 Open in App
Not now

# Find minimum value of y for the given x values in Q queries from all the given set of lines

• Last Updated : 16 Sep, 2022

Given a 2-dimensional array arr[][] consisting of slope(m) and intercept(c) for a large number of lines of the form y = mx + c and Q queries such that each query contains a value x. The task is to find the minimum value of y for the given x values from all the given sets of lines.

Examples:

Input: arr[][] ={ {1, 1}, {0, 0}, {-3, 3} }, Q = {-2, 2, 0}
Output: -1, -3, 0
Explanation:
For query x = -2, y values from the equations are -1, 0, 9. So the minimum value is -1
Similarly, for x = 2, y values are 3, 0, -3. So the minimum value is -3
And for x = 0, values of y = 1, 0, 3 so min value is 0.

Input: arr[][] ={ {5, 6}, {3, 2}, {7, 3} }, Q = { 1, 2, 30 }
Output: 5, 8, 92

Naive Approach: The naive approach is to substitute the values of x in every line and compute the minimum of all the lines. For each query, it will take O(N) time and so the complexity of the solution becomes O(Q * N) where N is the number of lines and Q is the number of queries.

Efficient approach: The idea is to use convex hull trick:

• From the given set of lines, the lines which carry no significance (for any value of x they never give the minimal value y) can be found and deleted thereby reducing the set.
• Now, if the ranges (l, r) can be found where each line gives the minimum value, then each query can be answered using binary search.
• Therefore, a sorted vector of lines with decreasing order of slopes is created and the lines are inserted in decreasing order of the slopes.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the above approach` `#include ``using` `namespace` `std;` `struct` `Line {``    ``int` `m, c;` `public``:``    ``// Sort the line in decreasing``    ``// order of their slopes``    ``bool` `operator<(Line l)``    ``{` `        ``// If slopes arent equal``        ``if` `(m != l.m)``            ``return` `m > l.m;` `        ``// If the slopes are equal``        ``else``            ``return` `c > l.c;``    ``}` `    ``// Checks if line L3 or L1 is better than L2``    ``// Intersection of Line 1 and``    ``// Line 2 has x-coordinate (b1-b2)/(m2-m1)``    ``// Similarly for Line 1 and``    ``// Line 3 has x-coordinate (b1-b3)/(m3-m1)``    ``// Cross multiplication will give the below result``    ``bool` `check(Line L1, Line L2, Line L3)``    ``{``        ``return` `(L3.c - L1.c) * (L1.m - L2.m)``               ``< (L2.c - L1.c) * (L1.m - L3.m);``    ``}``};` `struct` `Convex_HULL_Trick {` `    ``// To store the lines``    ``vector l;` `    ``// Add the line to the set of lines``    ``void` `add(Line newLine)``    ``{` `        ``int` `n = l.size();` `        ``// To check if after adding the new line``        ``// whether old lines are``        ``// losing significance or not``        ``while` `(n >= 2``               ``&& newLine.check(l[n - 2],``                                ``l[n - 1],``                                ``newLine)) {``            ``n--;``        ``}` `        ``l.resize(n);` `        ``// Add the present line``        ``l.push_back(newLine);``    ``}` `    ``// Function to return the y coordinate``    ``// of the specified line for the given coordinate``    ``int` `value(``int` `in, ``int` `x)``    ``{``        ``return` `l[in].m * x + l[in].c;``    ``}` `    ``// Function to Return the minimum value``    ``// of y for the given x coordinate``    ``int` `minQuery(``int` `x)``    ``{``        ``// if there is no lines``        ``if` `(l.empty())``            ``return` `INT_MAX;` `        ``int` `low = 0, high = (``int``)l.size() - 2;` `        ``// Binary search``        ``while` `(low <= high) {``            ``int` `mid = (low + high) / 2;` `            ``if` `(value(mid, x) > value(mid + 1, x))``                ``low = mid + 1;``            ``else``                ``high = mid - 1;``        ``}` `        ``return` `value(low, x);``    ``}``};` `// Driver code``int` `main()``{``    ``Line lines[] = { { 1, 1 },``                     ``{ 0, 0 },``                     ``{ -3, 3 } };``    ``int` `Q[] = { -2, 2, 0 };``    ``int` `n = 3, q = 3;``    ``Convex_HULL_Trick cht;` `    ``// Sort the lines``    ``sort(lines, lines + n);` `    ``// Add the lines``    ``for` `(``int` `i = 0; i < n; i++)``        ``cht.add(lines[i]);` `    ``// For each query in Q``    ``for` `(``int` `i = 0; i < q; i++) {``        ``int` `x = Q[i];``        ``cout << cht.minQuery(x) << endl;``    ``}` `    ``return` `0;``}`

