Equivalent Serial Schedule of Conflict Serializable Schedule in DBMS (original) (raw)

Last Updated : 23 Jul, 2025

Prerequisite: Conflict Serializability, Precedence Graph

Conflict Serializable Schedule: A schedule is called conflict serializable if it can be transformed into a serial schedule by swapping non-conflicting operations. The serial schedule of the Conflict Serializable Schedule can be found by applying topological Sorting on the Precedence Graph of the Conflict Serializable Schedule.

Note: Precedence Graph of Conflict Serial Schedule is always directed acyclic graph.

Approach: Follow to below steps to find topological sorting of Precedence Graph:

Find the indegree of all nodes for the given Precedence Graph and store it in an auxiliary array.
Now For each node having indegree 0 perform the following:
- Print the current node T as the order of the topological sort.
- Let the node T be the node with in-degree 0.
- Remove T and all edges connecting to T from the graph.
- Update indegree of all nodes after the above steps.
After the above steps, the topological sort of the given precedence graph can be calculated.

Below is the illustration of the above approach:

Let, the Conflict Serial Schedule be S: R2(A) W2(A) R3(C) W2(B) W3(A) W3(C) R1(A) R2(B) W1(A) W2(B)

Here node T2 has indegree 0.
So, select T2 and remove T2 and all edges connecting from it.
Now T3 has indegree 0. So, select T3 and remove the edge T3→T1.
At the end select T3. So the topological Sorting is T2, T3, T1.
Hence, the equivalent serial schedule of given conflict serializable schedule is T2→T3→T1, i.e., S2: R2(A) W2(A) W2(B) R3(C) W3(A) W3(C) R1(A) R2(B) W1(A) W1(B).

There may be more than one equivalent serial schedule of the conflict serializable schedule.

Below is the implementation to get the Serial Schedule of CSS using Topological Sorting:

C++ `

// C++ program to print serial schedule // of CSS using Topological sorting #include <bits/stdc++.h> using namespace std;

class PrecedenceGraph {

// No. of vertices
int V;

// Pointer to an array containing
// adjacency vector
vector<int>* adj;

// Vector to store SerialSchedule
vector<int> serialSchedule;

// Function  declarations
void computeIndegree(int* indegree);
int findZeroIndegree(int* indegree,
                     bool* visited);

public: // Constructor PrecedenceGraph(int V);

// Function declarations
void addEdge(int v, int w);
void topologicalSort();
void printSchedule();

};

// Function to create the precedence // graph PrecedenceGraph::PrecedenceGraph(int V) { this->V = V; adj = new vector[V]; }

// Function to add the edge to the // precedence graph void PrecedenceGraph::addEdge(int v, int w) { adj[v].push_back(w); }

// Function to compute indegree of nodes void PrecedenceGraph::computeIndegree( int* indegree) { for (int i = 1; i < V; i++) {

    // Traverse the adjacency list
    // of node i
    for (int j = 0;
         j < adj[i].size(); j++) {
        int node = adj[i][j];

        // Update the indegree
        // of node
        indegree[node]++;
    }
}

}

// Function to find node with // zero indegree int PrecedenceGraph::findZeroIndegree( int* indegree, bool* visited) { for (int i = 1; i < V; i++) {

    // If node is not visited
    // and have indegree 0
    if (!visited[i]
        && indegree[i] == 0) {

        // Mark node visited
        visited[i] = true;

        // Return node
        return i;
    }
}

// All nodes are visited
return -1;

}

// Function to find the topological // Sorting of the given graph void PrecedenceGraph::topologicalSort() { // Create and initialize // visited array bool* visited = new boolV;

// Create and initialize
// indegree array
int* indegree = new int[V]();

computeIndegree(indegree);

// Check if the node with
// indegree 0 is available
int node = findZeroIndegree(
    indegree, visited);

bool nodeAvailable = false;

if (node != -1) {
    nodeAvailable = true;
}
while (nodeAvailable) {

    // Add node to serial schedule
    serialSchedule.push_back(node);

    for (int i = 0;
         i < adj[node].size(); i++) {

        // Delete all edges of current
        // node and update indegree
        indegree[adj[node][i]]--;
    }

    // Find next node with indegree 0
    node = findZeroIndegree(indegree,
                            visited);

    if (node == -1) {

        // Node with zero indegree
        // not available
        nodeAvailable = false;
    }
}

}

// Function to print the serial schedule void PrecedenceGraph::printSchedule() { for (int i : serialSchedule) { cout << "T" << i << " "; } }

// Driver Code int main() { // Create a precedence graph // given in the above diagram PrecedenceGraph graph(4); graph.addEdge(2, 1); graph.addEdge(2, 3); graph.addEdge(3, 1);

// Find topological ordereing
graph.topologicalSort();

// Print Schedule
cout << "Equivalent Serial"
     << " Schedule is :";
graph.printSchedule();

}

Java

// Java program to print serial schedule // of CSS using Topological sorting

import java.util.*;

class PrecedenceGraph { // No. of vertices private int V;

