Solution: Encrypted Packets and Network Resets¶
Codex (GPT-6): applies incremental nonnegative shortest paths with distances truncated at the specified threshold, including logical-edge accounting, efficient resets, and a complexity proof.
Keep only distances within the threshold¶
Store \(d_v=\min(\operatorname{dist}(1,v),T+1)\) , using \(T+1\) as a common unusable distance. With nonnegative weights, extending a path already longer than \(T\) cannot bring it back within the threshold. Truncation therefore preserves the reset decision.
Compute these distances after reading the initial graph, without resetting it. When inserting an arc \((u,v,x)\) , improve \(d_v\) and push it into a min-heap if \(d_u+x<d_v\) . Repeatedly pop a current distance and relax its outgoing arcs; discard stale heap entries. A bidirectional logical edge adds two arcs but increases the logical-edge count by only one.
Why propagating improvements is sufficient¶
Distances are correct before insertion. Any newly shorter path must use a new arc. Its first use triggers an endpoint relaxation, after which improvements propagate through existing arcs. Every finite stored label corresponds to a real path, so labels cannot underestimate shortest distances.
When propagation ends, every arc satisfies the truncated relaxation inequality. Induction along any path shows that a label is at most that path's length or \(T+1\) . Together with the real-path bound, this proves that labels equal the truncated shortest distances. Hence \(d_n\le T\) is exactly the reset condition.
Zero-weight edges and cycles are safe: only strict improvements enter the heap, so equal-length cycles do not propagate indefinitely. Nonnegative weights give each propagation the usual Dijkstra ordering of valid heap removals.
Decryption and reset accounting¶
Compute \((b+3)(c+37)\) in 64 bits, reduce modulo the encryption modulus, then XOR each field. Never use an encrypted value as a vertex identifier before decoding it.
Maintain a separate current logical-edge count \(e\) , initially \(m\) , increasing it once per packet. On reset, set \(b\gets b+e\) and \(e\gets0\) , then remove all arcs. This separate counter correctly handles initial and bidirectional edges.
A reset visits only adjacency heads actually used in this period and distances that actually became reachable. Restore the source distance to zero and other reached distances to \(T+1\) . Scanning every vertex at each reset could add \(O(nk)\) work when repeated direct edges trigger frequent resets.
Complexity and tradeoffs¶
Let \(A\le2(m+k)\) be the total number of directed arcs appearing in the input. Between consecutive resets, a vertex's integer distance can strictly decrease at most \(T+1\) times from its initial value \(T+1\) . Thus a live arc is scanned from its source at most \(T+1\) times, and every input arc belongs to only one such period.
There are \(O(A(T+1))\) arc scans and heap operations, giving \(O(n+A(T+1)\log(n+A))\) time. Within one propagation, nonnegative Dijkstra scans an arc at most once, producing at most one heap candidate per arc; space is \(O(n+A)\) . Total reset work is linear in the vertices, arcs, and periods actually used and is included in the bound.
Restarting Dijkstra for every packet is a useful small oracle but may rescan nearly the same graph \(k\) times. The bound relies on \(T\le100\) . Increasing that threshold requires a new performance assessment.
See the standard program, test recipes, and measurements.