## Java

 `// Java implementation of the above approach``import` `java.util.ArrayList;``import` `java.util.Arrays;` `class` `GFG{` `static` `class` `Line ``implements` `Comparable``{``    ``int` `m, c;` `    ``public` `Line(``int` `m, ``int` `c)``    ``{``        ``this``.m = m;``        ``this``.c = c;``    ``}` `    ``// Sort the line in decreasing``    ``// order of their slopes``    ``@Override``    ``public` `int` `compareTo(Line l)``    ``{``        ` `        ``// If slopes arent equal``        ``if` `(m != l.m)``            ``return` `l.m - ``this``.m;` `        ``// If the slopes are equal``        ``else``            ``return` `l.c - ``this``.c;``    ``}` `    ``// Checks if line L3 or L1 is better than L2``    ``// Intersection of Line 1 and``    ``// Line 2 has x-coordinate (b1-b2)/(m2-m1)``    ``// Similarly for Line 1 and``    ``// Line 3 has x-coordinate (b1-b3)/(m3-m1)``    ``// Cross multiplication will give the below result``    ``boolean` `check(Line L1, Line L2, Line L3)``    ``{``        ``return` `(L3.c - L1.c) * (L1.m - L2.m) <``               ``(L2.c - L1.c) * (L1.m - L3.m);``    ``}``}` `static` `class` `Convex_HULL_Trick``{``    ` `    ``// To store the lines``    ``ArrayList l = ``new` `ArrayList<>();` `    ``// Add the line to the set of lines``    ``void` `add(Line newLine)``    ``{``        ``int` `n = l.size();` `        ``// To check if after adding the new``        ``// line whether old lines are``        ``// losing significance or not``        ``while` `(n >= ``2` `&&``               ``newLine.check(l.get(n - ``2``),``                             ``l.get(n - ``1``), newLine))``        ``{``            ``n--;``        ``}` `        ``// l = new Line[n];` `        ``// Add the present line``        ``l.add(newLine);``    ``}` `    ``// Function to return the y coordinate``    ``// of the specified line for the given``    ``// coordinate``    ``int` `value(``int` `in, ``int` `x)``    ``{``        ``return` `l.get(in).m * x + l.get(in).c;``    ``}` `    ``// Function to Return the minimum value``    ``// of y for the given x coordinate``    ``int` `minQuery(``int` `x)``    ``{``        ` `        ``// If there is no lines``        ``if` `(l.isEmpty())``            ``return` `Integer.MAX_VALUE;` `        ``int` `low = ``0``, high = (``int``)l.size() - ``2``;` `        ``// Binary search``        ``while` `(low <= high)``        ``{``            ``int` `mid = (low + high) / ``2``;` `            ``if` `(value(mid, x) > value(mid + ``1``, x))``                ``low = mid + ``1``;``            ``else``                ``high = mid - ``1``;``        ``}``        ``return` `value(low, x);``    ``}``};` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``Line[] lines = { ``new` `Line(``1``, ``1``),``                     ``new` `Line(``0``, ``0``),``                     ``new` `Line(-``3``, ``3``) };``    ``int` `Q[] = { -``2``, ``2``, ``0` `};``    ``int` `n = ``3``, q = ``3``;``    ` `    ``Convex_HULL_Trick cht = ``new` `Convex_HULL_Trick();` `    ``// Sort the lines``    ``Arrays.sort(lines);` `    ``// Add the lines``    ``for``(``int` `i = ``0``; i < n; i++)``        ``cht.add(lines[i]);` `    ``// For each query in Q``    ``for``(``int` `i = ``0``; i < q; i++)``    ``{``        ``int` `x = Q[i];``        ``System.out.println(cht.minQuery(x));``    ``}``}``}` `// This code is contributed by sanjeev2552`

