109,013 research outputs found
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Throughput Scaling Of Convolution For Error-Tolerant Multimedia Applications
Convolution and cross-correlation are the basis of filtering and pattern or
template matching in multimedia signal processing. We propose two throughput
scaling options for any one-dimensional convolution kernel in programmable
processors by adjusting the imprecision (distortion) of computation. Our
approach is based on scalar quantization, followed by two forms of tight
packing in floating-point (one of which is proposed in this paper) that allow
for concurrent calculation of multiple results. We illustrate how our approach
can operate as an optional pre- and post-processing layer for off-the-shelf
optimized convolution routines. This is useful for multimedia applications that
are tolerant to processing imprecision and for cases where the input signals
are inherently noisy (error tolerant multimedia applications). Indicative
experimental results with a digital music matching system and an MPEG-7 audio
descriptor system demonstrate that the proposed approach offers up to 175%
increase in processing throughput against optimized (full-precision)
convolution with virtually no effect in the accuracy of the results. Based on
marginal statistics of the input data, it is also shown how the throughput and
distortion can be adjusted per input block of samples under constraints on the
signal-to-noise ratio against the full-precision convolution.Comment: IEEE Trans. on Multimedia, 201
Stable marriage with general preferences
We propose a generalization of the classical stable marriage problem. In our
model, the preferences on one side of the partition are given in terms of
arbitrary binary relations, which need not be transitive nor acyclic. This
generalization is practically well-motivated, and as we show, encompasses the
well studied hard variant of stable marriage where preferences are allowed to
have ties and to be incomplete. As a result, we prove that deciding the
existence of a stable matching in our model is NP-complete. Complementing this
negative result we present a polynomial-time algorithm for the above decision
problem in a significant class of instances where the preferences are
asymmetric. We also present a linear programming formulation whose feasibility
fully characterizes the existence of stable matchings in this special case.
Finally, we use our model to study a long standing open problem regarding the
existence of cyclic 3D stable matchings. In particular, we prove that the
problem of deciding whether a fixed 2D perfect matching can be extended to a 3D
stable matching is NP-complete, showing this way that a natural attempt to
resolve the existence (or not) of 3D stable matchings is bound to fail.Comment: This is an extended version of a paper to appear at the The 7th
International Symposium on Algorithmic Game Theory (SAGT 2014
When Hashing Met Matching: Efficient Spatio-Temporal Search for Ridesharing
Carpooling, or sharing a ride with other passengers, holds immense potential
for urban transportation. Ridesharing platforms enable such sharing of rides
using real-time data. Finding ride matches in real-time at urban scale is a
difficult combinatorial optimization task and mostly heuristic approaches are
applied. In this work, we mathematically model the problem as that of finding
near-neighbors and devise a novel efficient spatio-temporal search algorithm
based on the theory of locality sensitive hashing for Maximum Inner Product
Search (MIPS). The proposed algorithm can find near-optimal potential
matches for every ride from a pool of rides in time and space for a small . Our
algorithm can be extended in several useful and interesting ways increasing its
practical appeal. Experiments with large NY yellow taxi trip datasets show that
our algorithm consistently outperforms state-of-the-art heuristic methods
thereby proving its practical applicability
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