Given is an array of three numeric values and I'd like to know the middle value of the three.
The question is, what is the fastest way of finding the middle of the three?
My approach is this kind of pattern - as there are three numbers there are six permutations:
if (array[randomIndexA] >= array[randomIndexB] &&
array[randomIndexB] >= array[randomIndexC])
It would be really nice, if someone could help me out finding a more elegant and faster way of doing this.
There's an answer here using min/max and no branches (https://stackoverflow.com/a/14676309/2233603). Actually 4 min/max operations are enough to find the median, there's no need for xor's:
median = max(min(a,b), min(max(a,b),c));
Though, it won't give you the median value's index...
Breakdown of all cases:
a b c
1 2 3 max(min(1,2), min(max(1,2),3)) = max(1, min(2,3)) = max(1, 2) = 2
1 3 2 max(min(1,3), min(max(1,3),2)) = max(1, min(3,2)) = max(1, 2) = 2
2 1 3 max(min(2,1), min(max(2,1),3)) = max(1, min(2,3)) = max(1, 2) = 2
2 3 1 max(min(2,3), min(max(2,3),1)) = max(2, min(3,1)) = max(2, 1) = 2
3 1 2 max(min(3,1), min(max(3,1),2)) = max(1, min(3,2)) = max(1, 2) = 2
3 2 1 max(min(3,2), min(max(3,2),1)) = max(2, min(3,1)) = max(2, 1) = 2
It's possible to answer the query without branches if the hardware can answer min and max queries without branches (most CPUs today can do this).
The operator ^ denotes bitwise xor.
Input: triple (a,b,c)
1. mx=max(max(a,b),c)
2. mn=min(min(a,b),c)
3. md=a^b^c^mx^mn
4. return md
This is correct because:
The appropriate min/max functions should be chosen for int/float. If only positive floats are present then it's possible to use integer min/max directly on the floating point representation (this could be desirable, since integer operations are generally faster).
In the unlikely scenario that the hardware doesn't support min/max, it's possible to do something like this:
max(a,b)=(a+b+|a-b|)/2
min(a,b)=(a+b-|a-b|)/2
However, this isn't correct when using float operations since the exact min/max is required and not something that's close to it. Luckily, float min/max has been supported in hardware for ages (on x86, from Pentium III and onwards).
If you are looking for the most efficient solution, I would imagine that it is something like this:
if (array[randomIndexA] > array[randomIndexB]) {
if (array[randomIndexB] > array[randomIndexC]) {
return "b is the middle value";
} else if (array[randomIndexA] > array[randomIndexC]) {
return "c is the middle value";
} else {
return "a is the middle value";
}
} else {
if (array[randomIndexA] > array[randomIndexC]) {
return "a is the middle value";
} else if (array[randomIndexB] > array[randomIndexC]) {
return "c is the middle value";
} else {
return "b is the middle value";
}
}
This approach requires at least two and at most three comparisons. It deliberately ignores the possibility of two values being equal (as did your question): if this is important, the approach can be extended to check this also.
This can be done with two comparisons at most.
int median(int a, int b, int c) {
if ( (a - b) * (c - a) >= 0 ) // a >= b and a <= c OR a <= b and a >= c
return a;
else if ( (b - a) * (c - b) >= 0 ) // b >= a and b <= c OR b <= a and b >= c
return b;
else
return c;
}
And one more idea. There are three numbers {a,b,c}. Then:
middle = (a + b + c) - min(a,b,c) - max(a,b,c);
Of course, we have to remember about numeric limits...
Here's how you can express this using only conditionals:
int a, b, c = ...
int middle = (a <= b)
? ((b <= c) ? b : ((a < c) ? c : a))
: ((a <= c) ? a : ((b < c) ? c : b));
EDITS:
I didn't see a solution which implements swaps:
int middle(int a, int b, int c) {
// effectively sort the values a, b & c
// putting smallest in a, median in b, largest in c
int t;
if (a > b) {
// swap a & b
t = a;
a = b;
b = t;
}
if (b > c) {
// swap b & c
t = b;
b = c;
c = t;
if (a > b) {
// swap a & b
t = a;
a = b;
b = t;
}
}
// b always contains the median value
return b;
}
If you must find one out of X values satisfying some criteria you have to at least compare that value to each of the X-1 others. For three values this means at least two comparisons. Since this is "find the value that is not the smallest and not the largest" you can get away with only two comparisons.
