algorithm design techniques max min max min max figure game treewhich can be pruned write the remaining procedures for tic-tac-toe the one-dimensional circle packing problem is as followsyou have circles of radii rn these circles are packed in box such that each circle is tangent to the bottom of the box and are arranged in the original order the problem is to find the width of the minimum-sized box figure shows an example with circles of radii respectively the minimum-sized box has width suppose that the edges in an undirected graph satisfy the triangle inequalitycu, cv, >cu, show how to compute traveling salesman tour of cost at most twice optimal (hintconstruct minimum spanning tree you are tournament director and need to arrange round robin tournament among players in this tournamenteveryone plays exactly one game each dayafter daysa match has occurred between every pair of players give recursive algorithm to do this prove that in round robin tournament it is always possible to arrange the players in an order pi pi pin such that for all < npij has won the match against pij+ figure sample for circle packing problem