because the dutch national flag consists of three horizontal bandseach in different color as an exampleassuming that black precedes white and white precedes grayfigure (bis valid partitioning for figure (aif gray precedes black and black precedes whitefigure (cis valid partitioning for figure (ageneralizingsuppose iand the pivot index is then [ so is valid partitioning for the same arrayif the pivot index is then [ so the arrays as well as are valid partitionings (abefore partitioning (ba three-way partitioning re(canother three-way partitionsembling the dutch national flag ingthe russian national flag figure illustrating the dutch national flag problem write program that takes an array and an index into aand rearranges the elements such that all elements less than [ (the "pivot"appear firstfollowed by elements equal to the pivotfollowed by elements greater than the pivot hintthink about the partition step in quicksort solutionthe problem is trivial to solve with (nadditional spacewhere is the length of we form three listsnamelyelements less than the pivotelements equal to the pivotand elements greater than the pivot consequentlywe write these values into the time complexity is (nwe can avoid using (nadditional space at the cost of increased time complexity as follows in the first stagewe iterate through starting from index then index etc in each iterationwe seek an element smaller than the pivot--as soon as we find itwe move it to the subarray of smaller elements via an exchange this moves all the elements less than the pivot to the start of the array the second stage is similar to the first onethe difference being that we move elements greater than the pivot to the end of the array code illustrating this approach is shown below red white blue range ( def dutch_flag_partition pivot_index )pivot apivot_index first passgroup elements smaller than pivot for in range(len( ))look for smaller element for in range( len( ))if [jpivot [ ] [ja[ ] [ibreak second passgroup elements larger than pivot