Spice Routes: Editorial¶

Solutions and tradeoffs¶

Problem setter’s solution: path classification for at most two high-price vertices. Path classification exploits the limit of at most two high-price vertices and is used in the reference. Three-state tree DP also handles more high-price vertices, making it a peer method with wider scope. Both use \(O(n+U)\) time and space, where \(U\) is the largest input denominator.

Programs and credits: std.cpp implements the problem setter’s solution; Codex (GPT-6) contributed implementation details, boundary handling and proof exposition. For the three-state tree DP method used by Codex (GPT-6), see alternative.cpp and the peer method editorial.

Complement event¶

A trade route exists exactly when a high-price vertex connects to a low-price vertex. Compute the probability that every high-price vertex fails to connect to any low-price vertex, then subtract it from one.

With no high-price vertex, return zero. With one, the main path below is its singleton path. With two, let their unique connecting path be \(x_0,\ldots,x_t\) .

Compressing attached branches¶

For a branch vertex \(u\) , let \(f_u\) be the probability that it cannot reach a low-price vertex through its branch. If \(u\) is low-price, set \(f_u=0\) ; otherwise

\[ f_u=\prod_{v\text{ a branch child}}\bigl(q_{uv}+(1-q_{uv})f_v\bigr), \]

where \(q_{uv}\) is the edge-failure probability. The edge is either closed, or open with the child still failing. These cases are disjoint, and different branches are independent.

Compute the same product at each main-path vertex, excluding main-path children. Its resulting \(f_i\) describes failure through itself and its attached branches. The edge sets underlying different \(f_i\) are disjoint.

First and last closed edges¶

Let \(q_i\) be the failure probability of \((x_i,x_{i+1})\) . Define

\[ A_i=\prod_{k=0}^{i}f_k\prod_{k=0}^{i-1}(1-q_k),\qquad B_i=\prod_{k=i}^{t}f_k\prod_{k=i}^{t-1}(1-q_k). \]

If the whole path is open, failure has probability \(B_0\) . Otherwise let \(i\) be the first closed edge from the left:

If it is the only closed edge, the contribution is \(A_iq_iB_{i+1}\) .
If the last closed edge is \(j>i\) , the contribution is \(A_iq_iq_jB_{j+1}\) . Everything between these cuts is unrestricted and sums to probability one: that portion contains no high-price endpoint.

The total failure probability is therefore

\[ B_0+\sum_{i=0}^{t-1}A_iq_i\left(B_{i+1}+\sum_{j=i+1}^{t-1}q_jB_{j+1}\right). \]

The unique first and last cuts make these cases disjoint and exhaustive. Precompute \(B_i\) and the suffix sums, then update \(A_i\) incrementally. A low-price vertex on the main path has \(f_i=0\) , automatically eliminating invalid events. Overlapping high- and low-price vertices work the same way.

Modular arithmetic and complexity¶

For \(P=10^9+7\) , all input denominators are smaller than \(P\) . Precompute inverses through \(U\) and multiply each numerator by its denominator's inverse. Never divide by \(f_i\) or an open-edge probability, so zero probabilities and certain failures/openings are valid.

Explicit traversals build the path and processing order. Branch products and prefix/suffix calculations are linear, giving \(O(n+U)\) time and space.

Validation and performance¶

Timing and peak-memory measurements are available in environment and measurements.