java types
Consider expanding the linear search algorithm to other types such as double or String. These two types are very different. The type double, like int is a primitive type and String is an Object. With respect to double, or any of the other primitive types, we have to write separate functions for each even though they effectively do the same thing. Regarding Object types, such as String, we would want a version like this:
using objects Show
The problem with creating an example is how to "meaningfully"
generate an array of Strings and a search key.
A simple example (not terribly meaningful) is this:
package util; public class MyArrays { // ... public static int linearSearch(Object[] A, int from, int to, Object key) { if (from < 0  from > to) { throw new IllegalArgumentException(); } int i; for (i = from; i < to; ++i) { if (key.equals(A[i])) { return i; } } return 1; } public static int linearSearch(Object[] A, Object key) { return linearSearch(A, 0, A.length, key); } }
null values
We have implicitly assumed that the search key is not null; otherwise, this would cause an exception:key.equals(A[i])We could reverse it as:
A[i].equals(key)but then the implicit assumption is that the array has no null elements. A complete solution might be to treat key == null as a special case:
int i; if (key == null) { for (i = from; i < to; ++i) { if (A[i] == null) { return i; } } } else { for (i = from; i < to; ++i) { if (key.equals(A[i])) { return i; } } } return 1;
Abstract time for linear search
Regarding the notion of "abstract time" we usually say to "count the number of comparisons" to get a measure of the actual time. There are several deviations from reality: we're overlooking any setup, like initializing the variables found and i, as well as other repetitive features, like incrementing i comparing it to the size

We're ignoring the actual cost of a comparison.
The Object comparison
key.equals(A[i])
say, for String type, is not constant and is likely to be more timeconsuming than comparison of primitive types.
Analysis of Linear Search
The worst case is if the element is not found or is found at the end of the search in which case there are n comparisons. The best case is that the thing you're looking for is in the first slot and so there's only one comparison. We say: the best case is 1 comparison
 the worst case is n comparisons
Average case for linear search
A simple guess might be to take the average of the best and worst, getting (n+1)/2. This answer turns out to be correct, but we must derive it more methodically. First of all, we need to make the assumption that the search is successful. Why? Consider an array of size 10, holding arbitrary integers and the key is an arbitrary integer. Given that there are about 4 billion 32bit integers, the probability that our key is one of these 10 is effectively zero! In addition, we will assume that the key is equally likely to be found at each of the n positions. Using this notion, the average cost is:
( total cost for finds at all positions ) / number of positions
The cost for finding at position i is (i+1), for values
i = 0 ... n1. Therefore we derive:
1 + 2 + ⋅⋅⋅ + n  n*(n+1)/2  n + 1  
average cost =  
=  
=  
n  n  2 
Language of Asymptotics
Asymptotics has to do with describing the behavior of functions, specifically algorithmic timing functions, for arbitrarily large problem sizes. Often a problem size can be characterized by a single parameter n, e.g., the size of an array.BigO Notation
We would really like to present information about this algorithm which ignores the constants. The "big O notation" and other complexity terminology allows us do precisely that. We say:The bigO notation is simply a convenience for expressing the relation between an unknown "timing function", T(n), and a known reference function, f(n). In reality O(f(n)) is a class of functions, of which T(n) is a member. Nevertheless, it is convenient to be able to make statements like thisT(n) = O(f(n)) or T(n) is O(f(n)) if there are positive constants C, k, such that T(n) ≤ C * f(n) for all n ≥ k
"the worst case time for suchandsuchalgorithm is O(n*log(n))"
and have a rigorous basis for what you're saying.
The idea of "n ≥ k" in the definition means
eventually, i.e., if we ignore some initial finite portion.
For example, suppose
T(n) = n + 100We can say:
T(n) ≤ 101 * n, for all n ≥ 1but we can also say
T(n) ≤ 2 * n, for all n ≥ 100getting a "better" asymptotic constant, C, in the sense that it is smaller (we can always make it larger). In either case we have proved that
n + 100 = O(n)by finding constants C and k which make the definition statement work.
Linear Search in bigO terms
Going back to linear search we observed that when counting comparisons:the best case is 1In bigO terminology, we would say this about linear search:
the worst case is n
the average case is (n+1)/2 = ½ n + ½
the best case time is O(1)
the worst case time is O(n)
the average case time is O(n)
Binary Search
Binary search searches a sorted array in the most efficient way possible. This algorithm employs a simple example of a divideandconquer strategy in which we subdivide the problem into equalsized "subproblems". The idea is simple: compare the key to the "middle" element, if not equal, either look left or look right portion based on whether the key is less than, or greater than, the middle element.First Binary Search Implementation
This algorithm expresses itself most naturally in a recursive manner based on the way I say:
search the whole array invokes search of one half or another.
