▁may ▁assign ▁a ▁piece ▁to ▁partner ▁i , ▁only ▁if ▁it ▁is ▁longer ▁than ▁all ▁land mark - interval s ▁of ▁i . ▁At ▁least ▁n / 2 ▁partners ▁must ▁receive ▁an ▁interval ▁with ▁a ▁length ▁of ▁at ▁most ▁ 2 / n ; ▁hence ▁all ▁their ▁land mark - interval s ▁must ▁have ▁a ▁length ▁of ▁at ▁most ▁ 2 / n ; ▁hence ▁they ▁must ▁have ▁at ▁least ▁n / 2 ▁land mark - interval s ; ▁hence ▁they ▁must ▁have ▁at ▁least ▁n / 2 ▁land marks . ▁ ▁D . ▁Each ▁query ▁answered ▁by ▁partner ▁i ▁involves ▁at ▁most ▁two ▁new ▁end points , ▁so ▁it ▁increases ▁the ▁number ▁of ▁land marks ▁of ▁i ▁by ▁at ▁most ▁ 2 . ▁Hence , ▁in ▁the ▁case ▁described ▁by ▁paragraph ▁C , ▁the ▁algorithm ▁must ▁ask ▁each ▁of ▁n / 2 ▁partners , ▁at ▁least ▁n / 4 ▁queries . ▁The ▁total ▁number ▁of ▁queries ▁is ▁thus ▁at ▁least ▁n 2 / 8 ▁= ▁ Ω ( n 2 ). ▁ ▁It ▁is ▁an ▁open ▁question ▁whether ▁a ▁bounded ▁algorithm ▁exists ▁for ▁arbitrary ▁valu ation ▁functions . ▁Thus ▁there ▁is ▁a ▁huge ▁gap ▁between ▁the ▁lower ▁bound ▁of ▁ Ω ( n 2 ) ▁and ▁the ▁best ▁currently ▁known ▁algorithm ▁which ▁is ▁finite ▁but ▁un bounded . ▁ ▁Ex istence ▁proofs ▁and ▁variants ▁ ▁In ▁addition ▁to ▁the ▁general ▁existence ▁proofs ▁imp lied ▁by ▁the ▁algorithms ▁described ▁above , ▁there ▁are ▁proofs ▁for ▁the ▁existence ▁of ▁env y - free ▁partitions