I understand that a bitonic tour crosses all vertices with one monotonic path traveling from the left most point to the right most point, then monotonically from the left most point to the right most point, with no self-crossing. I have read (Introduction to Algorithms, 3rd Edition by Cormen et al.), that a solution can be obtained by dynamic programming (solving subproblems, and saving their solution for later solutions of larger subproblems). I have read several solutions online, but I still can't understand how to characterize the subproblems... and build the solution in a bottom-up fashion. Pictures would be nice, or an example.


Say, we are given $n$ points $x_1, \ldots, x_n$ in the plane. Let's assume them to be ordered by their x-coordinates. I.e. $x_1$ is leftmost and $x_n$ is rightmost.

Let's think a bit about the minimal bitonic tour on all vertices. We can be sure that the edge $(x_{n-1}, x_n)$ will be contained in such a tour, so it suffices to find the length of a minimal path going from $x_n$ strictly to the left upto $x_1$ - leaving out $x_{n-1}$ - and then from $x_1$ strictly to the right upto $x_{n-1}$ (and this is the problem I will solve by dynamic programming). I'll call such a path a bitonic path from $x_n$ to $x_{n-1}$. We observe the following:

  • Any bitonic path from $x_n$ to $x_{n-1}$ must start with a first edge $(x_n, x_k)$ for some $k<n-1$.

  • Since we must visit all points, and since - from $x_k$ - we can only continue to the left, all of the points $x_{k+1}, \, x_{k+2}, \, \ldots, \, x_{n-1}$ must necessarily be visited on the way from left to right (and in this order). So necessarily, our bitonic path ends with $x_{k+1} \to x_{k+2} \to \dots \to x_{n-1}$.

What have we figured out so far? A bitonic path from $x_n$ to $x_{n-1}$ has the form

$$x_n \to x_k \to\quad ??? \quad \to x_{k+1} \to x_{k+2} \to \dots \to x_{n-1}$$

where $x_k \to ??? \to x_{k+1}$ is itself a bitonic path from $x_k$ to $x_{k+1}$ with $k< n-1$.

If we want to minimize the length of the whole bitonic path from $x_n$ to $x_{n-1}$, we must also minimize the length of the bitonic path from $x_k$ to $x_{k+1}$ (which is the same as the length of a bitonic path from $x_{k+1}$ to $x_k$).

For any $i>1$, let $\ell(i)$ denote the length of a length-minimizing bitonic path from $x_i$ to $x_{i-1}$. The above tells us that

$$\ell(n) = d(x_n, x_k) + \ell(k+1) + \sum_{m = k+1}^{n-2} d(x_m, x_{m+1})$$

for some $k < n-1$. So we must have

$$\ell(n) = \min_{1< i < n} \left[ d(x_n, x_{i-1}) + \ell(i) + \sum_{m = i}^{n-2} d(x_m, x_{m+1})\right]$$

But the exact same reasoning on such a bitonic path applies also for $p < n$. I.e. we have obtained the recursion

$$\ell(p) = \min_{1< i < p} \left[ d(x_p, x_{i-1}) + \ell(i) + \sum_{m = i}^{p-2} d(x_m, x_{m+1})\right]$$

with $\ell(2) = d(x_2, x_1)$. We can use this recursion to successively calculate $\ell(p)$ for $p=2, \ldots, n$, and the length of a bitonic tour on all points $x_1, \ldots, x_n$ then calculates as

$$\ell(n) + d(x_{n-1},x_n)$$

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  • $\begingroup$ What about time complexity analysis? :) $\endgroup$ – Mantas Dec 10 '13 at 23:32

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