I'm really struggling to design an algorithm to find d, which is the lowest value that can be added or subtracted (at most) to make a given sequence strictly increasing.
For example.. say seq[] = [2,4,8,3,1,12]
given that sequence, the algorithm should return "5" as d because you can add or subtract at most 5 to each element such that the function is strictly increasing.
I've tried several approaches and can't seem to get a solid technique down.
I've tried looping through the seq. and checking if seq[i] < seq[i+1]. If not, it checks if d>0.. if it is, try to add/subtract it from seq[i+1]. Otherwise it calculates d by taking the difference of seq[i-1] - seq[i].
I can't get it to be stable though and Its like I keep adding if statements that are more "special cases" for unique input sequences. People have suggested using a binary search approach, but I can't make sense of applying it to this problem.
Any tips and suggestions are greatly appreciated. Thanks!
Here's my code in progress - using Python - v4
def ComputeMaxDelta3(seq):
# Create a copy to speed up comparison on modified values
aItems = seq[1:] #copies sequence elements from 1 (ignores seq[0])
# Will store the fix values for every item
# this should allocate 'length' times the 0 value
fixes = [0] * len(aItems)
print("fixes>>",fixes)
# Loop until no more fixes get applied
bNeedFix = True
while(bNeedFix):
# Hope will have no fix this turn
bNeedFix = False
# loop all subsequent item pairs (i should run from 0 to length - 2)
for i in range(0,len(aItems)-1):
# Left item
item1 = aItems[i]
# right item
item2 = aItems[i+1]
# Compute delta between left and right item
# We remember that (right >= left + 1
nDelta = item2 - (item1 + 1)
if(nDelta < 0):
# Fix the right item
fixes[i+1] -= nDelta
aItems[i+1] -= nDelta
# Need another loop
bNeedFix = True
# Compute the fix size (rounded up)
# max(s) should be int and the division should produce an int
nFix = int((max(fixes)+1)/2)
print("current nFix:",nFix)
# Balance all fixes
for i in range(len(aItems)):
fixes[i] -= nFix
print("final Fixes:",fixes)
print("d:",nFix)
print("original sequence:",seq[1:])
print("result sequence:",aItems)
return
Here's whats displayed:
Working with: [6, 2, 4, 8, 3, 1, 12]
[0]= 6 So the following numbers are the sequence:
aItems = [2, 4, 8, 3, 1, 12]
fixes>> [0, 0, 0, 0, 0, 0]
current nFix: 6
final Fixes: [-6, -6, -6, 0, 3, -6]
d: 1
original sequence: [2, 4, 8, 3, 1, 12]
result sequence: [2, 4, 8, 9, 10, 12]
d SHOULD be: 5
done!
~Note~
I start at 1 rather than 0 due to the first element being a key
As anticipated, here is (or should be) the Python version of my initial solution:
def ComputeMaxDelta(aItems):
# Create a copy to speed up comparison on modified values
aItems = aItems[:]
# Will store the fix values for every item
# this should allocate 'length' times the 0 value
fixes = [0] * len(aItems)
# Loop until no more fixes get applied
bNeedFix = True
while(bNeedFix):
# Hope will have no fix this turn
bNeedFix = False
# loop all subsequent item pairs (i should run from 0 to length - 2)
for i in range(0,len(aItems)-1):
# Left item
item1 = aItems[i]
# right item
item2 = aItems[i+1]
# Compute delta between left and right item
# We remember that (right >= left + 1
nDelta = item2 - (item1 + 1)
if(nDelta < 0):
# Fix the right item
fixes[i+1] -= nDelta
aItems[i+1] -= nDelta
# Need another loop
bNeedFix = True
# Compute the fix size (rounded up)
# max(s) should be int and the division should produce an int
nFix = (max(fixes)+1)/2 # corrected from **(max(s)+1)/2**
# Balance all fixes
for i in range(len(s)):
fixes[i] -= nFix
print("d:",nFix) # corrected from **print("d:",nDelta)**
print("s:",fixes)
return
I took your Python and fixed in order to operate exactly as my C# solution.
I don't know Python, but looking for some reference on the web, I should have found the points where your porting was failing.
If you compare your python version with mine you should find the following differences:
You saved a reference aItems into s and used it as my fixes, but fixes was meant to start as all 0.
