Amir Abboud - Academia.edu (original) (raw)

Papers by Amir Abboud

We consider the parity variants of basic problems studied in fine-grained complexity. We show tha... more We consider the parity variants of basic problems studied in fine-grained complexity. We show that finding the exact solution is just as hard as finding its parity (i.e. if the solution is even or odd) for a large number of classical problems, including All-Pairs Shortest Paths (APSP), Diameter, Radius, Median, Second Shortest Path, Maximum Consecutive Subsums, Min-Plus Convolution, and 0/10/10/1-Knapsack. A direct reduction from a problem to its parity version is often difficult to design. Instead, we revisit the existing hardness reductions and tailor them in a problem-specific way to the parity version. Nearly all reductions from APSP in the literature proceed via the (subcubic-equivalent but simpler) Negative Weight Triangle (NWT) problem. Our new modified reductions also start from NWT or a non-standard parity variant of it. We are not able to establish a subcubic-equivalence with the more natural parity counting variant of NWT, where we ask if the number of negative triangles is e...

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2018

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in... more The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k 2 log n) bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k = √ n, the upper bound is O(n) and their constructions imply an Ω(n 3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric intoÕ(min(k 2 , √ k • n)) bits, which is optimal up to log factors. Our data structure circumvents an Ω(k 2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs withÕ(√ k • n) space, andÕ(n 3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n 1/2) vs. Ω(n 1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.

We consider the parity variants of basic problems studied in fine-grained complexity. We show tha... more We consider the parity variants of basic problems studied in fine-grained complexity. We show that finding the exact solution is just as hard as finding its parity (i.e. if the solution is even or odd) for a large number of classical problems, including All-Pairs Shortest Paths (APSP), Diameter, Radius, Median, Second Shortest Path, Maximum Consecutive Subsums, Min-Plus Convolution, and 0/10/10/1-Knapsack. A direct reduction from a problem to its parity version is often difficult to design. Instead, we revisit the existing hardness reductions and tailor them in a problem-specific way to the parity version. Nearly all reductions from APSP in the literature proceed via the (subcubic-equivalent but simpler) Negative Weight Triangle (NWT) problem. Our new modified reductions also start from NWT or a non-standard parity variant of it. We are not able to establish a subcubic-equivalence with the more natural parity counting variant of NWT, where we ask if the number of negative triangles is e...

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2018

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in... more The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k 2 log n) bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k = √ n, the upper bound is O(n) and their constructions imply an Ω(n 3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric intoÕ(min(k 2 , √ k • n)) bits, which is optimal up to log factors. Our data structure circumvents an Ω(k 2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs withÕ(√ k • n) space, andÕ(n 3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n 1/2) vs. Ω(n 1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.