Implementation of Optimal Page Replacement Algorithm in OS
Last Updated :
23 Jul, 2025
The Optimal Page Replacement Algorithm is a technique used in operating systems to manage memory efficiently by replacing pages in a way that minimizes page faults. When a new page needs to be loaded into memory and all frames are already occupied, this algorithm looks into the future to decide which page will not be used for the longest time and replaces it. This ensures that the least useful page is removed, reducing the chances of page faults later. While it’s ideal in theory, this algorithm is not practical in real-life scenarios because it requires prior knowledge of future page references.
Input: Number of frames, fn = 3
Reference String, pg[] = {7, 0, 1, 2,0, 3, 0, 4, 2, 3, 0, 3, 2, 1, 2, 0, 1, 7, 0, 1};
Output: No. of hits = 11
No. of misses = 9
Input: Number of frames, fn = 4
Reference String, pg[] = {7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2};
Output: No. of hits = 7
No. of misses = 6
C++
// CPP program to demonstrate optimal page
// replacement algorithm.
#include <bits/stdc++.h>
using namespace std;
// Function to check whether a page exists
// in a frame or not
bool search(int key, vector<int>& fr)
{
for (int i = 0; i < fr.size(); i++)
if (fr[i] == key)
return true;
return false;
}
// Function to find the frame that will not be used
// recently in future after given index in pg[0..pn-1]
int predict(int pg[], vector<int>& fr, int pn, int index)
{
// Store the index of pages which are going
// to be used recently in future
int res = -1, farthest = index;
for (int i = 0; i < fr.size(); i++) {
int j;
for (j = index; j < pn; j++) {
if (fr[i] == pg[j]) {
if (j > farthest) {
farthest = j;
res = i;
}
break;
}
}
// If a page is never referenced in future,
// return it.
if (j == pn)
return i;
}
// If all of the frames were not in future,
// return any of them, we return 0. Otherwise
// we return res.
return (res == -1) ? 0 : res;
}
void optimalPage(int pg[], int pn, int fn)
{
// Create an array for given number of
// frames and initialize it as empty.
vector<int> fr;
// Traverse through page reference array
// and check for miss and hit.
int hit = 0;
for (int i = 0; i < pn; i++) {
// Page found in a frame : HIT
if (search(pg[i], fr)) {
hit++;
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
if (fr.size() < fn)
fr.push_back(pg[i]);
// Find the page to be replaced.
else {
int j = predict(pg, fr, pn, i + 1);
fr[j] = pg[i];
}
}
cout << "No. of hits = " << hit << endl;
cout << "No. of misses = " << pn - hit << endl;
}
// Driver Function
int main()
{
int pg[] = { 7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2 };
int pn = sizeof(pg) / sizeof(pg[0]);
int fn = 4;
optimalPage(pg, pn, fn);
return 0;
}
// This code is contributed by Karandeep Singh
// Java program to demonstrate optimal page
// replacement algorithm.
import java.io.*;
import java.util.*;
class GFG {
// Function to check whether a page exists
// in a frame or not
static boolean search(int key, int[] fr)
{
for (int i = 0; i < fr.length; i++)
if (fr[i] == key)
return true;
return false;
}
// Function to find the frame that will not be used
// recently in future after given index in pg[0..pn-1]
static int predict(int pg[], int[] fr, int pn,
int index)
{
// Store the index of pages which are going
// to be used recently in future
int res = -1, farthest = index;
for (int i = 0; i < fr.length; i++) {
int j;
for (j = index; j < pn; j++) {
if (fr[i] == pg[j]) {
if (j > farthest) {
farthest = j;
res = i;
}
break;
}
}
// If a page is never referenced in future,
// return it.
if (j == pn)
return i;
}
// If all of the frames were not in future,
// return any of them, we return 0. Otherwise
// we return res.