You didn't cloned aItems over itself, then every alteration to its items was reflected outside of the method.
Your for loop was starting at index 1, whereas mine started at 0 (the very first element).
After the check for nDelta you subtracted nDelta from both s and aItems, but as I stated at points 1 and 2 they were pointing to the same items.
The ceil instruction was unnedeed because the division between two integers produces an integer, as with C#.
Please remember that I fixed the Python code basing my knowledge only on online documentation, because I don't code in that language, so I'm not 100% sure about some syntax (my main doubt is about the fixes declaration).
Regards,
Daniele.
Here is my solution:
public static int ComputeMaxDelta(int[] aItems, out int[] fixes)
{
// Create a copy to speed up comparison on modified values
aItems = (int[])aItems.Clone();
// Will store the fix values for every item
fixes = new int[aItems.Length];
// Loop until no more fixes get applied
var bNeedFix = true;
while (bNeedFix)
{
// Hope will have no fix this turn
bNeedFix = false;
// loop all subsequent item pairs
for (int ixItem = 0; ixItem < aItems.Length - 1; ixItem++)
{
// Left item
var item1 = aItems[ixItem];
// right item
var item2 = aItems[ixItem + 1];
// Compute delta between left and right item
// We remember that (right >= left + 1)
var nDelta = item2 - (item1 + 1);
if (nDelta < 0)
{
// Fix the right item
fixes[ixItem + 1] -= nDelta;
aItems[ixItem + 1] -= nDelta;
//Need another loop
bNeedFix = true;
}
}
}
// Compute the fix size (rounded up)
var nFix = (fixes.Max() + 1) / 2;
// Balance all fixes
for (int ixItem = 0; ixItem < aItems.Length; ixItem++)
fixes[ixItem] -= nFix;
return nFix;
}
The function returns the maximum computed fix gap.
As a bounus, the parameter fixes will receive the fixes for every item. These are the delta to apply to each source value in order to be sure that they will be in ascending order: some fix can be reduced but some analysis loop is required to achieve that optimization.
The following is a code to test the algorithm. If you set a breakpoint at the end of the loop, you'll be able to check the result for sequence you provided in your example.
var random = new Random((int)Stopwatch.GetTimestamp());
for (int ixLoop = -1; ixLoop < 100; ixLoop++)
{
int nCount;
int[] aItems;
// special case as the provided sample sequence
if (ixLoop == -1)
{
aItems = new[] { 2, 4, 8, 3, 1, 12 };
nCount = aItems.Length;
}
else
{
// Generates a random amount of items based on my screen's width
nCount = 4 + random.Next(21);
aItems = new int[nCount];
for (int ixItem = 0; ixItem < nCount; ixItem++)
{
// Keep the generated numbers below 30 for easier human analysis
aItems[ixItem] = random.Next(30);
}
}
Console.WriteLine("***");
Console.WriteLine(" # " + GetText(Enumerable.Range(0, nCount).ToArray()));
Console.WriteLine(" " + GetText(aItems));
int[] aFixes;
var nFix = ComputeMaxDelta(aItems, out aFixes);
// Computes the new values, that will be always in ascending order
var aNew = new int[aItems.Length];
for (int ixItem = 0; ixItem < aItems.Length; ixItem++)
{
aNew[ixItem] = aItems[ixItem] + aFixes[ixItem];
}
Console.WriteLine(" = " + nFix.ToString());
Console.WriteLine(" ! " + GetText(aFixes));
Console.WriteLine(" > " + GetText(aNew));
}
Regards,
Daniele.
Does anyone know a simple algorithm to check if a Sudoku-Configuration is valid? The simplest algorithm I came up with is (for a board of size n) in Pseudocode
for each row
for each number k in 1..n
if k is not in the row (using another for-loop)
return not-a-solution
..do the same for each column
But I'm quite sure there must be a better (in the sense of more elegant) solution. Efficiency is quite unimportant.
You need to check for all the constraints of Sudoku :
check the sum on each row
check the sum on each column
check for sum on each box
check for duplicate numbers on each row
check for duplicate numbers on each column
check for duplicate numbers on each box
that's 6 checks altogether.. using a brute force approach.
