Given N point on a 2D plane as pair of (x, y) co-ordinates, we need to find maximum number of point which lie on the same line.

Examples:

Input : points[] = {-1, 1}, {0, 0}, {1, 1}, {2, 2}, {3, 3}, {3, 4} Output : 4 Then maximum number of point which lie on same line are 4, those point are {0, 0}, {1, 1}, {2, 2}, {3, 3}

We can solve above problem by following approach – For each point p, calculate its slope with other points and use a map to record how many points have same slope, by which we can find out how many points are on same line with p as their one point. For each point keep doing the same thing and update the maximum number of point count found so far.

Some things to note in implementation are:

1) if two point are (x1, y1) and (x2, y2) then their slope will be (y2 – y1) / (x2 – x1) which can be a double value and can cause precision problems. To get rid of the precision problems, we treat slope as pair ((y2 – y1), (x2 – x1)) instead of ratio and reduce pair by their gcd before inserting into map. In below code points which are vertical or repeated are treated separately.

2) If we use unordered_map in c++ or HashMap in Java for storing the slope pair, then total time complexity of solution will be O(n^2)

## C++

`/* C/C++ program to find maximum number of point` `which lie on same line */` `#include <bits/stdc++.h>` `#include <boost/functional/hash.hpp>` ` ` `using` `namespace` `std;` ` ` `// method to find maximum colinear point` `int` `maxPointOnSameLine(vector< pair<` `int` `, ` `int` `> > points)` `{` ` ` `int` `N = points.size();` ` ` `if` `(N < 2)` ` ` `return` `N;` ` ` ` ` `int` `maxPoint = 0;` ` ` `int` `curMax, overlapPoints, verticalPoints;` ` ` ` ` `// here since we are using unordered_map ` ` ` `// which is based on hash function ` ` ` `//But by default we don't have hash function for pairs` ` ` `//so we'll use hash function defined in Boost library` ` ` `unordered_map<pair<` `int` `, ` `int` `>, ` `int` `,boost::` ` ` `hash<pair<` `int` `, ` `int` `> > > slopeMap;` ` ` ` ` `// looping for each point` ` ` `for` `(` `int` `i = 0; i < N; i++)` ` ` `{` ` ` `curMax = overlapPoints = verticalPoints = 0;` ` ` ` ` `// looping from i + 1 to ignore same pair again` ` ` `for` `(` `int` `j = i + 1; j < N; j++)` ` ` `{` ` ` `// If both point are equal then just` ` ` `// increase overlapPoint count` ` ` `if` `(points[i] == points[j])` ` ` `overlapPoints++;` ` ` ` ` `// If x co-ordinate is same, then both` ` ` `// point are vertical to each other` ` ` `else` `if` `(points[i].first == points[j].first)` ` ` `verticalPoints++;` ` ` ` ` `else` ` ` `{` ` ` `int` `yDif = points[j].second - points[i].second;` ` ` `int` `xDif = points[j].first - points[i].first;` ` ` `int` `g = __gcd(xDif, yDif);` ` ` ` ` `// reducing the difference by their gcd` ` ` `yDif /= g;` ` ` `xDif /= g;` ` ` ` ` `// increasing the frequency of current slope` ` ` `// in map` ` ` `slopeMap[make_pair(yDif, xDif)]++;` ` ` `curMax = max(curMax, slopeMap[make_pair(yDif, xDif)]);` ` ` `}` ` ` ` ` `curMax = max(curMax, verticalPoints);` ` ` `}` ` ` ` ` `// updating global maximum by current point's maximum` ` ` `maxPoint = max(maxPoint, curMax + overlapPoints + 1);` ` ` ` ` `// printf("maximum colinear point ` ` ` `// which contains current point ` ` ` `// are : %d\n", curMax + overlapPoints + 1);` ` ` `slopeMap.clear();` ` ` `}` ` ` ` ` `return` `maxPoint;` `}` ` ` `// Driver code` `int` `main()` `{` ` ` `const` `int` `N = 6;` ` ` `int` `arr[N][2] = {{-1, 1}, {0, 0}, {1, 1}, {2, 2},` ` ` `{3, 3}, {3, 4}};` ` ` ` ` `vector< pair<` `int` `, ` `int` `> > points;` ` ` `for` `(` `int` `i = 0; i < N; i++)` ` ` `points.push_back(make_pair(arr[i][0], arr[i][1]));` ` ` ` ` `cout << maxPointOnSameLine(points) << endl;` ` ` ` ` `return` `0;` `}` |

## Python3

`def` `maxPoints(points):` ` ` `n` `=` `len` `(points)` ` ` ` ` `# upto two points all points will be part of the line` ` ` `if` `n<` `3` `:` ` ` `return` `n` ` ` ` ` `max_val` `=` `0` ` ` ` ` `# looping for each point` ` ` `for` `i ` `in` `points:` ` ` ` ` `# Creating a dictionary for every new` ` ` `# point to save memory` ` ` `d ` `=` `{} ` ` ` `dups ` `=` `0` ` ` `cur_max ` `=` `0` ` ` ` ` `# pairing with all other points ` ` ` `for` `j ` `in` `points:` ` ` `if` `i!` `=` `j:` ` ` `if` `j[` `0` `]` `=` `=` `i[` `0` `]: ` `#vertical line` ` ` `slope` `=` `'inf'` ` ` `else` `:` ` ` `slope` `=` `float` `(j[` `1` `]` `-` `i[` `1` `])` `/` `float` `(j[` `0` `]` `-` `i[` `0` `])` ` ` ` ` `# Increasing the frequency of slope and ` ` ` `# updating cur_max for current point(i) ` ` ` `d[slope] ` `=` `d.get(slope,` `0` `)` `+` `1` ` ` `cur_max` `=` `max` `(cur_max, d[slope])` ` ` ` ` `# if both points are equal same increase ` ` ` `# duplicates count.` ` ` `# Please note that this will also increment` ` ` `# when we map it with itself.` ` ` `# we still do it because we will not have to` ` ` `# add the extra one at the end.` ` ` `else` `:` ` ` `dups` `+` `=` `1` ` ` ` ` `max_val` `=` `max` `(max_val, cur_max` `+` `dups)` ` ` ` ` `return` `max_val` ` ` `# Driver code` `points ` `=` `[(` `-` `1` `, ` `1` `), (` `0` `, ` `0` `), (` `1` `, ` `1` `), (` `2` `, ` `2` `), (` `3` `, ` `3` `), (` `3` `, ` `4` `)]` `print` `(maxPoints(points))` |

This article is contributed by **Utkarsh Trivedi**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the **Essential Maths for CP Course** at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer **Complete Interview Preparation Course****.**