## Python3

 `# Python3 implementation of the above approach``class` `Line: ``    ` `    ``def` `__init__(``self``, a ``=` `0``, b ``=` `0``):``     ` `        ``self``.m ``=` `a;``        ``self``.c ``=` `b;``     ` `    ``# Sort the line in decreasing``    ``# order of their slopes``    ``def`  `__gt__(``self``, l):``     `  `        ``# If slopes arent equal``        ``if` `(``self``.m !``=` `l.m):``            ``return` `self``.m > l.m;` `        ``# If the slopes are equal``        ``else``:``            ``return` `self``.c > l.c;``     `  `    ``# Checks if line L3 or L1 is better than L2``    ``# Intersection of Line 1 and``    ``# Line 2 has x-coordinate (b1-b2)/(m2-m1)``    ``# Similarly for Line 1 and``    ``# Line 3 has x-coordinate (b1-b3)/(m3-m1)``    ``# Cross multiplication will give the below result``    ``def` `check(``self``, L1, L2, L3):``     ` `        ``return` `(L3.c ``-` `L1.c) ``*` `(L1.m ``-` `L2.m)  < (L2.c ``-` `L1.c) ``*` `(L1.m ``-` `L3.m);``     ` ` `  `class` `Convex_HULL_Trick  :` `    ``# To store the lines``    ``def` `__init__(``self``):``     ` `        ``self``.l ``=` `[];``     `  `    ``# Add the line to the set of lines``    ``def` `add(``self``, newLine):``     `  `        ``n ``=` `len``(``self``.l)` `        ``# To check if after adding the new line``        ``# whether old lines are``        ``# losing significance or not``        ``while` `(n >``=` `2` `and` `newLine.check((``self``.l)[n ``-` `2``], (``self``.l)[n ``-` `1``], newLine)):``            ``n ``-``=` `1``;``         ` `        ``# Add the present line``        ``(``self``.l).append(newLine);``     ` `    ``# Function to return the y coordinate``    ``# of the specified line for the given coordinate``    ``def` `value(``self``, ind, x):``     ` `        ``return` `(``self``.l)[ind].m ``*` `x ``+` `(``self``.l)[ind].c;``     ` `    ``# Function to Return the minimum value``    ``# of y for the given x coordinate``    ``def` `minQuery(``self``, x):``     ` `        ``# if there is no lines``        ``if` `(``len``(``self``.l) ``=``=` `0``):``            ``return` `99999999999``;` `        ``low ``=` `0``        ``high ``=` `len``(``self``.l) ``-` `2``;` `        ``# Binary search``        ``while` `(low <``=` `high)  :``            ``mid ``=` `int``((low ``+` `high) ``/` `2``);` `            ``if` `(``self``.value(mid, x) > ``self``.value(mid ``+` `1``, x)):``                ``low ``=` `mid ``+` `1``;``            ``else``:``                ``high ``=` `mid ``-` `1``;``         `  `        ``return` `self``.value(low, x);``     ` `# Driver code``lines ``=` `[ Line(``1``, ``1``), Line(``0``, ``0``), Line(``-``3``, ``3``)]` `Q ``=` `[ ``-``2``, ``2``, ``0` `];``n ``=` `3``q ``=` `3``;``cht ``=` `Convex_HULL_Trick();` `# Sort the lines``lines.sort()` `# Add the lines``for` `i ``in` `range``(n):``    ``cht.add(lines[i]);` `# For each query in Q``for` `i ``in` `range``(q):``    ``x ``=` `Q[i];``    ``print``(cht.minQuery(x));` `# This code is contributed by phasing17`

## C#

 `// C# implementation of the above approach` `using` `System;``using` `System.Collections.Generic;``using` `System.Collections;` `class` `Line : IComparable {``    ``public` `int` `m, c;` `    ``public` `Line(``int` `m, ``int` `c)``    ``{``        ``this``.m = m;``        ``this``.c = c;``    ``}` `    ``// Sort the line in decreasing``    ``// order of their slopes``    ``public` `int` `CompareTo(Line l)` `    ``{` `        ``// If slopes arent equal``        ``if` `(m != l.m)``            ``return` `l.m - ``this``.m;` `        ``// If the slopes are equal``        ``else``            ``return` `l.c - ``this``.c;``    ``}` `    ``// Checks if line L3 or L1 is better than L2``    ``// intersection of Line 1 and``    ``// Line 2 has x-coordinate (b1-b2)/(m2-m1)``    ``// Similarly for Line 1 and``    ``// Line 3 has x-coordinate (b1-b3)/(m3-m1)``    ``// Cross multiplication will give the below result``    ``public` `bool` `check(Line L1, Line L2, Line L3)``    ``{``        ``return` `(L3.c - L1.c) * (L1.m - L2.m)``            ``< (L2.c - L1.c) * (L1.m - L3.m);``    ``}``};` `class` `Convex_HULL_Trick {` `    ``// To store the lines``    ``public` `List l = ``new` `List();` `    ``// Add the line to the set of lines``    ``public` `void` `add(Line newLine)``    ``{``        ``int` `n = l.Count;` `        ``// To check if after adding the new``        ``// line whether old lines are``        ``// losing significance or not``        ``while` `(``            ``n >= 2``            ``&& newLine.check(l[n - 2], l[n - 1], newLine)) {``            ``n--;``        ``}` `        ``// l = new Line[n];` `        ``// Add the present line``        ``l.Add(newLine);``    ``}` `    ``// Function to return the y coordinate``    ``// of the specified line for the given``    ``// coordinate``    ``public` `int` `value(``int` `ind, ``int` `x)``    ``{``        ``return` `l[ind].m * x + l[ind].c;``    ``}` `    ``// Function to Return the minimum value``    ``// of y for the given x coordinate``    ``public` `int` `minQuery(``int` `x)``    ``{` `        ``// If there is no lines``        ``if` `(l.Count == 0)``            ``return` `Int32.MaxValue;` `        ``int` `low = 0, high = l.Count - 2;` `        ``// Binary search``        ``while` `(low <= high) {``            ``int` `mid = (low + high) / 2;` `            ``if` `(value(mid, x) > value(mid + 1, x))``                ``low = mid + 1;``            ``else``                ``high = mid - 1;``        ``}``        ``return` `value(low, x);``    ``}``};` `class` `GFG {` `    ``// Driver code``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``Line[] lines = { ``new` `Line(1, 1), ``new` `Line(0, 0),``                         ``new` `Line(-3, 3) };``        ``int``[] Q = { -2, 2, 0 };``        ``int` `n = 3, q = 3;` `        ``Convex_HULL_Trick cht = ``new` `Convex_HULL_Trick();` `        ``// Sort the lines``        ``Array.Sort(lines);` `        ``// Add the lines``        ``for` `(``int` `i = 0; i < n; i++)``            ``cht.add(lines[i]);` `        ``// For each query in Q``        ``for` `(``int` `i = 0; i < q; i++) {``            ``int` `x = Q[i];``            ``Console.WriteLine(cht.minQuery(x));``        ``}``    ``}``}` `// This code is contributed by phasing17`