Some sort of mathematical optimization can be used if you know the size of the board (ie 3x3 or 9x9)
Edit: explanation for the sum constraint: Checking for the sum first (and stoping if the sum is not 45) is much faster (and simpler) than checking for duplicates. It provides an easy way of discarding a wrong solution.
Peter Norvig has a great article on solving sudoku puzzles (with python),
https://norvig.com/sudoku.html
Maybe it's too much for what you want to do, but it's a great read anyway
Check each row, column and box such that it contains the numbers 1-9 each, with no duplicates. Most answers here already discuss this.
But how to do that efficiently? Answer: Use a loop like
result=0;
for each entry:
result |= 1<<(value-1)
return (result==511);
Each number will set one bit of the result. If all 9 numbers are unique, the lowest 9
bits will be set.
So the "check for duplicates" test is just a check that all 9 bits are set, which is the same as testing result==511.
You need to do 27 of these checks.. one for each row, column, and box.
Just a thought: don't you need to also check the numbers in each 3x3 square?
I'm trying to figure out if it is possible to have the rows and columns conditions satisfied without having a correct sudoku
This is my solution in Python, I'm glad to see it's the shortest one yet :)
The code:
def check(sud):
zippedsud = zip(*sud)
boxedsud=[]
for li,line in enumerate(sud):
for box in range(3):
if not li % 3: boxedsud.append([]) # build a new box every 3 lines
boxedsud[box + li/3*3].extend(line[box*3:box*3+3])
for li in range(9):
if [x for x in [set(sud[li]), set(zippedsud[li]), set(boxedsud[li])] if x != set(range(1,10))]:
return False
return True
And the execution:
sudoku=[
[7, 5, 1, 8, 4, 3, 9, 2, 6],
[8, 9, 3, 6, 2, 5, 1, 7, 4],
[6, 4, 2, 1, 7, 9, 5, 8, 3],
[4, 2, 5, 3, 1, 6, 7, 9, 8],
[1, 7, 6, 9, 8, 2, 3, 4, 5],
[9, 3, 8, 7, 5, 4, 6, 1, 2],
[3, 6, 4, 2, 9, 7, 8, 5, 1],
[2, 8, 9, 5, 3, 1, 4, 6, 7],
[5, 1, 7, 4, 6, 8, 2, 3, 9]]
print check(sudoku)
Create an array of booleans for every row, column, and square. The array's index represents the value that got placed into that row, column, or square. In other words, if you add a 5 to the second row, first column, you would set rows[2][5] to true, along with columns[1][5] and squares[4][5], to indicate that the row, column, and square now have a 5 value.
Regardless of how your original board is being represented, this can be a simple and very fast way to check it for completeness and correctness. Simply take the numbers in the order that they appear on the board, and begin building this data structure. As you place numbers in the board, it becomes a O(1) operation to determine whether any values are being duplicated in a given row, column, or square. (You'll also want to check that each value is a legitimate number: if they give you a blank or a too-high number, you know that the board is not complete.) When you get to the end of the board, you'll know that all the values are correct, and there is no more checking required.
Someone also pointed out that you can use any form of Set to do this. Arrays arranged in this manner are just a particularly lightweight and performant form of a Set that works well for a small, consecutive, fixed set of numbers. If you know the size of your board, you could also choose to do bit-masking, but that's probably a little overly tedious considering that efficiency isn't that big a deal to you.
Create cell sets, where each set contains 9 cells, and create sets for vertical columns, horizontal rows, and 3x3 squares.
Then for each cell, simply identify the sets it's part of and analyze those.
You could extract all values in a set (row, column, box) into a list, sort it, then compare to '(1, 2, 3, 4, 5, 6, 7, 8, 9)
I did this once for a class project. I used a total of 27 sets to represent each row, column and box. I'd check the numbers as I added them to each set (each placement of a number causes the number to be added to 3 sets, a row, a column, and a box) to make sure the user only entered the digits 1-9. The only way a set could get filled is if it was properly filled with unique digits. If all 27 sets got filled, the puzzle was solved. Setting up the mappings from the user interface to the 27 sets was a bit tedious, but made the rest of the logic a breeze to implement.
It would be very interesting to check if:
when the sum of each row/column/box equals n*(n+1)/2
and the product equals n!
with n = number of rows or columns
this suffices the rules of a sudoku. Because that would allow for an algorithm of O(n^2), summing and multiplying the correct cells.
