TY - GEN

T1 - Dominance product and high-dimensional closest pair under L∞

AU - Gold, Omer

AU - Sharir, Micha

N1 - Funding Information:
∗ Work on this paper has been supported by Grant 892/13 from the Israel Science Foundation, by Grant 2012/229 from the U.S.-Israeli Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. 1 The Õ(·) notation hides poly-logarithmic factors.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - Given a set S of n points in Rd, the Closest Pair problem is to find a pair of distinct points in S at minimum distance. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L∞ Closest Pair problem, where d = nr for some r > 0, and the underlying metric is L∞. We improve and simplify previous results for L∞ Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in O(DP(n, d) log n) time, and by a randomized algorithm that runs in O(DP(n, d)) expected time, where DP(n, d) is the time bound for computing the dominance product for n points in Rd. That is a matrix D, such that D[i, j] = | {k | pi[k] ≤pj [k]} | ; this is the number of coordinates at which pj dominates pi. For integer coordinates from some interval [-M,M], we obtain an algorithm that runs in O min{Mnω(1,r,1), DP(n, d)} | time1, where ω(1, r, 1) is the exponent of multiplying an n × nr matrix by an nr × n matrix. We also give slightly better bounds for DP(n, d), by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).

AB - Given a set S of n points in Rd, the Closest Pair problem is to find a pair of distinct points in S at minimum distance. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L∞ Closest Pair problem, where d = nr for some r > 0, and the underlying metric is L∞. We improve and simplify previous results for L∞ Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in O(DP(n, d) log n) time, and by a randomized algorithm that runs in O(DP(n, d)) expected time, where DP(n, d) is the time bound for computing the dominance product for n points in Rd. That is a matrix D, such that D[i, j] = | {k | pi[k] ≤pj [k]} | ; this is the number of coordinates at which pj dominates pi. For integer coordinates from some interval [-M,M], we obtain an algorithm that runs in O min{Mnω(1,r,1), DP(n, d)} | time1, where ω(1, r, 1) is the exponent of multiplying an n × nr matrix by an nr × n matrix. We also give slightly better bounds for DP(n, d), by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).

KW - Closest pair

KW - Dominance product

KW - L∞

KW - Matrix multiplication

UR - http://www.scopus.com/inward/record.url?scp=85038577200&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2017.39

DO - 10.4230/LIPIcs.ISAAC.2017.39

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AN - SCOPUS:85038577200

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th International Symposium on Algorithms and Computation, ISAAC 2017

A2 - Tokuyama, Takeshi

A2 - Okamoto, Yoshio

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 28th International Symposium on Algorithms and Computation, ISAAC 2017

Y2 - 9 December 2017 through 22 December 2017

ER -