Here is a diagram:
In diagrams, searching for key in array, within the given range:
from  ...  mid  ...  to 
Check array[mid] where mid = (from+to)/2:
from  mid  
key? 
If key < array[mid], search left half (excluding mid):
from  ...  mid 
If key > array[mid], search right half (excluding mid):
from  mid  mid+1  ...  to  
binary search code Show
The else keywords are optional because of the
return statements.
One has to keep in mind that the division used to compute mid
is integer division (i.e., fractional truncation).
As is the case with linearSearch,
the second range parameter, toIndex, is not included in the search
range. Our textbook makes, in my opinion, the unfortunate
decision to make the second parameter position in his arraybased
algorithms the rightmost index, so beware.
int binarySearch(int[] A, int from, int to, int key) { if (from == to) { return  1; } int mid = (from + to) / 2; if (key == A[mid]) { return mid; } // else if (key < A[mid]) { return binarySearch(A, from, mid, key); } // else (key > A[mid]) return binarySearch(A, mid+1, to, key); }
Binary Search Visitation tree
Lay out the array positions that the algorithm would visit in a binary tree, where the root is the middle of the array, the left and right subtrees are generated by searches of the left and right subarrays, respectively. Here are the binary search trees for arrays of size 7 and 10, resp.:Algorithmic correctness by induction
The proof uses a form of induction called strong induction, whereby we assume true the thing we want to prove for all values (within a suitable range) up to that point. Consider the execution of: The execution ofint pos = binarySearch(A, from, to, key)We want to say that:
 if pos ≥ 0 then from ≤ pos < to and A[pos] == key
 if pos < 0 then A[i] != key, for all from ≤ i < to.
a/b = floor(a/b)where floor which truncates any decimal part. The base case is when len = 0. This means that to == from and the array is empty. The key cannot be present in the array and the algorithm indicates failure. Inductively, suppose len ≥ 1 and the argument has been proved for values less than len. Because the array is sorted, it is obvious that the algorithm will work so long as:
 from ≤ mid < to
 the left (from,mid) and right (mid+1,to) ranges both have fewer than len elements.
to = from + lenand compute:
mid = (from + to)/2 = (2*from + len)/2 = from + len/2, if len is even = from + (len1)/2, if len is oddSummarizing, we see that
 when len is even, the split is unequal, the left side having len/2 elements and the right side one less.
 when len is odd, the split is equal, both sides having (len1)/2 elements
 both "halves" have length less than the full array range
Logarithms in analysis
Logarithms, particularly base2 logarithms, are important because they represent the number of times n things can be halved. In particular, there are roughly log_{2}n term in this sequence:n, n/2, n/4, ..., 1The definition of logarithm to the base b is:
x = log_{b}(n) means b^{x} = n, i.e., b^{logb(n)} = nThe most common bases are these:
 base 2, used in computer science
 base 10, used in other sciences
 base e = 2.718..., used in mathematics, where e is the base of the natural logarithm; this socalled Euler's number is one of the fundamental constants of mathematics, like π
b^{logb(2)} = 2 (b^{logb(2)})^{x} = b^{logb(2)*x} = 2^{x}Using x = log_{2}(n),
b^{logb(2) * log2(n)} = 2^{log2(n)} = nThis means, according to what the definition of log:
log_{b}(n) = log_{b}(2) * log_{2}(n) (think of the 2's as "canceling out"), or simplifying,
log_{2}(n) = log_{b}(n) / log_{b}(2)
Often the basee logarithm is written as ln(n) to clarify the base.
