Pinaki Mitra Dept. of CSE IIT Guwahati

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1 Pak Mtra Dept. of CSE IIT Guwahat

2 Hero s Problem HIGHWAY FACILITY LOCATION Faclty Hgh Way Farm A Farm B

3 Illustrato of the Proof of Hero s Theorem p q s r r l d(p,r) + d(q,r) = d(p,q) p d(p,r ) + d(q,r ) = d(p,r ) + d(q,r ) >= d(p,q)

4 MINIMAX FACILITY LOCATION Gve pots the plae represetg customers (plats, schools, tows etc..) t s desred to determe the locato X (aother pot the plae) where a faclty should be located so as to mmze the dstace from X to ts furthest customer. The problem has a elegat ad succct geometrcal terpretato: Fd the smallest crcle that ecloses a gve set of pots. The ceter of the crcle s Precsely the locato of X + The smallest crcle eclosg a set of pots

5 The mmum eclosg crcle ca be computed from the Furthest Neghbor Voroo Dagram. [Shamos] The correctess of the above algorthm was establshed by [Bhattacharya & Toussat]. Ther argumet s based o the followg property of the mmum eclosg crcle : The mmum eclosg crcle of a set S of pots s ether determed by the dameter of the pot set S or by three pots o the covex hull of S, where those three pots form a acute agled tragle.

6 Furthest Neghbor Voroo Dagram ca be computed from the proecto of the upper hull of the pot set lfted o the parabolod z = x 2 + y 2. z q (x,y, x 2 +y 2 )) y p (x,y ) (x,y) (x,y, x 2 +y 2 ) x

7 But the computato of Furthest Neghbor Voroo Dagram s lower bouded by the computato of the covex hull of pots.e., Ω (log). But ths lower boud does t hold for the mmum eclosg crcle. I fact we ca compute the mmum eclosg crcle of a set of pots θ() tme by solvg three varable Covex Program. [Meggdo] The result holds for ay fxed dmesoal pot set.

8 Here (x,y ), (x 2, y 2 ),., (x,y ) are the set of pots for whch we have to compute the mmum eclosg crcle. Let (x,y) be the ceter of the mmum eclosg crcle ad let r be ts radus. The we have the mmum eclosg crcle problem formulated as the followg optmzato problem: Mmze r 2 Subect to: (x x ) 2 + (y y ) 2 r 2 =, 2,, If we substtute z = x 2 + y 2 r 2 the we have the followg covex programmg formulato of the problem: Mmze x 2 + y 2 z Subect to: 2xx 2yy + z + (x 2 + y 2 ) 0 =, 2,, Thus we have to mmze a quadratc fucto over a set of lear costrats volvg 3 varables.

9 Fermat Weber Problem Problem Defto: For a gve set of m pots x, x 2,, x wth each x R d fd a pot y from where the sum of all Eucldea dstaces to the x s s the mmum. Ths problem s also kow as Geometrc Meda problem : arg m d y R = x y The specal case whe = 3 ad d = 2 arses the costructo of Mmal Steer Trees ad was orgally posed as problem by Torrcell.

10 a f 20 b c For 3 coplaar pots a, b, c f each agle of the tragle abc s less ta 20 the Fermat Pot f s a pot sde the tragle whch subteds 20 a agle to all three pars of pots. Otherwse f ay agle of the tragle abc s greater tha the Fermat Pot s the pot makg that agle. For 4 coplaar pots f a pot s sde the tragle formed by the other three pots the geometrc meda s that pot. Otherwse all 4 pots form a covex quadrlateral ad the geometrc meda s the tersecto pot of both the dagoals ad also kow as Rado Pot. med

11 If we restrct d =,.e., for the dmesoal case the the geometrc meda cocdes wth the meda. Aga f we cosder the problem L metrc wth d = 2,.e., 2 dmeso we ca combe the optmal solutos of two dmesoal problems to obta the optmal soluto of the orgal problem. x The method exteds for ay value of d L metrc,.e., we ca decompose the problem d oe dmesoal problems ad combe ther optmal soluto to obta the optmal soluto of the orgal problem. I L 2 metrc Weszfeld s algorthm teratvely computes the geometrc meda problem : x y + y = = = x x y y