Looking at n = 9, the sums should be 45, the products 362880.
You would do something like:
for i = 0 to n-1 do
boxsum[i] := 0;
colsum[i] := 0;
rowsum[i] := 0;
boxprod[i] := 1;
colprod[i] := 1;
rowprod[i] := 1;
end;
for i = 0 to n-1 do
for j = 0 to n-1 do
box := (i div n^1/2) + (j div n^1/2)*n^1/2;
boxsum[box] := boxsum[box] + cell[i,j];
boxprod[box] := boxprod[box] * cell[i,j];
colsum[i] := colsum[i] + cell[i,j];
colprod[i] := colprod[i] * cell[i,j];
rowsum[j] := colsum[j] + cell[i,j];
rowprod[j] := colprod[j] * cell[i,j];
end;
end;
for i = 0 to n-1 do
if boxsum[i] <> 45
or colsum[i] <> 45
or rowsum[i] <> 45
or boxprod[i] <> 362880
or colprod[i] <> 362880
or rowprod[i] <> 362880
return false;
Some time ago, I wrote a sudoku checker that checks for duplicate number on each row, duplicate number on each column & duplicate number on each box. I would love it if someone could come up one with like a few lines of Linq code though.
char VerifySudoku(char grid[81])
{
for (char r = 0; r < 9; ++r)
{
unsigned int bigFlags = 0;
for (char c = 0; c < 9; ++c)
{
unsigned short buffer = r/3*3+c/3;
// check horizontally
bitFlags |= 1 << (27-grid[(r<<3)+r+c])
// check vertically
| 1 << (18-grid[(c<<3)+c+r])
// check subgrids
| 1 << (9-grid[(buffer<<3)+buffer+r%3*3+c%3]);
}
if (bitFlags != 0x7ffffff)
return 0; // invalid
}
return 1; // valid
}
if the sum and the multiplication of a row/col equals to the right number 45/362880
First, you would need to make a boolean, "correct". Then, make a for loop, as previously stated. The code for the loop and everything afterwards (in java) is as stated, where field is a 2D array with equal sides, col is another one with the same dimensions, and l is a 1D one:
for(int i=0; i<field.length(); i++){
for(int j=0; j<field[i].length; j++){
if(field[i][j]>9||field[i][j]<1){
checking=false;
break;
}
else{
col[field[i].length()-j][i]=field[i][j];
}
}
}
I don't know the exact algorithim to check the 3x3 boxes, but you should check all the rows in field and col with "/*array name goes here*/[i].contains(1)&&/*array name goes here*/[i].contains(2)" (continues until you reach the length of a row) inside another for loop.
def solution(board):
for i in board:
if sum(i) != 45:
return "Incorrect"
for i in range(9):
temp2 = []
for x in range(9):
temp2.append(board[i][x])
if sum(temp2) != 45:
return "Incorrect"
return "Correct"
board = []
for i in range(9):
inp = raw_input()
temp = [int(i) for i in inp]
board.append(temp)
print solution(board)
Here's a nice readable approach in Python:
from itertools import chain
def valid(puzzle):
def get_block(x,y):
return chain(*[puzzle[i][3*x:3*x+3] for i in range(3*y, 3*y+3)])
rows = [set(row) for row in puzzle]
columns = [set(column) for column in zip(*puzzle)]
blocks = [set(get_block(x,y)) for x in range(0,3) for y in range(0,3)]
return all(map(lambda s: s == set([1,2,3,4,5,6,7,8,9]), rows + columns + blocks))
Each 3x3 square is referred to as a block, and there are 9 of them in a 3x3 grid. It is assumed as the puzzle is input as a list of list, with each inner list being a row.
Let's say int sudoku[0..8,0..8] is the sudoku field.
bool CheckSudoku(int[,] sudoku)
{
int flag = 0;
// Check rows
for(int row = 0; row < 9; row++)
{
flag = 0;
for (int col = 0; col < 9; col++)
{
// edited : check range step (see comments)
if ((sudoku[row, col] < 1)||(sudoku[row, col] > 9))
{
return false;
}
// if n-th bit is set.. but you can use a bool array for readability
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
// set the n-th bit
flag |= (1 << sudoku[row, col]);
}
}
// Check columns
for(int col= 0; col < 9; col++)
{
flag = 0;
for (int row = 0; row < 9; row++)
{
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
flag |= (1 << sudoku[row, col]);
}
}
// Check 3x3 boxes
for(int box= 0; box < 9; box++)
{
flag = 0;
for (int ofs = 0; ofs < 9; ofs++)
{
int col = (box % 3) * 3;
int row = ((int)(box / 3)) * 3;
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
flag |= (1 << sudoku[row, col]);
}
}
return true;
}
Let's assume that your board goes from 1 - n.