In computer language library functions, log(n) often means
the natural logarithm and base2 logarithm is usually written with
an explicit base. For example, in Java, the function Math.log
is the natural logarithm, but we can easily write the base2 logarithm:
static double log_2(double x) { return Math.log(x) / Math.log(2); }
log(n) = log_{2}(n)
Integer logarithms
In computational settings in which logarithms appear, they always appear in these integer formats:flr(log(n)) = the largest power of 2 so that 2^{power} ≤ n ceil(log(n)) = the smallest power of 2 so that n ≤ 2^{power}For example,
flr(log(8)) = 3, ceil(log(8)) = 3
flr(log 10) = 3, ceil(log 10) = 4
flr(log 10) = 3, ceil(log 10) = 4
Binary Search Worst Case
Let T(n) = worst case number of comparisons in binary search of an array of size nFrom an analysis perspective, when binary search is used with an array of size n > 1, the sizes of the two halves are:
n even => left side n/2, right side n/2  1 n odd => both sides have size (n1)/2 = n/2In the worst case, we would end up consistently going to the left side with n/2 elements. Couting 1 for the middle element comparison we get this recurrence equation for the worstcase number of comparisons:
T(1) = 1 T(n) = 1 + T( n/2 ), n > 1Our claim is that T(n) = O(log(n)). In fact we want to prove:
T(n) ≤ 2 * log(n), n ≥ 2Look at a comparison of some computed values of T(n) and log(n)
T(2) = 2, log(2) = 1 T(3) = 2, log(3) = 1.x T(4) = 3, log(4) = 2 T(5) = 3, log(5) = 2.xObserve that the multiplier 2 satisfies the inequality we claim for each value.
Proof by induction.
We use socalled strong induction. We have verified that for base cases n = 2, 3, 4, 5 that:T(n) ≤ 2 * log(n), n ≥ 2Assume valid up to (but not including) n. This means that we can make the inductive assumption and assume this to be true:
T(n/2) ≤ 2 * log(n/2), n/2 < nThen the proof goes like this:
T(n) = 1 + T(n/2) (the recurrence) ≤ 1 + 2 * log( n/2 ) (substitute from inductive assumption) = 1 + 2 * (log(n) – 1) (properties of log) = 2 * log(n) – 1 (simple algebra) ≤ 2 * log(n) (becoming larger)The key algebraic step relies on the property:
log(a/b) = log(a) – log(b)which we are using like this:
log(n/2) = log(n) – log(2) = log(n) – 1However, because n/2 is truncated division, this last statement is not technically correct when n is odd. For simplicity, we'll ignore this technicality.
Demo program
It's useful to see a programmatic comparison of T(n) and log(n) as can be done with the following program:TimeBS
package search; public class TimeBS { static int T(int n) { if (n == 1 ) { return 1; } return 1 + T( n/2 ); } // base2 logarithm static double log(double x) { return Math.log(x) / Math.log(2); } public static void main(String[] args) { for (int n = 2; n <= 32; ++n) { int t_val = T(n); double log_val = log(n); System.out.printf("T(%s)/log(%s)\t%.3f\n", n, n, t_val/log_val); } } }
T(2)/log(2) 2.000 T(3)/log(3) 1.262 T(4)/log(4) 1.500 T(5)/log(5) 1.292 T(6)/log(6) 1.161 T(7)/log(7) 1.069 T(8)/log(8) 1.333 T(9)/log(9) 1.262 T(10)/log(10) 1.204 T(11)/log(11) 1.156 T(12)/log(12) 1.116 T(13)/log(13) 1.081 T(14)/log(14) 1.051 T(15)/log(15) 1.024 T(16)/log(16) 1.250
T(17)/log(17) 1.223 T(18)/log(18) 1.199 T(19)/log(19) 1.177 T(20)/log(20) 1.157 T(21)/log(21) 1.138 T(22)/log(22) 1.121 T(23)/log(23) 1.105 T(24)/log(24) 1.091 T(25)/log(25) 1.077 T(26)/log(26) 1.064 T(27)/log(27) 1.052 T(28)/log(28) 1.040 T(29)/log(29) 1.029 T(30)/log(30) 1.019 T(31)/log(31) 1.009 T(32)/log(32) 1.200
Improvement on the worstcase order constant
In the above sections we proved thatT(n) ≤ 2 * log(n), n ≥ 2Can we do better? It is not true that T(n) ≤ log(n) per se, but the proof above can easily be adpated to prove this inequality:
T(n) ≤ log(n) + 1, n ≥ 2Thus, including the extra "1" term the constant multiplier is effectively 1, which is an improvement over the multiplier 2 indicated above.
