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Unformatted text preview: ach it with at most 2 joints "kinked": Only 2 joints among J1,...,Jn-1 have nonzero angles. The 2 joints can be chosen to be those at either end of the "median link": the link Lm such that l is at l most half the total link length but is more than half.
m -1 i =1
m i i =1 i Note this does not require reordering links.
Source: O'Rourke Separability Examples
Separable: movable to infinity without overlapping others u
separable using combination of different translation directions separable along some translation directions, but not for direction u unseparable via 2D translation Source: O'Rourke 2D Separability via Translation Guibas/Yao 1983: A collection of 2D convex polygons can be separated under these motion conditions (does not necessarily hold in 3D!): Translation: all motions are translations Unidirectional: all translations in same direction Moved once: each polygon moved only once One-at-a-time: only one polygon is moved at a time
Separating Disjoint Segments: Of subset of segments whose upper endpoint is illuminated from right, the segment with lowest upper endpoint is completely illuminated and therefore separable towards right. Separating Convex Polygons: Region swept by right boundary of convex shape moving horizontally is subset of region swept by line segment between its leftmost highest and lowest points. Separating these segments separates the polygons. O(nlogn) time, based on sorting Final Result: Any set of n 2D convex shapes can be separated via translations all parallel to any given fixed direction, with each shape moving only once. Moving order can be computed in O(nlogn) time. Developing this... 2D Separability Hardness An NP-Hard 2D Separability Problem: Translation: all motions are translations Polygons are moved one-at-a-time Each translation can be in a different direction Each polygon can be moved more than once
Create separability instance from arbitrary PARTITION instance Reduction from PARTITION Blocks of height 1 and widths from PARTITION numbers Requires them to be stacked and perfectly packed into rectangle of width = sum of PARTITION block widths Q can be moved down and right iff blocks can be packed into left part of orange shape Q
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- Spring '10