You should then concentrate on writing the code so you can very clearly see what goes on and keep it simple. Here this means nested if's. This will allow the JVM to optimize this comparison as much as possible at runtime.
See the solution provided by Tim (Fastest way of finding the middle value of a triple?) to see an example of this. The many code line does not necessarily turn out to be larger code than nested questionmark-colon's.
You might as well write this in the most straightforward, way. As you said, there are only six possibilities. No reasonable approach is going to be any faster or slower, so just go for something easy to read.
I'd use min() and max() for conciseness, but three nested if/thens would be just as good, I think.
median = (a+b+c) - Math.min(Math.min(a,b),c) - Math.max(Math.max(a,b),c)
This is the basic one, i don't know how efficient this would work but these functions use if conditions after all. If you would like you can turn this statement into if-else statements, yet it will take time. Why so lazy?
Bumping up an old thread, but still it's the shortest solution, and nobody mentioned it.
int median2(int a, int b, int c) {
return (a > b) ^ (a > c) ? a : (a > b) ^ (b > c) ? c : b;
}
(tests cover all the possible combinations, all of them print 6)
public static void main(String[] args) {
System.out.println(median(3, 6, 9));
System.out.println(median(3, 9, 6));
System.out.println(median(6, 3, 9));
System.out.println(median(6, 9, 3));
System.out.println(median(9, 3, 6));
System.out.println(median(9, 6, 3));
System.out.println(median(6, 6, 3));
System.out.println(median(6, 6, 9));
System.out.println(median(6, 3, 6));
System.out.println(median(6, 9, 6));
System.out.println(median(3, 6, 6));
System.out.println(median(9, 6, 6));
System.out.println(median(6, 6, 6));
}
(a > b) ^ (a > c) false if either c > a > b or c < a < b - return a;
otherwise (a > b) ^ (b > c) false if either a > b > c or a < b < c - return b;
otherwise return c;
Let's assume p = a > b; q = b > c; s = a > c;
Let's build a Karnaugh map.
| 00 01 11 10 (p, q)
---+----------------------
0 | b c * a
1 | * a b c
(s)|
* means that the combination is impossible (like a > b; b > c; a < c)
Notice that the right part is a mirrored left part, and the map can be simplified by introducing t = p ^ q; u = s ^ p
| 0 1 (t)
---+---------
0 | b c
1 | * a
(u)|
So the function may be written as
private static int median(int a, int b, int c) {
boolean t = (a > b) ^ (b > c);
boolean u = (a > b) ^ (a > c);
if (u)
return a;
else if (t)
return c;
else
return b;
}
Inlining variables and replacing ifs with ?: gives the answer
int median2(int a, int b, int c) {
return (a > b) ^ (a > c) ? a : (a > b) ^ (b > c) ? c : b;
}
The solution works fine even if some on the inputs are equal, which may be not evident, but quite logical.
Based on the excellent answer from Gyorgy, you can get the median's index without branches by replacing min/max with conditional moves:
int i = (array[A] >= array[B]) ? A : B;
int j = (array[A] <= array[B]) ? A : B;
int k = (array[i] <= array[C]) ? i : C;
int median_idx = (array[j] >= array[k]) ? j : k;
javac should generate a ConditionalNode for each of these ternary assignments, which translate to cmp/cmov pairs in assembly. Also note the comparisons were chosen such that in case of equality, the first index in alphabetical order is returned.
The easiest way is through sorting. For example consider this code :
import java.util.Arrays;
int[] x = {3,9,2};
Arrays.sort(x); //this will sort the array in ascending order
//so now array x will be x = {2,3,9};
//now our middle value is in the middle of the array.just get the value of index 1
//Which is the middle index of the array.
int middleValue = x[x.length/2]; // 3/2 = will be 1
That's it.It's that much simple.
In this way you don't need to consider the size of the array.So if you have like 47 different values then you can also use this code to find the middle value.