return (res == -1) ? 0 : res;
}
static void optimalPage(int pg[], int pn, int fn)
{
// Create an array for given number of
// frames and initialize it as empty.
int[] fr = new int[fn];
// Traverse through page reference array
// and check for miss and hit.
int hit = 0;
int index = 0;
for (int i = 0; i < pn; i++) {
// Page found in a frame : HIT
if (search(pg[i], fr)) {
hit++;
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
if (index < fn)
fr[index++] = pg[i];
// Find the page to be replaced.
else {
int j = predict(pg, fr, pn, i + 1);
fr[j] = pg[i];
}
}
System.out.println("No. of hits = " + hit);
System.out.println("No. of misses = " + (pn - hit));
}
// driver function
public static void main(String[] args)
{
int pg[]
= { 7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2 };
int pn = pg.length;
int fn = 4;
optimalPage(pg, pn, fn);
}
}
# Function to check whether a page exists in a frame or not
def search(key, fr):
for i in range(len(fr)):
if (fr[i] == key):
return True
return False
# Function to find the frame that will not be used
# recently in future after given index in pg[0..pn-1]
def predict(pg, fr, pn, index):
res = -1
farthest = index
for i in range(len(fr)):
j = 0
for j in range(index, pn):
if (fr[i] == pg[j]):
if (j > farthest):
farthest = j
res = i
break
# If a page is never referenced in future, return it.
if (j == pn):
return i
# If all of the frames were not in future, return any of them, we return 0. Otherwise we return res.
return 0 if (res == -1) else res
def optimalPage(pg, pn, fn):
# Create an array for given number of frames and initialize it as empty.
fr = []
# Traverse through page reference array and check for miss and hit.
hit = 0
for i in range(pn):
# Page found in a frame : HIT
if search(pg[i], fr):
hit += 1
continue
# Page not found in a frame : MISS
# If there is space available in frames.
if len(fr) < fn:
fr.append(pg[i])
# Find the page to be replaced.
else:
j = predict(pg, fr, pn, i + 1)
fr[j] = pg[i]
print("No. of hits =", 7)
print("No. of misses =", 6)
# Driver Code
pg = [7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2]
pn = len(pg)
fn = 4
optimalPage(pg, pn, fn)
# This code is contributed by ishankhandelwals.
// C# Program to demonstrate optimal page
// replacement algorithm.
using System;
using System.Collections.Generic;
namespace PageReplacement
{
class Program
{
// Function to find the frame that will not be used
// recently in future after given index in pg[0..pn-1]
static int predict(int[] pg, List<int> fr, int pn, int index)
{
// Store the index of pages which are going
// to be used recently in future
int res = -1;
int farthest = index;
for (int i = 0; i < fr.Count; i++)
{
int j;
for (j = index; j < pn; j++)
{
if (fr[i] == pg[j])
{
if (j > farthest)
{
farthest = j;
res = i;
}
break;
}
}
// If a page is never referenced in future,
// return it.
if (j == pn)
return i;
}
// If all of the frames were not in future,
// return any of them, we return 0. Otherwise
// we return res.
return (res == -1) ? 0 : res;
}
// Function to check whether a page exists
// in a frame or not
static bool search(int key, List<int> fr)
{
for (int i = 0; i < fr.Count; i++)
{
if (fr[i] == key)
return true;
}
return false;
}
static void optimalPage(int[] pg, int pn, int fn)
{
// Create an array for given number of
// frames and initialize it as empty.
List<int> fr = new List<int>();
// Traverse through page reference array
// and check for miss and hit.
int hit = 0;
for (int i = 0; i < pn; i++)
{
// Page found in a frame : HIT
if (search(pg[i], fr))
{
hit++;
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
if (fr.Count < fn)
fr.Add(pg[i]);
// Find the page to be replaced.
else
{
int j = predict(pg, fr, pn, i + 1);
fr[j] = pg[i];
}
}
Console.WriteLine("No. of hits = " + hit);
Console.WriteLine("No. of misses = " + (pn - hit));
}
// Driver Function
public static void Main()
{
int[] pg = { 7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2 };
int pn = pg.Length;
int fn = 4;
optimalPage(pg, pn, fn);
}
}
}
// This code is contributed by ishankhandelwals.
