Ternary search

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A ternary search algorithm^[1] is a technique in computer science for finding the minimum or maximum of a unimodal 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 ${\displaystyle f(a)>f(b)}$.

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)

def ternary_search(f, left, right, absolute_precision) -> float:
    """Left and right are the current bounds;
    the maximum is between them.
    """
    if abs(right - left) < absolute_precision:
        return (left + right) / 2

    left_third = (2*left + right) / 3
    right_third = (left + 2*right) / 3

    if f(left_third) < f(right_third):
        return ternary_search(f, left_third, right, absolute_precision)
    else:
        return ternary_search(f, left, right_third, absolute_precision)

def ternary_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.
    """
    while abs(right - left) >= absolute_precision:
        left_third = left + (right - left) / 3
        right_third = right - (right - left) / 3

        if f(left_third) < f(right_third):
            left = left_third
        else:
            right = right_third

     # Left and right are the current bounds; the maximum is between them
     return (left + right) / 2

• 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