// Pointer to an array containing // adjacency vector private ArrayList[] adj;

// Vector to store SerialSchedule private ArrayList serialSchedule;

// Function to create the precedence // graph PrecedenceGraph(int V) { this.V = V; adj = new ArrayList[V]; for (int i = 0; i < V; i++) { adj[i] = new ArrayList<>(); } serialSchedule = new ArrayList<>(); }

// Function to add the edge to the // precedence graph void addEdge(int v, int w) { adj[v].add(w); }

// Function to compute indegree of nodes void computeIndegree(int[] indegree) { for (int i = 1; i < V; i++) { // Traverse the adjacency list // of node i for (int j = 0; j < adj[i].size(); j++) { int node = adj[i].get(j);

        // Update the indegree
        // of node
        indegree[node]++;
    }
}

}

// Function to find node with // zero indegree int findZeroIndegree(int[] indegree, boolean[] visited) { for (int i = 1; i < V; i++) { // If node is not visited // and have indegree 0 if (!visited[i] && indegree[i] == 0) { // Mark node visited visited[i] = true; // Return node return i; } } // All nodes are visited return -1; }

// Function to find the topological // Sorting of the given graph void topologicalSort() { // Create and initialize // visited array boolean[] visited = new boolean[V];

// Create and initialize 
// indegree array
int[] indegree = new int[V];

computeIndegree(indegree);

// Check if the node with indegree 0 is available
int node = findZeroIndegree(indegree, visited);

boolean nodeAvailable = false;

if (node != -1) {
    nodeAvailable = true;
}
while (nodeAvailable) {
    // Add node to serial schedule
    serialSchedule.add(node);

    for (int i = 0; i < adj[node].size(); i++) {
        // Delete all edges of current
        // node and update indegree
        indegree[adj[node].get(i)]--;
    }

    // Find next node with indegree 0
    node = findZeroIndegree(indegree, visited);

    if (node == -1) {
        // Node with zero indegree 
        // not available
        nodeAvailable = false;
    }
}

}

// Function to print the serial schedule void printSchedule() { for (int i : serialSchedule) { System.out.print("T" + i + " "); } System.out.println(); } }

// Driver Code class Main { public static void main(String[] args) { // Create a precedence graph // given in the above diagram PrecedenceGraph graph = new PrecedenceGraph(4); graph.addEdge(2, 1); graph.addEdge(2, 3); graph.addEdge(3, 1);

// Find topological ordereing
graph.topologicalSort();

// Print Schedule
System.out.print("Equivalent Serial Schedule is :");
graph.printSchedule();

} }

// This code is contributed by Pushpesh Raj.

Python3

Python program to print serial schedule of CSS using Topological sorting

from collections import defaultdict

class PrecedenceGraph: # Constructor def init(self, V): self.V = V self.adj = defaultdict(list) self.serialSchedule = []

# Function to add the edge to the precedence graph
def addEdge(self, v, w):
    self.adj[v].append(w)

# Function to compute indegree of nodes
def computeIndegree(self, indegree):
    for i in range(1, self.V):
        for node in self.adj[i]:
            # Update the indegree of node
            indegree[node] += 1

# Function to find node with zero indegree
def findZeroIndegree(self, indegree, visited):
    for i in range(1, self.V):
        # If node is not visited and have indegree 0
        if not visited[i] and indegree[i] == 0:
            # Mark node visited
            visited[i] = True
            # Return node
            return i
    # All nodes are visited
    return -1

# Function to find the topological Sorting of the given graph
def topologicalSort(self):
    # Create and initialize visited array
    visited = [False for i in range(self.V)]

    # Create and initialize indegree array
    indegree = [0 for i in range(self.V)]

    self.computeIndegree(indegree)

    # Check if the node with indegree 0 is available
    node = self.findZeroIndegree(indegree, visited)
    nodeAvailable = False
    if node != -1:
        nodeAvailable = True

    while nodeAvailable:
        # Add node to serial schedule
        self.serialSchedule.append(node)
        for next_node in self.adj[node]:
            # Delete all edges of current node and update indegree
            indegree[next_node] -= 1

        # Find next node with indegree 0
        node = self.findZeroIndegree(indegree, visited)

        if node == -1:
            # Node with zero indegree not available
            nodeAvailable = False

# Function to print the serial schedule
def printSchedule(self):
    print("Equivalent Serial Schedule is: ", end="")
    for i in self.serialSchedule:
        print("T{} ".format(i), end="")

Driver Code

if name == "main": # Create a precedence graph given in the above diagram graph = PrecedenceGraph(4) graph.addEdge(2, 1) graph.addEdge(2, 3) graph.addEdge(3, 1)

# Find topological ordereing
graph.topologicalSort()

# Print Schedule
graph.printSchedule()

C#

using System; using System.Collections.Generic;

public class PrecedenceGraph {

// No. of vertices
private int V;

// Pointer to an array containing
// adjacency vector
private List<int>[] adj;