We'll create a verification array, fill it and then verify it.
grid [0-(n-1)][0-(n-1)]; //this is the input grid
//each verification takes n^2 bits, so three verifications gives us 3n^2
boolean VArray (3*n*n) //make sure this is initialized to false
for i = 0 to n
for j = 0 to n
/*
each coordinate consists of three parts
row/col/box start pos, index offset, val offset
*/
//to validate rows
VArray( (0) + (j*n) + (grid[i][j]-1) ) = 1
//to validate cols
VArray( (n*n) + (i*n) + (grid[i][j]-1) ) = 1
//to validate boxes
VArray( (2*n*n) + (3*(floor (i/3)*n)+ floor(j/3)*n) + (grid[i][j]-1) ) = 1
next
next
if every array value is true then the solution is correct.
I think that will do the trick, although i'm sure i made a couple of stupid mistakes in there. I might even have missed the boat entirely.
array = [1,2,3,4,5,6,7,8,9]
sudoku = int [][]
puzzle = 9 #9x9
columns = map []
units = map [] # box
unit_l = 3 # box width/height
check_puzzle()
def strike_numbers(line, line_num, columns, units, unit_l):
count = 0
for n in line:
# check which unit we're in
unit = ceil(n / unit_l) + ceil(line_num / unit_l) # this line is wrong - rushed
if units[unit].contains(n): #is n in unit already?
return columns, units, 1
units[unit].add(n)
if columns[count].contains(n): #is n in column already?
return columns, units, 1
columns[count].add(n)
line.remove(n) #remove num from temp row
return columns, units, line.length # was a number not eliminated?
def check_puzzle(columns, sudoku, puzzle, array, units):
for (i=0;i< puzzle;i++):
columns, units, left_over = strike_numbers(sudoku[i], i, columns, units) # iterate through rows
if (left_over > 0): return false
Without thoroughly checking, off the top of my head, this should work (with a bit of debugging) while only looping twice. O(n^2) instead of O(3(n^2))
Here is paper by math professor J.F. Crook: A Pencil-and-Paper Algorithm for Solving Sudoku Puzzles
This paper was published in April 2009 and it got lots of publicity as definite Sudoku solution (check google for "J.F.Crook Sudoku" ).
Besides algorithm, there is also a mathematical proof that algorithm works (professor admitted that he does not find Sudoku very interesting, so he threw some math in paper to make it more fun).
I'd write an interface that has functions that receive the sudoku field and returns true/false if it's a solution.
Then implement the constraints as single validation classes per constraint.
To verify just iterate through all constraint classes and when all pass the sudoku is correct. To speedup put the ones that most likely fail to the front and stop in the first result that points to invalid field.
Pretty generic pattern. ;-)
You can of course enhance this to provide hints which field is presumably wrong and so on.
First constraint, just check if all fields are filled out. (Simple loop)
Second check if all numbers are in each block (nested loops)
Third check for complete rows and columns (almost same procedure as above but different access scheme)
Here is mine in C. Only pass each square once.
