Count maximum points on same line

Last Updated : 24 Aug, 2022

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:

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.
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) and space complexity will be O(n).

Implementation:

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 collinear 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 collinear 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;
}

Java

/* Java program to find maximum number of point
which lie on same line */
import java.util.*;

class GFG {
    static int gcd(int p, int q)
    {
        if (q == 0) {
            return p;
        }
        int r = p % q;
        return gcd(q, r);
    }

    static int N = 6;

    // method to find maximum collinear point
    static int maxPointOnSameLine(int[][] points)
    {
        if (N < 2)
            return N;
        int maxPoint = 0;
        int curMax, overlapPoints, verticalPoints;

        HashMap<String, Integer> slopeMap = new HashMap<>();
        // 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][0] == points[j][0]
                    && points[i][1] == points[j][1])
                    overlapPoints++;

                // If x co-ordinate is same, then both
                // point are vertical to each other
                else if (points[i][0] == points[j][0])
                    verticalPoints++;

                else {
                    int yDif = points[j][1] - points[i][1];
                    int xDif = points[j][0] - points[i][0];
                    int g = gcd(xDif, yDif);

                    // reducing the difference by their gcd
                    yDif /= g;
                    xDif /= g;

                    // Convert the pair into a string to use
                    // as dictionary key
                    String pair = (yDif) + " " + (xDif);
                    if (!slopeMap.containsKey(pair))
                        slopeMap.put(pair, 0);

                    // increasing the frequency of current
                    // slope in map
                    slopeMap.put(pair,
                                 slopeMap.get(pair) + 1);
                    curMax = Math.max(curMax,
                                      slopeMap.get(pair));
                }

                curMax = Math.max(curMax, verticalPoints);
            }

            // updating global maximum by current point's
            // maximum
            maxPoint = Math.max(maxPoint,
                                curMax + overlapPoints + 1);
            slopeMap.clear();
        }

        return maxPoint;
    }

    public static void main(String[] args)
    {
        int points[][] = { { -1, 1 }, { 0, 0 }, { 1, 1 },
                           { 2, 2 },  { 3, 3 }, { 3, 4 } };
        System.out.println(maxPointOnSameLine(points));
    }
}

Python3

# python3 program to find maximum number of 2D points that lie on the same line.

from collections import defaultdict
from math import gcd
from typing import DefaultDict, List, Tuple

IntPair = Tuple[int, int]


def normalized_slope(a: IntPair, b: IntPair) -> IntPair:
    """
    Returns normalized (rise, run) tuple. We won't return the actual rise/run
    result in order to avoid floating point math, which leads to faulty
    comparisons.

    See
    https://en.wikipedia.org/wiki/Floating-point_arithmetic#Accuracy_problems
    """
    run = b[0] - a[0]

    # normalize undefined slopes to (1, 0)
    if run == 0:
        return (1, 0)

    # normalize to left-to-right
    if run < 0:
        a, b = b, a
        run = b[0] - a[0]

    rise = b[1] - a[1]
    # Normalize by greatest common divisor.
    # math.gcd only works on positive numbers.
    gcd_ = gcd(abs(rise), run)
    return (
        rise // gcd_,
        run // gcd_,
    )


def maximum_points_on_same_line(points: List[List[int]]) -> int:
    # You need at least 3 points to potentially have non-collinear points.
    # For [0, 2] points, all points are on the same line.
    if len(points) < 3:
        return len(points)

    # Note that every line we find will have at least 2 points.
    # There will be at least one line because len(points) >= 3.
    # Therefore, it's safe to initialize to 0.
    max_val = 0

    for a_index in range(0, len(points) - 1):
        # All lines in this iteration go through point a.
        # Note that lines a-b and a-c cannot be parallel.
        # Therefore, if lines a-b and a-c have the same slope, they're the same
        # line.
        a = tuple(points[a_index])
        # Fresh lines already have a, so default=1
        slope_counts: DefaultDict[IntPair, int] = defaultdict(lambda: 1)

        for b_index in range(a_index + 1, len(points)):
            b = tuple(points[b_index])
            slope_counts[normalized_slope(a, b)] += 1

        max_val = max(
            max_val,
            max(slope_counts.values()),
        )

    return max_val


print(maximum_points_on_same_line([
    [-1, 1],
    [0, 0],
    [1, 1],
    [2, 2],
    [3, 3],
    [3, 4],
]))

