Ternary search

A ternary search algorithm ^[1] is a technique in computer science for finding the minimum or maximum of a unimodal function.

The function

Assume we are looking for a maximum of ${\displaystyle f(x)}$ and that we know the maximum lies somewhere between ${\displaystyle A}$ and ${\displaystyle B}$. For the algorithm to be applicable, there must be some value ${\displaystyle x}$ such that

• for all ${\displaystyle a,b}$ with ${\displaystyle A\leq a<b\leq x}$, we have ${\displaystyle f(a)<f(b)}$, and
• for all ${\displaystyle a,b}$ with ${\displaystyle x\leq a<b\leq B}$, we have ${\displaystyle f(a)>f(b)}$.

Algorithm

Let ${\displaystyle f(x)}$ be a unimodal function on some interval ${\displaystyle [l;r]}$. Take any two points ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ in this segment: ${\displaystyle l<m_{1}<m_{2}<r}$. Then there are three possibilities:

• if ${\displaystyle f(m_{1})<f(m_{2})}$, then the required maximum can not be located on the left side – ${\displaystyle [l;m_{1}]}$. It means that the maximum further makes sense to look only in the interval ${\displaystyle [m_{1};r]}$
• if ${\displaystyle f(m_{1})>f(m_{2})}$, that the situation is similar to the previous, up to symmetry. Now, the required maximum can not be in the right side – ${\displaystyle [m_{2};r]}$, so go to the segment ${\displaystyle [l;m_{2}]}$
• if ${\displaystyle f(m_{1})=f(m_{2})}$, then the search should be conducted in ${\displaystyle [m_{1};m_{2}]}$, but this case can be attributed to any of the previous two (in order to simplify the code). Sooner or later the length of the segment will be a little less than a predetermined constant, and the process can be stopped.

choice points ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$:

• ${\displaystyle m_{1}=l+(r-l)/3}$
• ${\displaystyle m_{2}=r-(r-l)/3}$

Run time order

${\displaystyle T(n)=T(2n/3)+O(1)=\Theta (\log n)}$ (by the Master Theorem)

Recursive algorithm

defternary_search(f,left,right,absolute_precision)->float:"""Left and right are the current bounds;    the maximum is between them.    """ifabs(right-left)<absolute_precision:return(left+right)/2left_third=(2*left+right)/3right_third=(left+2*right)/3iff(left_third)<f(right_third):returnternary_search(f,left_third,right,absolute_precision)else:returnternary_search(f,left,right_third,absolute_precision)

Iterative algorithm

defternary_search(f,left,right,absolute_precision)->float:"""Find maximum of unimodal function f() within [left, right].    To find the minimum, reverse the if/else statement or reverse the comparison.    """whileabs(right-left)>=absolute_precision:left_third=left+(right-left)/3right_third=right-(right-left)/3iff(left_third)<f(right_third):left=left_thirdelse:right=right_third# Left and right are the current bounds; the maximum is between themreturn(left+right)/2

See also

• Newton's method in optimization (can be used to search for where the derivative is zero)
• Golden-section search (similar to ternary search, useful if evaluating f takes most of the time per iteration)
• Binary search algorithm (can be used to search for where the derivative changes in sign)
• Interpolation search
• Exponential search
• Linear search

References

1. "Ternary Search". cp-algorithms.com. Retrieved 21 August 2023.