int checkSudoku(int board[]) {
int i;
int check[13] = { 0 };
for (i = 0; i < 81; i++) {
if (i % 9 == 0) {
check[9] = 0;
if (i % 27 == 0) {
check[10] = 0;
check[11] = 0;
check[12] = 0;
}
}
if (check[i % 9] & (1 << board[i])) {
return 0;
}
check[i % 9] |= (1 << board[i]);
if (check[9] & (1 << board[i])) {
return 0;
}
check[9] |= (1 << board[i]);
if (i % 9 < 3) {
if (check[10] & (1 << board[i])) {
return 0;
}
check[10] |= (1 << board[i]);
} else if (i % 9 < 6) {
if (check[11] & (1 << board[i])) {
return 0;
}
check[11] |= (1 << board[i]);
} else {
if (check[12] & (1 << board[i])) {
return 0;
}
check[12] |= (1 << board[i]);
}
}
}
Here is what I just did for this:
boolean checkers=true;
String checking="";
if(a.length/3==1){}
else{
for(int l=1; l<a.length/3; l++){
for(int n=0;n<3*l;n++){
for(int lm=1; lm<a[n].length/3; lm++){
for(int m=0;m<3*l;m++){
System.out.print(" "+a[n][m]);
if(a[n][m]<=0){
System.out.print(" (Values must be positive!) ");
}
if(n==0){
if(m!=0){
checking+=", "+a[n][m];
}
else{
checking+=a[n][m];
}
}
else{
checking+=", "+a[n][m];
}
}
}
System.out.print(" "+checking);
System.out.println();
}
}
for (int i=1;i<=a.length*a[1].length;i++){
if(checking.contains(Integer.toString(i))){
}
else{
checkers=false;
}
}
}
checkers=checkCol(a);
if(checking.contains("-")&&!checking.contains("--")){
checkers=false;
}
System.out.println();
if(checkers==true){
System.out.println("This is correct! YAY!");
}
else{
System.out.println("Sorry, it's not right. :-(");
}
}
private static boolean checkCol(int[][]a){
boolean checkers=true;
int[][]col=new int[][]{{0,0,0},{0,0,0},{0,0,0}};
for(int i=0; i<a.length; i++){
for(int j=0; j<a[i].length; j++){
if(a[i][j]>9||a[i][j]<1){
checkers=false;
break;
}
else{
col[a[i].length-j][i]=a[i][j];
}
}
}
String alia="";
for(int i=0; i<col.length; i++){
for(int j=1; j<=col[i].length; j++){
alia=a[i].toString();
if(alia.contains(""+j)){
alia=col[i].toString();
if(alia.contains(""+j)){}
else{
checkers=false;
}
}
else{
checkers=false;
}
}
}
return checkers;
}
You can check if sudoku is solved, in these two similar ways:
Check if the number is unique in each row, column and block.
A naive solution would be to iterate trough every square and check if the number is unique in the row, column block that number occupies.
But there is a better way.
Sudoku is solved if every row, column and block contains a permutation of the numbers (1 trough 9)
This only requires to check every row, column and block, instead of doing that for every number. A simple implementation would be to have a bitfield of numbers 1 trough 9 and remove them when you iterate the columns, rows and blocks. If you try to remove a missing number or if the field isn't empty when you finish then sudoku isn't correctly solved.
Here's a very concise version in Swift, that only uses an array of Ints to track the groups of 9 numbers, and only iterates over the sudoku once.
import UIKit
func check(_ sudoku:[[Int]]) -> Bool {
var groups = Array(repeating: 0, count: 27)
for x in 0...8 {
for y in 0...8 {
groups[x] += 1 << sudoku[x][y] // Column (group 0 - 8)
groups[y + 9] += 1 << sudoku[x][y] // Row (group 9 - 17)
groups[(x + y * 9) / 9 + 18] += 1 << sudoku[x][y] // Box (group 18 - 27)
}
}
return groups.filter{ $0 != 1022 }.count == 0
}
let sudoku = [
[7, 5, 1, 8, 4, 3, 9, 2, 6],
[8, 9, 3, 6, 2, 5, 1, 7, 4],
[6, 4, 2, 1, 7, 9, 5, 8, 3],
[4, 2, 5, 3, 1, 6, 7, 9, 8],
[1, 7, 6, 9, 8, 2, 3, 4, 5],
[9, 3, 8, 7, 5, 4, 6, 1, 2],
[3, 6, 4, 2, 9, 7, 8, 5, 1],
[2, 8, 9, 5, 3, 1, 4, 6, 7],
[5, 1, 7, 4, 6, 8, 2, 3, 9]
]
if check(sudoku) {
print("Pass")
} else {
print("Fail")
}
One minor optimization you can make is that you can check for duplicates in a row, column, or box in O(n) time rather than O(n^2): as you iterate through the set of numbers, you add each one to a hashset. Depending on the language, you may actually be able to use a true hashset, which is constant time lookup and insertion; then checking for duplicates can be done in the same step by seeing if the insertion was successful or not. It's a minor improvement in the code, but going from O(n^2) to O(n) is a significant optimization.