Order class hierarchy
The functions which characterize algorithm timing tend to fall into a few common ones:O(1) — constant time O(log n) — logarithmic time O(n) — linear time O(n * log n) O(n^{2}) — quadratic time O(n^{2} * log n) O(b^{n}) — each base b generates a distinct order classThese order classes are upwardly inclusive, i.e., if T(n) = O(log(n), then of course, T(n) = O(n). We're usually interested in the "best fit" in the sense of finding the smallest order class to which T(n) belongs. In order to characterize the "best fit" of an order class, we need two other notions:
Lower bound: Ω
We say:T(n) = Ω(f(n)) (or T(n) is Ω(f(n)))if there are positive constants C and k, such that
T(n) ≥ C * f(n), for all n ≥ k
Exact bound: Θ
We say:T(n) = Θ(f(n)) (or T(n) is Θ(f(n)))if
T(n) = O(f(n)) and T(n) = Ω(f(n))
This means that there are positive constants
C_{1}, C_{2} and k, such that
C_{1} * f(n) ≤ T(n) ≤ C_{2} * f(n), for all n ≥ kThe Θ concept gives the precise sense to the notion of "order class" because it completely characterizes the behavior of a timing function relative to a reference function up to a constant multiple.
Order Summary
Officially there are three considerations. O: upper bound, meaning "we can do at least this well" up to a constant factor
 Ω: lower bound, ignoring constant factors, meaning, "we cannot expect to do better than this" up to a constant factor
 Θ: characterization of the runtime behavior, up to a constant factor
Other algorithmic terminology
Asymptotic dominance of one function by another is expressed by the littleo notation:T(n) = o(f(n))This means that for every c (no matter how small), there is a k such that
T(n) ≤ c * f(n), n ≥ kFor the most part this means the following:
lim_{n ⇾ ∞} T(n)/f(n) = 0Asymptotic dominance expresses the relationship of the reference functions in the order class hierarchy above:
1 = o(log(n)) log(n) = o(n) n = o(n^{2}) n^{2} = o(2^{n}) 2^{n} = o(3^{n}) ...The first of these relations, log(n) = o(n), is proved using L'Hôpital's rule from calculus, substituting a continous variable x for the integer n:
lim_{x ⇾ ∞} log_{2}(x)/x = lim_{x ⇾ ∞} log_{2}(e) * ln(x)/x = lim_{x ⇾ ∞} log_{2}(e) * ln′(x) / x′ = lim_{x ⇾ ∞} log_{2}(e) * 1/x / 1 = lim_{x ⇾ ∞} log_{2}(e) / x = 0Asymptotic equality is written this way:
T(n) ≈ f(n)and it means: lim_{n ⇾ ∞} T(n)/f(n) = 1 The "wavy" equal lines suggest that these two functions are essentially the same for large values. For example, in a polynomial function, we can effectively ignore all but the highest order term. For example, if
T(n) = 100 * n + 200 * n^{2} + 3 * n^{3}then
T(n) ≈ 3 * n^{3}Unfortunately the Weiss textbook does not define this relation. Asymptotic equality is, in some sense, similar to the exact bound Θ, except that it gives a precise order constant, which is often of interest when you want to compare two timing functions within the same order class. For example, let
W(n) = worst case time for linear search
A(n) = average case time for linear search
Both functions are exactly linear time and we would write:
A(n) = average case time for linear search
W(n) = Θ(n) A(n) = Θ(n)However, the order constants are different and this is expressed using ≈:
W(n) ≈ n A(n) ≈ ½ n
Binary Search average case
The average case timing is more complicated. As in the case of linear search, we assume a successful search, and that each of the array positions are equally likely to hold the search key. We are mostly interested in getting some sense about how much better the average case might be, and proving that it is still logarithmic. Technically we want to prove the lower bound:Average binary search time(n) = Ω(log(n))
Combined with the fact that the average time can only be better
than the worstcase time, which is
O(log(n))
we can then conclude that
Average binary search time(n) = Θ(log(n))
Additionally we want to get some idea about what the order constant might be.
Counting the total number of comparisons
In order to compute the average number of comparisons, we need to find a way to compute the total number of comparison for all possible nodes in the positional visitation tree. The level of a node is its distance from the root. The root, at level 0, counts for 1 comparison. Both of its children count for 2 comparisons each, etc. Thus,Total comparisons = ∑_{(all nodes at level i)} (i+1)In general, at level i, if it is full, there will be 2^{i} children, each contributing (i+1) comparisons. A binary tree is perfect if every level is full. In general, the binary search position visitation tree will not be perfect, but it can be argued that
 The levels 0 to flr(log n)1 are all full: thus a total of L = flr(log n) levels are full.
