6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs (Fall '14)

Prof. Erik Demaine     TAs: Sarah Eisenstat, Jayson Lynch


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[+] 3SUM & APSP. k-SUM upper and lower bounds, reductions. Distinct 3SUM, 3SUM′, GeomBase, 3 points on a line, point on 3 lines, Separator, Strips cover box, Triangles cover triangle, Hole in union, Triangle measure, Point covering, Visibility between segments, Visible triangle, Planar motion planning, 3D motion planning, fixed-angle chains. Nonquadratic lower bounds for triangle finding. Conjectured cubic graph problems: diameter, all-pairs shortest paths, negative triangle, radius, median. [Scribe Notes] [src]

This lecture is about hardness of problems that can be solved in polynomial time but where that polynomial seems to be substantially superlinear. Our focus will be on 3SUM (given n integers, do any 3 sum to zero?) which is conjectured to require roughly Θ(n2) time to solve. It has been the basis for many “3SUM-hardness” reductions, so if any of the problems can be solved in O(n2 − ε) time, then so can 3SUM. In class, we'll prove many problems 3SUM-hard, including a lot of computational geometry (where 3SUM originally comes from). We'll also talk about the generalization k-SUM, which has fairly strong lower bounds assuming ETH, and the conjectured-cubic graph problems all-pairs shortest paths and diameter.

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Handwritten notes, page 1/8[previous page][next page][PDF]

Handwritten notes, page 1/8[previous page][next page][PDF]

Slides, page 1/13[previous page][next page][PDF]

Figure drawn by Erik Demaine based on Figure 11 of http://​dx.doi.org/​10.1016/​0925-7721(95)00022-2

Slides, page 1/13[previous page][next page][PDF]

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