This one will work:
template<typename T> T median3_1_gt_2(const T& t1, const T& t2, const T& t3) {
if (t3>t1) {
return t1;
} else {
return std::max(t2, t3);
}
}
template<typename T> T median3(const T& t1, const T& t2, const T& t3) {
if (t1>t2) {
return median3_1_gt_2(t1, t2, t3);
} else {
return median3_1_gt_2(t2, t1, t3);
}
}
if(array[aIndex] > array[bIndex]) {
if(array[bIndex] > array[cIndex]) return bIndex;
if(array[aIndex] > array[cIndex]) return cIndex;
return aIndex;
} else {
if(array[bIndex] < array[cIndex]) return bIndex;
if(array[aIndex] < array[cIndex]) return cIndex;
return aIndex;
}
largest=(a>b)&&(a>c)?a:(b>c?b:c);
smallest=(a<b)&&(a<c)?a:(b<c?b:c);
median=a+b+c-largest-smallest;
Method 1
int a,b,c,result;
printf("enter three number");
scanf("%d%d%d",&a,&b,&c);
result=a>b?(c>a?a:(b>c?b:c)):(c>b?b:(a>c?a:c));
printf("middle %d",result);
Method 2
int a=10,b=11,c=12;
//Checking for a is middle number or not
if( b>a && a>c || c>a && a>b )
{
printf("a is middle number");
}
//Checking for b is middle number or not
if( a>b && b>c || c>b && b>a )
{
printf("b is middle number");
}
//Checking for c is middle number or not
if( a>c && c>b || b>c && c>a )
{
printf("c is middle number");
}
Method 3
if(a>b)
{
if(b>c)
{
printf("b is middle one");
}
else if(c>a)
{
printf("a is middle one");
}
else
{
printf("c is middle one");
}
}
else
{
if(b<c)
{
printf("b is middle one");
}
else if(c<a)
{
printf("a is middle one");
}
else
{
printf("c is middle one");
}
}
I got appropriate ans of finding the middle value of a triple
// Compute median of three values, no branches
int median3(int V[3])
{
unsigned int A,B,C;
A=(V[0] < V[1]);
B=(V[1] < V[2]);
C=(V[0] < V[2]);
return V[(B^C)<<1 | (A^B^1)];
}
Using idxA to idxC in ary,
int ab = ary[idxA] < ary[idxB] ? idxA : idxB;
int bc = ary[idxB] < ary[idxC] ? idxB : idxC;
int ac = ary[idxA] < ary[idxC] ? idxA : idxC;
int idxMid = ab == bc ? ac : ab == ac ? bc : ab;
indexMiddle points to the middle value.
Explanation: from the 3 minima 2 are the overall minimum and the other value must be the middle. Because we check equality we can compare the indices in the last line instead of having to compare the array values.
You can use array, like this:
private static long median(Integer i1, Integer i2, Integer i3) {
List<Integer> list = Arrays.asList(
i1 == null ? 0 : i1,
i2 == null ? 0 : i2,
i3 == null ? 0 : i3);
Collections.sort(list);
return list.get(1);
}
Here is the answer in Python, but same logic applies to the Java program.
def middleOfThree(a,b,c):
middle = a
if (a < b and b < c) or (c < b and b < a):
middle = b
elif (a < c and c < b) or (b < c and c < a):
middle = c
print 'Middle of a=%d, b=%d, c=%d is %d' % (a,b,c,middle)
middleOfThree(1,2,3)
middleOfThree(1,3,2)
middleOfThree(2,1,3)
middleOfThree(2,3,1)
middleOfThree(3,2,1)
middleOfThree(3,1,2)
100% branch-free version for integers:
int mid(const int a, const int b, const int c) {
const int d0 = b - a;
const int m = (d0 >> 31);
const int min_ab = a + (d0 & m);
const int max_ab = a + (d0 & ~m);
const int d1 = c - max_ab;
const int min_max_ab_c = max_ab + (d1 & (d1 >> 31));
const int d2 = min_ab - min_max_ab_c;
return min_ab - (d2 & (d2 >> 31));
}
Constructed using the branch-free min / max functions:
int min(const int a, const int b) { const int d = b - a; return a + (d & (d >> 31)); }
int max(const int a, const int b) { const int d = a - b; return a - (d & (d >> 31)); }
It may not look pretty but the machine code might prove more efficient on some architectures. Especially those without min / max instructions. But I have not made any benchmarks to confirm it.
or a one liner for finding the index in the array containing the middle value:
int middleIndex = (a[0]<a[1]) ? ((a[0]<a[2) ? a[2] : a[0]) : ((a[1]<a[2) ? a[2] : a[1]);
A lot of these seem to be using pretty complex if statements. I've found a really simple workaround using the Math library.
Math.max(Math.min(array[start], array[mid]), Math.min(array[start], array[mid], array[end]))
Works out quite nicely.
It can be solved in one line by the ternary operator
int middle(int A, int B, int C) {
return (A>B&&A>C)?B>C?B:C:(B>C&&B>A)?A>C?A:C:B;
}