 The maximum level of a node is flr(log n) (which may not be full)
flr(log(7)) = 2  flr(log(10)) = 3 
Average number of comparisons for perfect trees
We rely on a known formula, which gives an expression when counting comparisons in a perfect tree with L levels:Comparisons(L) = 1*2^{0} + 2*2^{1} + 3*2^{2} + ... + L*2^{L1} = ( L – 1 ) * 2^{L} + 1A perfect visitation tree occurs for the values n = 1, 3, 7, 15, ... which are powers of 2 minus 1. In these cases, the visitation tree has exactly L = log(n+1) levels. Using the above formula, we compute the total number of comparisons as:
= ( log(n+1)  1 ) * 2^{log(n+1)} + 1 = ( log(n+1)  1 ) * (n + 1) + 1 = n * log(n+1)  n + log(n+1)  1 + 1 = n * log(n+1)  n + log(n+1)Compute the averge by dividing by n:
average = log(n+1)  1 + log(n+1)/n
Averge comparisons lower bound
In general, the visitation tree is not perfect. To get a lower bound, simply omit all nodes on the last level. So we compute comparisons for a total of L = flr(log(n)) levels. Using the same formula, we gettotal comparisons ≥ Comparisons( flr(log(n)) ) = ( flr(log(n))  1 ) * 2^{flr(log n)} + 1The flr(log(n)) expression truncates the decimal part of log(n) and so it will subtract away less than 1 from log(n), i.e.,
flr(log(n)) ≥ log(n)1Replacing flr(log n) by log(n)1 and doing the algebra, we get
total comparisons ≥ ( log(n)  2 ) * 2^{log(n)  1} + 1 = log(n) * 2^{log(n)  1}  2^{log(n)} + 1 = ½ * n * log(n)  n + 1Dividing this total by n gives the average, i.e.,
average comparisons ≥ ½ * log(n)  1 + 1/n > ½ * log(n)  1From this we can say:
The average number of comparisons is Ω(log(n)).
We also know that average number must be less that the worst, which we know is
worst case comparisons ≤ log(n) + 1Therefore, the average number of comparisons for any n is somewhere between:
½*log(n)  1 and 1*log(n) + 1
Binary Search Implementations
First of all, create a simple test program to see what Java does. Create the main class JavaBinarySearch with this content:util.MyArrays
public static int binarySearch(int[] A, int from, int to, int key) { if (from == to) { return  from  1; } int mid = (from + to) / 2; if (key == A[mid]) { return mid; } if (key < A[mid]) { return binarySearch(A, from, mid, key); } // else key > A[mid] return binarySearch(A, mid+1, to, key); } public static int binarySearch(int[] A, int key) { return binarySearch(A, 0, A.length, key); }
Failed search value
Of what use is the negative return value? We have modified the return value in case of failure. Before we wrote simplyreturn 1;
return  fromIndex  1;
int insert_pos =  ret  1;We can then correctly add key to the array, A (maintaining sortedness) if we
 shift the array contents in positions insert_pos to toIndex one to the right,
 set A[insert_pos] = key
0  10  20  30  40 
0  1  2  3  4 
(0,5), (3,5), (3,4), (3,3)The return value would be 4 which would be correctly interpreted as an insert position of 3. It's also important to ensure that the "extreme" cases work out correctly:
 A failed search for key = 10 will maintain fromIndex = 0 on all steps. At the end, the return value will be 1, signifying that the correct insert position is 0.
 A failed search for key = 50 will search this sequence of ranges: (0,5), (3,5), (5,5). At the end, the return value will be 6, signifying that the correct insert position is 5 (off the right end).
Error Checking
We've left out error checking code from binarySearch:public static int binarySearch(int[] A, int from, int to, int key) { if (from < 0  from > to) { throw new IllegalArgumentException(); } //... }
public static int binarySearch(int[] A, int from, int to, int key) { if (from < 0  from > to) { throw new IllegalArgumentException(); } return _binarySearch(A, from, to, key); } private static int _binarySearch(int[] A, int from, int to, int key) { if (from == to) { return  from  1; } int mid = (from + to) / 2; if (key == A[mid]) { return mid; } if (key < A[mid]) { return _binarySearch(A, from, mid, key); } // else key > A[mid] return _binarySearch(A, mid+1, to, key); }
Comparable objects
The situation is more complicated than that of linear search for object types because binary search is based on the ability to make comparisons between objects obj1 and obj2 in order to establish: if obj1 is less than obj2 (look left)
 if obj1 equals obj2 (found)
 if obj1 is greater than obj2 (look right)
class MyClass implements Comparable<MyClass>This generic interface requires the definition of the one public member function:
int compareTo(MyClass other)This is the case for String, Integer, Double, etc., allowing the member function call between two MyClass objects:
 obj1.compareTo(obj2) < 0 means obj1 is less than obj2
 obj1.compareTo(obj2) = 0 means obj1 equals obj2
 obj1.compareTo(obj2) > 0 means obj1 is greater than obj2
 Require the classes to be comparable at compile time
as suggested by the Weiss textbook:
public static <T extends Comparable<? super T>> int binarySearch( T[] A, int from, int to, T key) { if (from == to) { return  from  1; } int mid = (from + to) / 2; int comp = key.compareTo(A[mid]); if (comp == 0) { return mid; } if (comp < 0) { return binarySearch(A, from, mid, key); } // else comp > 0 return binarySearch(A, mid+1, to, key); }

Require the classes to be comparable at run time.