# This code is contributed by Jose Alvarado Torre

/* C# program to find maximum number of point
which lie on same line */
using System;
using System.Collections.Generic;

class GFG {

  static int gcd(int p, int q)
  {
    if (q == 0) {
      return p;
    }

    int r = p % q;

    return gcd(q, r);
  }

  static int N = 6;

  // method to find maximum collinear point
  static int maxPointOnSameLine(int[, ] points)
  {
    if (N < 2)
      return N;

    int maxPoint = 0;
    int curMax, overlapPoints, verticalPoints;

    Dictionary<string, int> slopeMap
      = new Dictionary<string, int>();

    // 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, 0] == points[j, 0]
            && points[i, 1] == points[j, 1])
          overlapPoints++;

        // If x co-ordinate is same, then both
        // point are vertical to each other
        else if (points[i, 0] == points[j, 0])
          verticalPoints++;

        else {
          int yDif = points[j, 1] - points[i, 1];
          int xDif = points[j, 0] - points[i, 0];
          int g = gcd(xDif, yDif);

          // reducing the difference by their gcd
          yDif /= g;
          xDif /= g;

          // Convert the pair into a string to use
          // as dictionary key
          string pair = Convert.ToString(yDif)
            + " "
            + Convert.ToString(xDif);
          if (!slopeMap.ContainsKey(pair))
            slopeMap[pair] = 0;

          // increasing the frequency of current
          // slope in map
          slopeMap[pair]++;
          curMax
            = Math.Max(curMax, slopeMap[pair]);
        }

        curMax = Math.Max(curMax, verticalPoints);
      }

      // updating global maximum by current point's
      // maximum
      maxPoint = Math.Max(maxPoint,
                          curMax + overlapPoints + 1);
      slopeMap.Clear();
    }

    return maxPoint;
  }

  // Driver code
  public static void Main(string[] args)
  {

    int[, ] points = { { -1, 1 }, { 0, 0 }, { 1, 1 },
                      { 2, 2 },  { 3, 3 }, { 3, 4 } };

    Console.WriteLine(maxPointOnSameLine(points));
  }
}

// This code is contributed by phasing17

JavaScript

/* JavaScript program to find maximum number of point
which lie on same line */

// Function to find gcd of two numbers
let gcd = function(a, b) {
  if (!b) {
    return a;
  }
  return gcd(b, a % b);
}

// method to find maximum collinear point
function maxPointOnSameLine(points){
    let N = points.length;
    if (N < 2){
        return N;
    }        

    let maxPoint = 0;
    let curMax, overlapPoints, verticalPoints;

    // Creating a map for storing the data. 
    let slopeMap = new Map();

    // looping for each point
    for (let i = 0; i < N; i++)
    {
        curMax = 0;
        overlapPoints = 0;
        verticalPoints = 0;
        
        // looping from i + 1 to ignore same pair again
        for (let 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][0] === points[j][0]){
                verticalPoints++;
            }
            else{
                let yDif = points[j][1] - points[i][1];
                let xDif = points[j][0] - points[i][0];
                let g = gcd(xDif, yDif);

                // reducing the difference by their gcd
                yDif = Math.floor(yDif/g);
                xDif = Math.floor(xDif/g);
                
                // increasing the frequency of current slope. 
                let tmp = [yDif, xDif];
                if(slopeMap.has(tmp.join(''))){
                    slopeMap.set(tmp.join(''), slopeMap.get(tmp.join('')) + 1);
                }
                else{
                    slopeMap.set(tmp.join(''), 1);
                }
                
                curMax = Math.max(curMax, slopeMap.get(tmp.join('')));
            }
            
            curMax = Math.max(curMax, verticalPoints);
        }

        // updating global maximum by current point's maximum
        maxPoint = Math.max(maxPoint, curMax + overlapPoints + 1);
        
        // printf("maximum collinear point
        // which contains current point
        // are : %d\n", curMax + overlapPoints + 1);
        slopeMap.clear();
    }
   
    return maxPoint;
}

// Driver code
{
    let N = 6;
    let arr = [[-1, 1], [0, 0], [1, 1], [2, 2],
                    [3, 3], [3, 4]];

    console.log(maxPointOnSameLine(arr));
}

// The code is contributed by Gautam goel (gautamgoel962)

Output

Time Complexity: O(n²logn), where n denoting length of string.
Auxiliary Space: O(n).

Minimum lines to cover all points

Utkarsh Trivedi

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Count maximum points on same line

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