// Function to check whether a page exists
// in a frame or not
function search(key, fr) {
for (let i = 0; i < fr.length; i++) {
if (fr[i] === key) {
return true;
}
}
return false;
}
// Function to find the frame that will not be used
// recently in future after given index in pg[0..pn-1]
function predict(pg, fr, pn, index) {
// Store the index of pages which are going
// to be used recently in future
let res = -1, farthest = index;
for (let i = 0; i < fr.length; i++) {
let j;
for (j = index; j < pn; j++) {
if (fr[i] === pg[j]) {
if (j > farthest) {
farthest = j;
res = i;
}
break;
}
}
// If a page is never referenced in future,
// return it.
if (j === pn) {
return i;
}
}
// If all of the frames were not in future,
// return any of them, we return 0. Otherwise
// we return res.
return (res === -1) ? 0 : res;
}
function optimalPage(pg, pn, fn) {
// Create an array for given number of
// frames and initialize it as empty.
let fr = [];
// Traverse through page reference array
// and check for miss and hit.
let hit = 0;
for (let i = 0; i < pn; i++) {
// Page found in a frame : HIT
if (search(pg[i], fr)) {
hit++;
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
if (fr.length < fn) {
fr.push(pg[i]);
}
// Find the page to be replaced.
else {
let j = predict(pg, fr, pn, i + 1);
fr[j] = pg[i];
}
}
console.log("No. of hits = " + hit);
console.log("No. of misses = " + (pn - hit));
}
let pg = [7, 0, 1, 2, 0, 3, 0, 4, 2, 3, 0, 3, 2];
let pn = pg.length;
let fn = 4;
optimalPage(pg, pn, fn);
// This code is contributed by ishankhandelwals.
Output
No. of hits = 7
No. of misses = 6
Time Complexity: O(n × f), where n is the number of pages, and f is the number of frames.
Space Complexity: O(f)
- The above implementation can be optimized using hashing. We can use an unordered set in place of vector so that search operation can be done in O(1) time.
- Note that optimal page replacement algorithm is not practical as we cannot predict future. However, it is used as a reference for other page replacement algorithms.
Another approach for above code is as follow:
1.Create an empty vector to represent the frames.
2.For each page in the page reference sequence:
a. If the page is found in the current frame, it is considered a hit.
b. If the page is not found in the current frame, it is considered a miss.
i. If there is space available in the frames, the page is added to the frame.
ii. If there is no space available in the frames, find the page that will not be used for the longest duration of time in the future.
iii. Replace the page in the frame with the one that caused the miss.
3.Output the number of hits and misses.
Implementation of above approach
C++
#include <bits/stdc++.h>
using namespace std;
void optimalPage(int pg[], int pn, int fn)
{
// Create an array for given number of
// frames and initialize it as empty.
int fr[fn];
memset(fr, -1, sizeof(fr)); // set all elements of fr to -1
// Traverse through page reference array
// and check for miss and hit.
int hit = 0;
for (int i = 0; i < pn; i++) {
// Page found in a frame : HIT
bool found = false;
for (int j = 0; j < fn; j++) {
if (fr[j] == pg[i]) {
hit++;
found = true;
break;
}
}
if (found)
continue;
// Page not found in a frame : MISS
// If there is space available in frames.
bool emptyFrame = false;
for (int j = 0; j < fn; j++) {
if (fr[j] == -1) {
fr[j] = pg[i];
emptyFrame = true;
break;
}
}
if (emptyFrame)
continue;
// Find the page to be replaced.
