Extra&Credit&Homework&Assignment&
Due$May$13$
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These$problems$are$taken$from$de$Berg$et$al.$Computational+Geometry.$They$are$extra$
credit,$worth$the$number$of$points$assigned$to$them.$They$are$added$to$your$homework$
score$after$the$curve$is$computed.$
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1. (4pts)$Show$that$a$priority$search$tree$on$n$points$in$the$plane$can$be$constructed$in$
O(n)$time$if$the$points$are$already$sorted$by$their$y‐coordinate.$
2. (8pts)$Describe$a$data$structure$to$store$n$intervals$on$the$real$line$so$that$we$can$
find$the$intervals$that$are$completely+contained$in$a$query$range$[x1,$x2]$in$time$$
O(log$n$+$k),$where$k$is$the$number$of$matching$intervals.$Your$data$structure$
should$take$O(n$log$n)$space.$(Hint:$use$a$range$tree).$
3. (10pts)$Give$an$example$of$a$set$of$line$segments$in$the$plane$where$the$greedy$
method$of$constructing$an$auto‐partition$(where$the$splitting$line$is$taken$that$
induces$the$fewest$number$of$cuts)$results$in$a$BSP$of$quadratic$size.$
4. (a$lot)$[Open$Problem!]$Show$that$there$exists$a$BSP$of$size$O(n)$for$any$set$of$n$
non‐intersection$line$segments$in$the$plane.$
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