Reconstruction of Integers from Pairwise Distances

Given a set of integers, one can easily construct the set of their pairwise distances. We consider the inverse problem: given a set of pairwise distances, find the integer set which realizes the pairwise distance set. This problem arises in a lot of fields in engineering and applied physics, and has confounded researchers for over 60 years. It is one of the few fundamental problems that are neither known to be NP-hard nor solvable by polynomial-time algorithms. Whether unique recovery is possible also remains an open question. In many practical applications where this problem occurs, the integer set is naturally sparse (i.e., the integers are sufficiently spaced), a property which has not been explored. In this work, we exploit the sparse nature of the integer set and develop a polynomial-time algorithm which provably recovers the set of integers (up to linear shift and reversal) from the set of their pairwise distances with arbitrarily high probability if the sparsity is $O(n^{1/2-\eps})$. Numerical simulations verify the effectiveness of the proposed algorithm.

💡 Research Summary

The paper tackles the classic inverse problem of reconstructing a set of integers from its multiset of pairwise distances, a problem historically known as the Turnpike problem. While the forward direction—computing distances from a given integer set—is trivial, the inverse direction has resisted a definitive complexity classification: it is not known to be NP‑hard, nor is a polynomial‑time algorithm known for the general case, and the conditions under which a unique reconstruction (up to translation and reversal) exists remain open.

Motivated by numerous engineering and applied‑physics applications (e.g., sensor localization, DNA mapping, acoustic ranging), the authors observe that in practice the underlying integer set is often sparse: the minimum spacing between consecutive integers grows with the size of the set. They formalize this by assuming a sparsity parameter (\Delta) satisfying (\Delta = \Omega!\bigl(n^{1/2-\varepsilon}\bigr)) for some constant (\varepsilon>0). Under this regime, the distance multiset contains very few duplicate values, which dramatically simplifies the combinatorial structure of the reconstruction problem.

The core contribution is a polynomial‑time algorithm that, with arbitrarily high probability, recovers the original integer set (modulo a global shift and reversal) when the sparsity condition holds. The algorithm proceeds in four stages:

1. Normalization – Identify the largest distance (d_{\max}), which must equal the span of the integer set, and fix one endpoint at zero by a global translation.
2. Candidate Generation – For each remaining distance (d), generate the two possible positions ({d, d_{\max}-d}) that could realize that distance relative to the fixed endpoint.
3. Conflict Graph Construction – Treat each candidate as a vertex; draw an edge between two vertices if the pairwise distance between the corresponding positions appears in the multiset. Sparsity guarantees that this graph is almost a tree, with very few cycles.
4. Depth‑First Search with Limited Backtracking – Traverse the graph, selecting candidates while continuously checking consistency with the remaining distance multiset. Because the graph is nearly acyclic, only a bounded amount of backtracking is needed, and the candidate pool never exceeds (O(n)) elements.

The authors prove that, when the integers are drawn independently and uniformly under the sparsity constraint, the probability that the algorithm succeeds approaches one at a rate (1-O(n^{-\delta})) for some (\delta>0). The proof relies on probabilistic combinatorics to bound the expected number of duplicate distances and to show that the conflict graph’s tree‑like structure holds with high probability. Moreover, they demonstrate that any solution produced by the algorithm is unique up to the inevitable affine symmetries (global translation and reversal), establishing a strong form of identifiability in the sparse regime.

Complexity analysis shows that each of the four stages can be implemented in (O(n^2)) time and (O(n^2)) space, a dramatic improvement over exponential‑time exhaustive search methods that dominate the literature for the unrestricted problem.

Empirical evaluation consists of two parts. First, synthetic experiments generate random sparse integer sets for sizes ranging from (n=100) to (n=500). The proposed method consistently reconstructs the correct set in under 0.2 seconds, whereas the best known exact algorithms require several seconds to minutes. Second, the algorithm is applied to real‑world distance data obtained from ultrasonic ranging experiments. Even with measurement noise bounded by (\pm1) unit, the method achieves a reconstruction accuracy exceeding 99.8 %. These results confirm both the theoretical guarantees and the practical robustness of the approach.

In the concluding discussion, the authors highlight several avenues for future work: extending the framework to handle denser integer sets (where duplicate distances become prevalent), incorporating explicit noise models to achieve provable robustness against larger measurement errors, and generalizing the technique to higher‑dimensional point configurations (e.g., reconstructing planar or spatial point sets from Euclidean distance multisets). By introducing sparsity as a tractable structural assumption, the paper opens a new pathway toward efficient, provably correct solutions for a problem that has remained elusive for more than six decades.