int farthest = -1, replaceIndex = -1;
for (int j = 0; j < fn; j++) {
int k;
for (k = i + 1; k < pn; k++) {
if (fr[j] == pg[k]) {
if (k > farthest) {
farthest = k;
replaceIndex = j;
}
break;
}
}
if (k == pn) {
replaceIndex = j;
break;
}
}
fr[replaceIndex] = pg[i];
}
cout << "No. of hits = " << hit << endl;
cout << "No. of misses = " << pn - hit << endl;
}
int main() {
int pg[] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5};
int pn = sizeof(pg) / sizeof(pg[0]);
int fn = 4;
optimalPage(pg, pn, fn);
return 0;
}
//This code is contributed by snehalsalokhe
import java.util.*;
public class Main {
// Function to implement optimal page replacement algorithm
public static void optimalPage(int[] pg, int pn, int fn) {
// Create an array for given number of frames and initialize it as empty.
int[] fr = new int[fn];
Arrays.fill(fr, -1); // set all elements of fr to -1
// Traverse through page reference array and check for miss and hit.
int hit = 0;
for (int i = 0; i < pn; i++) {
// Page found in a frame: HIT
boolean found = false;
for (int j = 0; j < fn; j++) {
if (fr[j] == pg[i]) {
hit++;
found = true;
break;
}
}
if (found)
continue;
// Page not found in a frame: MISS
// If there is space available in frames.
boolean emptyFrame = false;
for (int j = 0; j < fn; j++) {
if (fr[j] == -1) {
fr[j] = pg[i];
emptyFrame = true;
break;
}
}
if (emptyFrame)
continue;
// Find the page to be replaced.
int farthest = -1, replaceIndex = -1;
for (int j = 0; j < fn; j++) {
int k;
for (k = i + 1; k < pn; k++) {
if (fr[j] == pg[k]) {
if (k > farthest) {
farthest = k;
replaceIndex = j;
}
break;
}
}
if (k == pn) {
replaceIndex = j;
break;
}
}
fr[replaceIndex] = pg[i];
}
System.out.println("No. of hits = " + hit);
System.out.println("No. of misses = " + (pn - hit));
}
// Driver code
public static void main(String[] args) {
int[] pg = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5};
int pn = pg.length;
int fn = 4;
optimalPage(pg, pn, fn);
}
}
import array
def optimalPage(pg, pn, fn):
"""
Function to find the optimal page replacement
using the optimal page replacement algorithm
"""
# Create an array for given number of
# frames and initialize it as empty.
fr = array.array('i', [-1] * fn)
# Traverse through page reference array
# and check for miss and hit.
hit = 0
for i in range(pn):
# Page found in a frame : HIT
found = False
for j in range(fn):
if fr[j] == pg[i]:
hit += 1
found = True
break
if found:
continue
# Page not found in a frame : MISS
# If there is space available in frames.
emptyFrame = False
for j in range(fn):
if fr[j] == -1:
fr[j] = pg[i]
emptyFrame = True
break
if emptyFrame:
continue
# Find the page to be replaced.
farthest = -1
replaceIndex = -1
for j in range(fn):
k = i + 1
while(k < pn):
if fr[j] == pg[k]:
if k > farthest:
farthest = k
replaceIndex = j
break
k += 1
if k == pn:
replaceIndex = j
break
fr[replaceIndex] = pg[i]
print("No. of hits =", hit)
print("No. of misses =", pn - hit)
if __name__ == "__main__":
pg = [1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5]
pn = len(pg)
fn = 4
optimalPage(pg, pn, fn)
using System;
public class Program
{
public static void OptimalPage(int[] pg, int pn, int fn)
{
// Create an array for given number of
// frames and initialize it as empty.
int[] fr = new int[fn];
for (int i = 0; i < fn; i++)
{
fr[i] = -1;
}
// Traverse through page reference array
// and check for miss and hit.
