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Tight Bounds for Dynamic Convex Hull Queries (Again) Erik DemaineMihai Pătraşcu

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linear programming tangents Dynamic Convex Hull Set S, |S|≤n points in 2d: insert point delete point update time t u query time t q

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History tutu tqtq [Overmars, van Leeuwen] STOC’80 O(lg 2 n )O(lg n ) [Chan] FOCS’99 O(lg 1+ n )O(lg n ) [Brodal, Jacob] SWAT’99 O(lg n lglg n )O(lg n ) [Brodal, Jacob] FOCS’99 O(lg n ) [Demaine, Pătraşcu] SoCG’07 So what are you going to improve? π π π

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O(lg n ) = Optimal? NO! radix sort, hashing, closest pair in O( n )… Sorting:O( n√ lglg n ) n·2 O( √ lglg n ) Voronoi, segment intersection etc. Searching:O ( min { lg w n, lg w}) O ( min { lg n/ lglg n, √w/ lg w}) 1d2d Pătraşcu FOCS’06 Chan FOCS’06 Chan, P. STOC’07 predecessor searchpoint location bounded precision say, w bits

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Motivation: Information binary search in each step, reduce entropy by 1 bit => O(lg n ) fusion trees: a sketch of w bits allows search among √w values => each step reduces entropy by ½ lg w => O(lg w n ) different information concepts H(s 1,s 2 )= lg ℓ + lg r can sketch k segments, if all H(s i,s i+1 )≥H(s 1,s k )/k 1d 2d ℓ r O(lg n ) s1s1 s2s2

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Dynamic Convex Hull linear programming => predecessor search e.g. O ( lg w) <= [Chazelle] tangents => planar point location e.g. O (√w) Static 1 2 34 5 6 1 2 45 3 6

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History tutu tqtq [Overmars, van Leeuwen] STOC’80 O(lg 2 n )O(lg n ) all queries (tangents) NEW O(lg 2 n )O(lg n/ lglg n ) [Chan] FOCS’99 O(lg 1+ n )O(lg n ) [Brodal, Jacob] SWAT’99 O(lg n lglg n )O(lg n ) some queries (LP) NEW O(lg n lglg n )O(lg w n ) all queries NEW lg O(1) n Ω (lg w n ) [Brodal, Jacob] FOCS’99 O(lg n ) Updating

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Review of [Overmars, van Leeuwen] split with vertical line compute 2 hulls recursively => O(lg n ) levels find bridges -- O(lg n ) cut+merge hull trees -- O(lg n ) => t u =O(lg 2 n ) examine bridges recurse left or right => t q =O(lg n )

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Proof sketch split into lg n subhulls => depth O(lg n /lglg n ) query: remember: “ can sketch k segments, if all H(s i,s i+1 )≥w/k ” => superconstant time/level if some H is small information efficiency: H only decreases through recursion info efficiency => cannot be slow too many times H acts as potential, bounding running time locate among 2 lg n bridges recurse

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Summary: Our Contribution “dynamic geometry with bounded precision” lots of geometry => [Overmars, van Leeuwen] is informationally efficient lower bound 1d-like structure for LP OPEN: [Chan], [Brodal-Jacob] not info efficient… OPEN: O(lg n /lglg n ) vs. Ω (lg w n ) OPEN: Improve updates. Can t u << lg n ??

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