// Vector to store SerialSchedule
private List<int> serialSchedule = new List<int>();

// Constructor
public PrecedenceGraph(int V)
{
    this.V = V;
    adj = new List<int>[ V ];
    for (int i = 0; i < V; i++) {
        adj[i] = new List<int>();
    }
}

// Function to add the edge to the
// precedence graph
public void addEdge(int v, int w) { adj[v].Add(w); }

// Function to compute indegree of nodes
private void computeIndegree(int[] indegree)
{
    for (int i = 1; i < V; i++) {
        // Traverse the adjacency list
        // of node i
        foreach(int node in adj[i])
        {
            // Update the indegree
            // of node
            indegree[node]++;
        }
    }
}

// Function to find node with
// zero indegree
private int findZeroIndegree(int[] indegree,
                             bool[] visited)
{
    for (int i = 1; i < V; i++) {
        // If node is not visited
        // and have indegree 0
        if (!visited[i] && indegree[i] == 0) {
            // Mark node visited
            visited[i] = true;

            // Return node
            return i;
        }
    }

    // All nodes are visited
    return -1;
}

// Function to find the topological
// Sorting of the given graph
public void topologicalSort()
{
    // Create and initialize
    // visited array
    bool[] visited = new bool[V];

    // Create and initialize
    // indegree array
    int[] indegree = new int[V];

    computeIndegree(indegree);

    // Check if the node with
    // indegree 0 is available
    int node = findZeroIndegree(indegree, visited);

    bool nodeAvailable = false;

    if (node != -1) {
        nodeAvailable = true;
    }

    while (nodeAvailable) {
        // Add node to serial schedule
        serialSchedule.Add(node);

        foreach(int i in adj[node])
        {
            // Delete all edges of current
            // node and update indegree
            indegree[i]--;
        }

        // Find next node with indegree 0
        node = findZeroIndegree(indegree, visited);

        if (node == -1) {
            // Node with zero indegree
            // not available
            nodeAvailable = false;
        }
    }
}

// Function to print the serial schedule
public void printSchedule()
{
    foreach(int i in serialSchedule)
    {
        Console.Write("T" + i + " ");
    }
}

}

public class Program { public static void Main(string[] args) { // Create a precedence graph // given in the above diagram PrecedenceGraph graph = new PrecedenceGraph(4); graph.addEdge(2, 1); graph.addEdge(2, 3); graph.addEdge(3, 1);

    // Find topological ordering
    graph.topologicalSort();

    // Print Schedule
    Console.Write("Equivalent Serial Schedule is: ");
    graph.printSchedule();
}

}

` JavaScript ``

class PrecedenceGraph { // Constructor constructor(V) { this.V = V; this.adj = new Map(); this.serialSchedule = []; }

// Function to add the edge to the precedence graph addEdge(v, w) { if (!this.adj.has(v)) { this.adj.set(v, []); } this.adj.get(v).push(w); }

// Function to compute indegree of nodes computeIndegree(indegree) { for (let i = 1; i < this.V; i++) { // Check if the current node exists in the adjacency list if (!this.adj.has(i)) { continue; } // Update the indegree of all adjacent nodes for (const node of this.adj.get(i)) { indegree[node] += 1; } } }

// Function to find node with zero indegree findZeroIndegree(indegree, visited) { for (let i = 1; i < this.V; i++) { // If node is not visited and have indegree 0 if (!visited[i] && indegree[i] === 0) { // Mark node visited visited[i] = true; // Return node return i; } } // All nodes are visited return -1; }

// Function to find the topological Sorting of the given graph topologicalSort() { // Create and initialize visited array const visited = new Array(this.V).fill(false);

// Create and initialize indegree array
const indegree = new Array(this.V).fill(0);

// Compute indegree of nodes
this.computeIndegree(indegree);

// Check if the node with indegree 0 is available
let node = this.findZeroIndegree(indegree, visited);
let nodeAvailable = false;
if (node !== -1) {
  nodeAvailable = true;
}

while (nodeAvailable) {
  // Add node to serial schedule
  this.serialSchedule.push(node);
  // Delete all edges of current node and update indegree
  if (this.adj.has(node)) {
    for (const nextNode of this.adj.get(node)) {
      indegree[nextNode] -= 1;
    }
  }
  // Find next node with indegree 0
  node = this.findZeroIndegree(indegree, visited);
  if (node === -1) {
    // Node with zero indegree not available
    nodeAvailable = false;
  }
}

}

// Function to print the serial schedule printSchedule() { let result = "Equivalent Serial Schedule is: "; for (const i of this.serialSchedule) { result += T${i} ; } console.log(result); } }

// Driver Code const graph = new PrecedenceGraph(4); graph.addEdge(2, 1); graph.addEdge(2, 3); graph.addEdge(3, 1); graph.topologicalSort(); graph.printSchedule();

Output:

Equivalent Serial Schedule is :T2 T3 T1

Time Complexity: O(N)
Auxiliary Space: O(N)