## Javascript

 `// JavaScript implementation of the above approach` `class Line {``    ``constructor(a = 0, b = 0)``    ``{``        ``this``.m = a;``        ``this``.c = b;``    ``}` `    ``// Sort the line in decreasing``    ``// order of their slopes``    ``cmp()``    ``{` `        ``// If slopes arent equal``        ``if` `(``this``.m != l.m)``            ``return` `this``.m > l.m;` `        ``// If the slopes are equal``        ``else``            ``return` `this``.c > l.c;``    ``}` `    ``// Checks if line L3 or L1 is better than L2``    ``// Intersection of Line 1 and``    ``// Line 2 has x-coordinate (b1-b2)/(m2-m1)``    ``// Similarly for Line 1 and``    ``// Line 3 has x-coordinate (b1-b3)/(m3-m1)``    ``// Cross multiplication will give the below result``    ``check(L1, L2, L3)``    ``{``        ``return` `(L3.c - L1.c) * (L1.m - L2.m)``               ``< (L2.c - L1.c) * (L1.m - L3.m);``    ``}``};` `class Convex_HULL_Trick {` `    ``// To store the lines``    ``constructor()``    ``{``        ``this``.l = [];``    ``}` `    ``// Add the line to the set of lines``     ``add(newLine)``    ``{` `        ``let n = (``this``.l).length` `        ``// To check if after adding the new line``        ``// whether old lines are``        ``// losing significance or not``        ``while` `(n >= 2``               ``&& newLine.check((``this``.l)[n - 2],``                                ``(``this``.l)[n - 1],``                                ``newLine)) {``            ``n--;``        ``}``        ` `        `  `        ``// Add the present line``        ``(``this``.l).push(newLine);``    ``}` `    ``// Function to return the y coordinate``    ``// of the specified line for the given coordinate``     ``value( ind, x)``    ``{``        ``return` `(``this``.l)[ind].m * x + (``this``.l)[ind].c;``    ``}` `    ``// Function to Return the minimum value``    ``// of y for the given x coordinate``     ``minQuery( x)``    ``{``        ``// if there is no lines``        ``if` `((``this``.l).length == 0)``            ``return` `99999999999;` `        ``let low = 0, high = (``this``.l).length - 2;` `        ``// Binary search``        ``while` `(low <= high) {``            ``let mid = Math.floor((low + high) / 2);` `            ``if` `(``this``.value(mid, x) > ``this``.value(mid + 1, x))``                ``low = mid + 1;``            ``else``                ``high = mid - 1;``        ``}` `        ``return` `this``.value(low, x);``    ``}``};` `// Driver code``let lines = [ ``new` `Line(1, 1), ``new` `Line(0, 0), ``new` `Line(-3, 3)]` `let Q = [ -2, 2, 0 ];``let n = 3, q = 3;``let cht = ``new` `Convex_HULL_Trick();` `// Sort the lines``lines.sort()` `// Add the lines``for` `(``var` `i = 0; i < n; i++)``    ``cht.add(lines[i]);` `// For each query in Q``for` `(``var` `i = 0; i < q; i++) {``    ``let x = Q[i];``    ``console.log(cht.minQuery(x));``}` `// This code is contributed by phasing17`

Output

```-1
-3
0```

Time Complexity: O(max(N*logN, Q*logN)), as we are using an inbuilt sort function to sort an array of size N which will cost O(N*logN) and we are using a loop to traverse Q times and in each traversal, we are calling the method minQuery which costs logN time. Where N is the number of lines and Q is the number of queries.
Auxiliary Space: O(N), as we are using extra space.

My Personal Notes arrow_drop_up