TAOCP 7.1.4 Exercise 228

Exercise 228. [21] [21] And mutate it once more, for Hamiltonian paths from s to any other vertex.

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Solution

Let $G = (V, E)$ with distinguished start vertex $s$. Exercise 227 constructs a ZDD that represents all Hamiltonian paths from $s$ to a fixed target vertex $t \in V \setminus {s}$. Denote this ZDD by $P_{s,t}$.

A Hamiltonian path from $s$ to an unspecified terminal vertex is exactly a Hamiltonian path whose endpoint is allowed to vary over $V \setminus {s}$. For each admissible terminal vertex $t$, the set of such paths with endpoint $t$ is disjoint from the corresponding set for any other $t' \neq t$, since a Hamiltonian path has a unique endpoint different from $s$.

Hence the desired family of all Hamiltonian paths starting at $s$ and ending anywhere in $V \setminus {s}$ is the disjoint union

$$ \bigcup_{t \in V \setminus {s}} P_{s,t}. $$

A ZDD represents unions of families via the standard apply operation for disjunction, preserving ordering and reduction. Therefore the construction reduces to building each $P_{s,t}$ using the method of Exercise 227, and then combining them into a single ZDD by repeated union.

Let $T = V \setminus {s}$. Define

$$ P_s = P_{s,t_1} \lor P_{s,t_2} \lor \cdots \lor P_{s,t_{|T|}}. $$

The correctness rests on two properties inherited from Exercise 227. First, every $P_{s,t}$ encodes exactly the edge-sets of Hamiltonian paths with fixed endpoints $s$ and $t$. Second, every Hamiltonian path from $s$ has exactly one terminal vertex $t$, so it is represented in exactly one $P_{s,t}$ and hence appears exactly once in the union. No non-Hamiltonian edge set is introduced by the union, since each operand already enforces the Hamiltonian constraints.

The resulting ZDD $P_s$ is reduced and ordered because the APPLY operation preserves the ordering constraints of the underlying decision diagram, and reduction merges identical substructures across different $P_{s,t}$ subgraphs. The size of $P_s$ is therefore at most the sum of the sizes of the individual $P_{s,t}$ ZDDs, minus any shared isomorphic subgraphs created by identical constraint propagation.

This completes the construction of a ZDD representing all Hamiltonian paths from $s$ to any vertex in $V \setminus {s}$. ∎