12 K Meda Gve a set S of pots the plae we have to locate a set F of k facltes that parttos the set S k parttos N(f ) =,2,, k so that for ay ste s N(f ) [eghborhood of f ] beg served by the faclty f the followg sum s mmzed: k = s N ( ) f dst( f, s ) where S = k N( f ) = ad N( f ) N( f ) = φ, < k

13 I k meda problem the elemets for the set F ca be ay k stes R d the urestrcted case. Sometmes we restrct the k meda problem such that F S,.e., put pot stes. Both of these problems are NP Hard. [Meggdo & Supowt] So our goal should be the desg of effcet approxmato algorthms for these problems. Here we wll restrct our atteto to the restrcted verso of the problem. Restrcted verso of our faclty locato ca stll be classfed to two subcategores: (a) Ucapactated k meda problem (b) Capactated k meda problem

14 A commo assumpto used whe dealg wth locato problems s that facltes are ucapactated ad ca thus servce ay umber of demad destatos. But the assumpto may be urealstc may applcatos ad thus capactated faclty locato a upper boud s provded o the umber of demad destatos servced by each faclty. Now we ca formulate these problems usg ILP as follows: Mmze = = dst ( s, s ) x Subect to: x = = x = C =,2,..., =,2,..., y = k = y x 0 =,2,..., =,2,...,

15 y {0,} x {0,} =,2,..., =,2,..., The dcator varable y deotes f the ste s s selected as the faclty or ot. The dcator varable x deotes f the faclty at the ste s serves the customer at ste s. Sce ILP s NP Complete the usual techque s to relax the tegralty costrat,.e., replace the above two costrats by: 0 y 0 x =,2,..., =,2,..., =,2,...,

16 The we solve the LP relaxato of the ILP. After that the fractoal soluto s cleverly rouded matag the feasblty ad some boud s establshed w.r.t Z* LP Z* ILP For rectlear meda problem Ω(log) lower boud was establshed by [Baa]. As far as upper bouds are cocered there are O(.5 log 4 ) tme algorthm for arbtrary weghted pots ad O(log 4 ) tme algorthm for equally weghted pot set was exhbted by [ElGdy & Kel] Thus eve L metrc there are bg gaps betwee these upper ad lower bouds. All these pose several ew drectos of future research.

17 Le Faclty I may practcal applcatos we may be terested locatg a le faclty rather tha locatg a pot faclty. For example we may have to costruct a road some resdetal area so that t s coveet to most of the resdets. The ths road desg s the same as locatg a le faclty amog a set of stes for resdets. There ca be may varatos of these problems varous metrc. The L 2 approxmato of ths problem,.e., where we have to mmze the sum of the squares of the perpedcular dstaces s the well kow Regresso Le Problem.

18 Gve a set of data pots (x, y ), (x 2, y 2 ),., (x, y ) we have to fd the equato of the straght le y = ax + b that mmzes the L 2 error,.e., Mmze Here the varables are fact a ad b. So we have two ukows ad we requre at least two equatos to solve. They are obtaed from the mma crtero after takg partal dervatves w.r.t. a ad b respectvely: The result ca be exteded for ay dmesoal data. For example d dmesos we have to fd the Regresso Hyper plae a x + a 2 x a d x d = b. Thus we have to Mmze = 2 [ y ( ax + b)] = = d [ b = a E a x ] 2 E E = 0, = 0 b = E

19 Thus we have d ukows a, a 2,, a d. They are solved from the followg system of equatos: E a E = 0, a 2 E = 0,..., a d = 0 For L approxmato of the problem we have to Mmze = b d = a x = E Ths absolute value fucto s ot a lear fucto. But we ca easly covert t to a stadard LP problem as follows [Chvatal]: Mmze Subect to: = b e d = b + d = a a x x e e =,2,..., =,2,...,

20 For L approxmato of the problem we have to Mmze max b d = a x Ths ca be drectly formulated to LP as follows: Mmze z Subect to: z d +a x b = =,2,..., z d = a x b =,2,..., The problem of L ad L approxmato problem was frst posed By Fourer. Subsequetly may good algorthms for computg L ad L approxmato were proposed by [Bloomfeld & Steger].

21 Smlar type of problems for le faclty cocers: ) Locatg a le faclty that mmzes the maxmum dstace from a set of stes. ) Locatg a le faclty that mmzes the sum of the dstaces from a set of stes. There are several possble extesos of these problems.

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