Algorithm 2 Fuzzy Quicksort

Input: a sequence of $n$ intervals $A$ , where $A[i].a$ and $A[i].b$ denotes the lower and upper bound of the interval

Output: a sorted sequence of input intervals

1:FuzzyQuicksort( $A, 1, n$ )

3:procedure FuzzyQuicksort( $A, p, r$ )

4:if $p < r$ then

5: $(q, t) =$ FuzzyPartition( $A, p, r$ )

6:FuzzyQuicksort( $A, p, q-1$ )

7:FuzzyQuicksort( $A, t+1, r$ )

8:end if

9:end procedure

10:

11:procedure FuzzyPartition( $A, p, r$ )

12:exchange $A[p]$ with $A[RANDOM(p, r)]$

13: $x = A[p]$

14: $q = p$

15: $t = p$

16:for $i = p + 1$ to $r$ do

17:if $A[i].b < x.a$ then // interval less than condition

18:exchange $A[q]$ with $A[i]$

19: $q = q + 1$

20:exchange $A[t+1]$ with $A[i]$

21: $t = t + 1$

22:else if $A[i].b \ge x.a$ and $A[i].a \le x.b$ then // interval equal condition

23:exchange $A[t+1]$ with $A[i]$

24: $t = t + 1$

25:end if

26:end for

27:return $(q, t)$

28:end procedure