Knowledge-Driven Mastermind Planning

• Knowledge-driven Mastermind planning is an algorithmic framework that leverages accumulated problem knowledge to efficiently reduce query complexity.
• It employs a two-phase strategy with signed queries and an information-tree token algorithm that achieves Θ(n) query complexity for k = n.
• Its integration of combinatorial reductions and adaptive query design sets tight theoretical bounds and offers a blueprint for other combinatorial search challenges.

Knowledge-driven planning in the context of Mastermind encompasses algorithmic frameworks that use accumulated knowledge—such as candidate sets, information-theoretic metrics, and combinatorial reductions—to devise optimal or near-optimal query strategies for code discovery. Recent advances establish optimal bounds and provide algorithmic blueprints with provably minimal query complexity, culminating in tight results for all regimes of code length $n$ and color count $k$. Below, the main architectural, combinatorial, and algorithmic constructs of knowledge-driven Mastermind planning are systematically presented.

1. Formal Problem Statement and Complexity Landscape

The standard Mastermind problem features a secret code $z = (z_1, ..., z_n) \in [k]^n$, and an adaptive interrogator (Codebreaker) that proposes queries $x^{(t)} \in [k]^n$ in rounds. The oracle returns feedback—classically, the number of correct positions (“black pegs”) denoted $eq(z, x^{(t)}) = |\{i \mid x^{(t)}_i = z_i\}|$—with “white pegs” providing additional permutation-invariant cues when allowed. The objective is to minimize the number of queries until $z$ is uniquely determined (Martinsson et al., 2020).

Classical bounds for query complexity are:

• Information-theoretic lower bound: Using the entropy in code space ($n \log_2 k$ bits) and the per-query information yield (at most $\log_2(n+1)$ bits), the lower bound is $\Omega(n \log k / \log n)$. For $k=n$, this simplifies to $\Omega(n)$.
• Upper bound (pre-2020, $k=n$): $O(n\log\log n)$ queries (Doerr et al., 2012).
• State-of-the-art (2020): $\Theta(n)$ query complexity for $k=n$, closing the previous gap (Martinsson et al., 2020).

Full query complexity (randomized, black-peg only):

$$bmm(n,k) = \Theta\left( n \frac{\log k}{\log n} + k \right)$$

For black and white pegs:

$$bwmm(n,k) = \Theta\left( n \frac{\log k}{\log n} + \frac{k}{n} \right)$$

In particular, $k=n$ gives $\Theta(n)$ in both regimes.

2. Algorithmic Blueprint for Linear-Query Mastermind ($k=n$)

The breakthrough $\Theta(n)$-query algorithm for black-peg Mastermind with $k=n$ colors integrates two primary phases (Martinsson et al., 2020):

A. Reduction to Signed Permutation Mastermind

1. All-wrong reference: Identify a guess $z$ with $eq(z, \text{secret}) = 0$ via $n+1$ deterministic or $O(1)$ random queries.
2. Permutation mapping: Generate $n$ queries, each with exactly one correct peg, to deduce a bijection $\varphi$ making the code a permutation w.r.t. $\varphi$.
3. Signed queries: Encode general black-peg queries in an auxiliary domain $\{-n, ..., n\}^n$; simulate these via pairs of standard queries plus response differencing.

B. Information-Tree Token Algorithm

• Construct a complete binary tree $T$ of depth $\lceil \log_2 n \rceil$ with $n$ leaves (each representing a code position).
• Initially, each color corresponds to a token at the root.
• At each step, the task is to localize tokens (colors) through $O(1)$-cost “zero-one” queries: “Is token $t$ in the left half of interval $v$?”
• Preprocessing moves tokens along the tree, splitting “big” intervals to create independent subproblems.
• The main “Solve” routine interleaves recursive calls and merges query requests using the following combination:

$$w^{(1)} = q^{(1)} + q^{(2)} + s, \quad w^{(2)} = q^{(1)} - q^{(2)}$$

Responses on $w^{(1)}$, $w^{(2)}$ suffice to recover all component answers per Lemma 3.4.

Total number of signed queries: $\leq 9n$ (including preprocessing and main solve phase). This matches the information-theoretic lower bound up to constant factors (Martinsson et al., 2020).

3. Combinatorial and Information-Theoretic Underpinnings

• Entropy argument: Code space size is $n^n \implies n \log_2 n$ bits, each black-peg query yields $\leq \log_2(n+1)$ bits $\implies \Omega(n)$ required.
• Coin-weighing analogy: Binary search on each color is $n \log n$, but the token algorithm leverages overlapping subproblems and batch (group testing) reductions to amortize determination across tasks.
• Adaptive query design: Overlapping intervals are exploited using a merge of token-spanning tasks, mimicking the Cantor–Mills sign-scheme to achieve adaptive speed-up (Martinsson et al., 2020).

4. Generalization to Arbitrary $k$ and $n$

The result extends to all regimes of $k$ as follows (Martinsson et al., 2020):

• Small $k \leq n^{1-\epsilon}$: Non-adaptive random guessing suffices; $\Omega(n \log k/\log n)$ queries match the lower bound.
• Intermediate $k$ ($\sqrt n \leq k \leq n$): The same entropy bound is dominant; run the $k=n$ strategy, treating surplus “blanks” as dummy colors.
• Large $k \geq n$: Use color-partitioning and subset-finding, reducing to the $bmm(n,n)$ or $bwmm(n,n)$ regime plus an additive term in $k$:

$$bmm(n,k) = \Theta(bmm(n,n) + k), \quad bwmm(n,k) = \Theta(bmm(n,n) + k/n)$$

All cases admit tight, constructive algorithms matching the information lower bounds.

5. Algorithm Correctness and Complexity Analysis

Proofs rest on:

• Correctness: Inductive tracking of token position through the tree structure demonstrates that each color is uniquely localized.
• Complexity: The recursion for query count $c(n)$ is dominated by

$c(n) \leq 2 \max(c(n/4), a(n/2)) + c(n/2)$

with $a(m) \leq 3m$, leading to $c(n) \leq 6n$. Additional $3n$ for preprocessing gives total $9n$ queries (Martinsson et al., 2020).

• Adaptivity efficiency: The recursion leverages parallelization of subproblems and efficient combination of queries, ensuring linear scaling.

6. Strategic Insights and Operational Intuition

• Early phase: Low-information queries dissociate large blocks, maximizing parallel progress by rapidly bisecting the task.
• Chunked resolution: “Pack” multiple subproblems into each query, resolving order $\log n$ tasks per round on average.
• Concrete illustration: For $n=8$, preprocessing uses $3n=24$ queries, with batch rounds of “Solve” phase combining independently localized tasks into a small number of aggregate queries—demonstrating the amortized information gain per query is nearly optimal (Martinsson et al., 2020).

7. Impact and Open Directions

This framework resolves the previously open asymptotic gap for $k=n$ and tightly characterizes the knowledge-driven planning regime in Mastermind for all $(n,k)$. The approach establishes a blueprint applicable to other combinatorial search problems with structured feedback and suggests that adaptivity, combinatorial reductions, and careful knowledge representation (e.g., candidate sets, interval tokens) are central to optimal planning in related domains.

References

• “Mastermind with a Linear Number of Queries” (Martinsson et al., 2020)