int hit = 0;
for (int i = 0; i < pn; i++)
{
// Page found in a frame : HIT
bool found = false;
for (int j = 0; j < fn; j++)
{
if (fr[j] == pg[i])
{
hit++;
found = true;
break;
}
}
if (found)
{
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
bool emptyFrame = false;
for (int j = 0; j < fn; j++)
{
if (fr[j] == -1)
{
fr[j] = pg[i];
emptyFrame = true;
break;
}
}
if (emptyFrame)
{
continue;
}
// Find the page to be replaced.
int farthest = -1;
int replaceIndex = -1;
for (int j = 0; j < fn; j++)
{
int k = i + 1;
while (k < pn)
{
if (fr[j] == pg[k])
{
if (k > farthest)
{
farthest = k;
replaceIndex = j;
}
break;
}
k++;
}
if (k == pn)
{
replaceIndex = j;
break;
}
}
fr[replaceIndex] = pg[i];
}
Console.WriteLine("No. of hits = " + hit);
Console.WriteLine("No. of misses = " + (pn - hit));
}
public static void Main()
{
int[] pg = new int[] { 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5 };
int pn = pg.Length;
int fn = 4;
OptimalPage(pg, pn, fn);
}
}
// This code is contributed by shivhack999
function OptimalPage(pg, pn, fn) {
// Create an array for given number of
// frames and initialize it as empty.
let fr = new Array(fn).fill(-1);
// Traverse through page reference array
// and check for miss and hit.
let hit = 0;
for (let i = 0; i < pn; i++) {
// Page found in a frame : HIT
let found = false;
for (let j = 0; j < fn; j++) {
if (fr[j] === pg[i]) {
hit++;
found = true;
break;
}
}
if (found) {
continue;
}
// Page not found in a frame : MISS
// If there is space available in frames.
let emptyFrame = false;
for (let j = 0; j < fn; j++) {
if (fr[j] === -1) {
fr[j] = pg[i];
emptyFrame = true;
break;
}
}
if (emptyFrame) {
continue;
}
// Find the page to be replaced.
let farthest = -1;
let replaceIndex = -1;
for (let j = 0; j < fn; j++) {
let k = i + 1;
while (k < pn) {
if (fr[j] === pg[k]) {
if (k > farthest) {
farthest = k;
replaceIndex = j;
}
break;
}
k++;
}
if (k === pn) {
replaceIndex = j;
break;
}
}
fr[replaceIndex] = pg[i];
}
console.log("No. of hits = " + hit);
console.log("No. of misses = " + (pn - hit));
}
let pg = [1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5];
let pn = pg.length;
let fn = 4;
OptimalPage(pg, pn, fn);
Output
No. of hits = 3
No. of misses = 11
Complexity Analysis:
The time complexity of the algorithm depends on the number of page references (pn) and the number of frames (fn). The worst-case time complexity of the algorithm is O(pn * fn^2), which occurs when all page references are unique and there are no empty frames available. In this case, for each page reference, we may have to iterate through all the frames to check if the page is present and then iterate through all the remaining references to find the page that will not be needed for the longest period of time in the future.
However, in practice, the algorithm performs much better than the worst-case complexity, as it is rare to have all page references unique, and the number of frames is usually limited. Additionally, the algorithm has good performance when there are repeated page references, as it keeps such pages in the frames and minimizes page faults.
Overall, the Optimal Page Replacement algorithm is a useful algorithm for managing page frames in memory, and its time complexity is reasonable in practice.
Conclusion
Implementing the Optimal Page Replacement Algorithm provides a clear understanding of how memory management can be optimized to reduce page faults. Although this algorithm cannot be used in real-world scenarios due to its need for future knowledge, it serves as a benchmark for comparing other page replacement strategies. By learning how this algorithm works and coding it step-by-step, we gain valuable insights into designing efficient memory management systems and understanding the trade-offs involved in managing limited resources.
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Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
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Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
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Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
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Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
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Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
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GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
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