This is what is done in the Java binarySearch:
public static int binarySearch(Object[] A, int from, int to, Object key) { if (fromIndex == toIndex) { return  from  1; } int mid = (from + to) / 2; int comp = ((Comparable) key).compareTo(A[mid]); if (comp == 0) { return mid; } if (comp < 0) { return binarySearch(A, from, mid, key); } // else comp > 0 return binarySearch(A, mid+1, to, key); }
<T extends Comparable<? super T>>
Using Comparators
In many cases we want to compare objects of a class which is not innately comparable by dictating the manner in which the objects are to be compared. This achieved by passing an external comparator object, i.e., one which implements this generic interface:interface Comparator<MyClass>
int compare(MyClass first, MyClass second)
public static <T> int binarySearch( T[] A, int from, int to, T key, Comparator<? super T> c) { if (fromIndex == to) { return from  1; } int mid = (from + to) / 2; int comp = c.compare(key, A[mid]); if (comp == 0) { return mid; } if (comp < 0) { return binarySearch(A, from, mid, key, c); } // else comp > 0 return binarySearch(A, mid + 1, to, key, c); }
Comparator<? super T> c
int comp = key.compareTo(A[mid]); int comp = ((Comparable) key).compareTo(A[mid]); int comp = c.compare(key, A[mid]);
Binary vs. Linear search
The problem with binary search is that, although the search time is much faster, the array must be sorted for it work. The best algorithms for sorting a random array have a run time of O(n * log n). So there is no advantage of binary search over linear search if every search is on a fresh array. Here is how we compare the two algorithms:Linear Search  Binary Search  



create array:  O(n)  O(n) 
prepare array:    O(n*log(n)) 
search array:  O(n)  O(log(n)) 
Integer exponentiation
We want to write an algorithm to compute x^{n} for an integer n ≥ 0. There is an obvious iterative way to do so:private static double powIter(double x, int n) { if (n == 0) { return 1; } double ans = x; for (int i = 1; i < n; ++i) { ans *= x; } return ans; }
powIter(x, 65);This algorithm would require 64 multiplications. We can do a lot less. In fact, we can compute the answer with only 7 multiplications:
ans = x; ans = ans * ans; // x^2 ans = ans * ans; // x^4 ans = ans * ans; // x^8 ans = ans * ans; // x^16 ans = ans * ans; // x^32 ans = ans * ans; // x^64 ans = ans * x; // x^65The basis of the more efficient algorithm we want is this:
n = 2 * n/2, if n even n = 2 * n/2 + 1, if n oddRecall that in Java, integer division truncates the decimal part, i.e., 5/2 = 2. Using properties of exponents, we can write these "recurrence equations"
x^{0} = 1 x^{1} = x for n > 1: n even: x^{n} = (x^{2})^{n/2} n odd: x^{n} = x * (x^{2})^{n/2}We can use the reduction in exponent to make a recursive function:
public static double powRecur(double x, int n) { if (n == 0) { return 1; } if (n == 1) { return x; } if (n % 2 == 0) { return powRecur(x * x, n/2); // n even } else { return x * powRecur(x * x, n/2); // n odd } }
 1 multiplication to compute x * x,
 T(n/2) for the recursive call to n/2
 1 multiplication afterwards
T(0) = 0 T(1) = 0 T(n) = T(n/2) + 2, n ≥ 2It can be proved by induction that:
T(n) = O( log(n) )The worst case can be seen to be when n in binary is all 1s, meaning that n = 2^{k}1 = 3, 7, 15, 31,...