CSE 5311 Notes 16: Matrices

Transcription

1 CSE 5311 Notes 16: Matrices STRASSEN S MATRIX MULTIPLICATION Matrix additio: takes scalar additios. Everyday atrix ultiply: p p Let = = p. takes p scalar ultiplies ad -1)p scalar additios. Best lower boud is Ω 2 ). For = 2: " A11 A12 %" B 11 B12% " = C 11 C12% A 21 A 22 & B 21 B 22 & C 21 C 22 & is doe by everyday ethod usig: 8 scalar ultiplies 4 scalar additios but Strasse s ethod CLRS, p.735) uses: 7 scalar ultiplies 18 scalar additios

2 2 Whe = 2 k : A ij ad B ij are 2 k-1 x 2 k-1 subatrices that are ultiplied recursively usig Strasse s ethod. Suppose = 4. Everyday ethod will take 4 3 = 64 scalar ultiplies ad 163 = 48 scalar additios. Strasse s ethod will use: 7 recursive 2 x 2 atrix ultiplies, each usig 7 scalar ultiplies ad 18 scalar additios x 2 atrix additios, each usig 4 scalar additios. This gives 49 scalar ultiplies ad 198 scalar additios. Let Mk) = uber of scalar ultiplies for 2 k x 2 k = x : ) =1 ) = 7 M 0 M 1 M k) = 7M k "1) = 7 k = 7 lg = lg Note that the costat is 1. Let Pk) be the uber of additios icludig subtractios) doe for 2 k x 2 k = x : ) = 0 ) =18 P 0 P 1 ) =18% 2 k"1 P k ) = ) P k $ P % 2 k & P ) ) =18 2 Master Method : & 2 + 7P k "1 ' ) =18% 2k 2 $ & 2 ' ) ) ) = ) ) P 2 P 2 a = 7 b = 2 logb a = lg 7 * 2.81 f ) = = O% 2.81"+ & Case 1: P ) $ ) =,% 2.81 & ' + 7P k "1)

3 CSE 5311 Notes 17: Coputatioal Geoetry FUNDAMENTAL PREDICATES Twice the siged) area of a triagle AT) is give by: 2A T) = xa ya 1 x b y b 1 xc yc 1 = x b " x a y b " y a xc " xa yc " ya = x b " x a ) y c " y a ) " x c " x a ) y b " y a ) If positive, the poits a, b, ad c ake a left tur couter-clockwise). If egative, the poits a, b, ad c ake a right tur clockwise). If zero, the poits a, b, ad c are colliear. Relatioship of a poit a to couter-clockwise circle of poits b, c, ad d? Based o tetrahedral volue.) xa ya xa 2 + ya 2 1 xb yb x 2 b + yb 2 1 xc yc xc 2 + yc 2 1 xd yd x 2 + yd 2 1 d Zero : o circle Positive : outside Negative : iside PROXIMITY Closest poits i 1-d space 1dclosest.c) 1. Fid edia of poit set. 2. Recursively deterie closest pair o left side ad right side. 3. Check whether rightost i left side ad leftost i right side are a closer pair tha 2. Worst-case: " log)

4 Closest poits i 2-d space 2dclosest.c) 2 Brute-force: "% 2 & Divide-ad-coquer: 1. Draw vertical lie to divide ito equal-size subsets. 2. Recursively fid closest pair for left ad right sides. Let δ be the saller of the two distaces. 3. Fid closest pair aog poits withi δ of the dividig lie. Sice the poit set is ot rado, details ust assure that "% 2 & behavior is avoided. Base Case: If 3 or soe other costat), use brute-force. To support the divides ad the sea processig, the set of poits is preprocessed: 1. Create array with poits sorted by x-coordiate. 2. Create secod array with poits sorted by y-coordiate. Also iclude cross-refereces to x- ordered array. Whe a divide by a vertical lie is eeded, the first array is trivial to split ad the secod array is split by usig the cross-refereces. The y-ordered array facilitates fidig the closest pair across the sea.

5 !! 3!! For a give left-side sea poit, the distaces to at ost six right-side sea poits are eeded. T ) = 2T 2) + " ) = " log) CONVEX HULLS Deterie sallest covex polygo that icludes all poits i a 2-d set. Graha sca - Based o agular sweep w.r.t. the leftost) botto poit X ad aitaiig stack with covex hull. 1. Fid X. 2. Sort by agle w.r.t. X. Coparisos by testig turs ad breakig colliear cases by takig farthest poit first. 3. Push X ad first two sweep poits. 4. for each poit P i sorted order while top-of-stack, ext-to-top-of-stack, ad P do ot ake a left tur Push P Discard top-of-stack it s ot i covex hull)

6 4 X Jarvis arch rubberbadig or gift-wrappig) Rus i " h ) where h is the uber of hull poits Good for cases like: Sae iitial poit X as Graha sca. Also eed leftost) top poit Y. Each successive hull poit is foud by fidig iiu agle WRT two previous poits. Covex hull ay be used to fid the diaeter of a set usig " h ) additioal tie.

7 5 SWEEP-LINE ALGORITHMS Siple exaple: Itersectio of rectiliear rectagles Idea: Sweep a vertical lie fro left-to-right ad store vertical cross-sectio sweep-lie status). Preprocessig: Sort left ad right edges by x-coordiate evet-poit schedule). Algorith: Sweep across x diesio Left edge: Check for itersectio. Isert i iterval tree CLRS, 14.3) Right edge: Delete fro iterval tree Rus i " log + log ) = " log ) is ax rects i tree) Difficulty: What if two rectagles touch? Treat as itersectig or ot by how ties are hadled. More sigificat exaple: 2-d closest pairs Idea: Icreetally deterie δ for the leftost k poits. Maitai y-ordered BST of poits whose x-distace fro poit k + 1 is < δ. Preprocessig: Sort poits by x-coordiate. Processig poit k + 1: 1. Delete BST poits that are at least δ to the left. 2. Isert poit k + 1 ito BST. 3. Exaie BST predecessors of poit k + 1 util δ away i y. Check for iprovig δ. 4. Like 3., but with BST successors. Tie: " log) Referece: K. Hirichs, J. Nievergelt, ad P. Schor, Plae-Sweep Solves the Closest Pair Proble Elegatly, Iforatio Processig Letters ),

8 6 EUCLIDEAN MINIMUM SPANNING TREES Vorooi Diagra - post office proble. Divides plae ito covex regios, each cotaiig poits closest to soe give poit blue lies). Fortue s Sweep-Lie achieves " log ) - applet Delauay Triagulatio Coects vertices for adjacet Vorooi regios black lies betwee iput poits). May trasfor a arbitrary triagulatio to a DT i "% 2 & tie usig flips based o icircle test ad the followig property: Three poits are vertices of a Delauay triagle iff the circle that passes through the three poits is epty.