BACKTRACKING & BRANCH AND BOUND – FULL EXAM PACKAGE

(15–25 Marks Guaranteed – Most Important Unit IV Questions!)

BACKTRACKING & BRANCH AND BOUND – FULL EXAM PACKAGE

(15–25 Marks Guaranteed – Most Important Unit IV Questions!)

Here are ALL 5 Classic Problems – Complete with Algorithm + Code + Diagram + Solved Exam Question

1. N-QUEENS PROBLEM (Backtracking)

Problem: Place N queens on N×N board – no two attack.

4-Queens Solution (Draw This!)

.  Q  .  .
.  .  .  Q
Q  .  .  .
.  .  Q  .

C Code (Best Version)

#include<stdio.h>
int board[20][20], n;

int isSafe(int row, int col) {
    for(int i=0; i<row; i++)
        if(board[i][col]) return 0;
    for(int i=row-1,j=col-1; i>=0&&j>=0; i--,j--)
        if(board[i][j]) return 0;
    for(int i=row-1,j=col+1; i>=0&&j<n; i--,j++)
        if(board[i][j]) return 0;
    return 1;
}

void print() {
    for(int i=0; i<n; i++) {
        for(int j=0; j<n; j++)
            printf("%c ", board[i][j]?'Q':'.');
        printf("\n");
    }
    printf("\n");
}

void nQueen(int row) {
    if(row == n) { print(); return; }
    for(int col=0; col<n; col++) {
        if(isSafe(row, col)) {
            board[row][col] = 1;
            nQueen(row+1);
            board[row][col] = 0;  // Backtrack
        }
    }
}

int main() {
    printf("Enter N: "); scanf("%d",&n);
    nQueen(0);
}

2. TRAVELING SALESMAN PROBLEM (TSP) – Branch and Bound

Classic 4-City Example (Most Asked!)

    20    30    10
A ───► B ───► C ───► D
 ↑              ↖    ↓
 15            35    25
 ↑                  ↙
 └──────────────────┘ 40

Cost Matrix:

	A	B	C	D
A	∞	20	30	10
B	15	∞	35	25
C	40	40	∞	25
D	25	30	20	∞

Optimal Tour: A → D → C → B → A → Cost = 80

Branch and Bound Tree (Draw This!)

Root: (A) → UB = ∞
  ↓
A→B → cost=20 → UB=95 → prune
A→C → cost=30 → UB=105 → prune
A→D → cost=10 → UB=85 → explore
  ↓
A→D→C → cost=35 → UB=90 → prune
A→D→B → cost=40 → UB=100 → prune
→ Final best = 80

C Code – TSP Branch and Bound

#include<stdio.h>
#define INF 9999
int cost[10][10], n, final_cost = INF;

void copyToFinal(int curr_path[], int final_path[]) {
    for(int i=0; i<n; i++) final_path[i] = curr_path[i];
    final_path[n] = curr_path[0];
}

int firstMin(int i) {
    int min = INF;
    for(int k=0; k<n; k++)
        if(cost[i][k] < min && i != k) min = cost[i][k];
    return min;
}

void TSP(int curr_bound, int curr_weight, int level, int curr_path[], int visited[]) {
    if(level == n) {
        if(cost[curr_path[level-1]][curr_path[0]] != INF) {
            int total = curr_weight + cost[curr_path[level-1]][curr_path[0]];
            if(total < final_cost) {
                final_cost = total;
                // copy path
            }
        }
        return;
    }

    for(int i=0; i<n; i++) {
        if(!visited[i]) {
            int temp = curr_bound;
            curr_bound -= ((level==1)?firstMin(curr_path[level-1]):0);
            if(curr_bound + curr_weight + cost[curr_path[level-1]][i] < final_cost) {
                curr_path[level] = i;
                visited[i] = 1;
                TSP(curr_bound, curr_weight + cost[curr_path[level-1]][i], level+1, curr_path, visited);
            }
            visited[i] = 0;
            curr_bound = temp;
        }
    }
}

3. GRAPH COLORING (Backtracking)

3-Coloring Example

A — B
|   | \
C — D  E

C Code

#include<stdio.h>
int G[50][50], x[50], n, m=3;  // m colors

int isSafe(int v, int c) {
    for(int i=0; i<n; i++)
        if(G[v][i] && x[i] == c) return 0;
    return 1;
}

void print() {
    for(int i=0; i<n; i++) printf("%d ", x[i]+1);
    printf("\n");
}

void graphColor(int v) {
    if(v == n) { print(); return; }
    for(int c=0; c<m; c++) {
        if(isSafe(v, c)) {
            x[v] = c;
            graphColor(v+1);
            x[v] = -1;
        }
    }
}

4. HAMILTONIAN CYCLE (Backtracking)

Find cycle visiting each vertex exactly once

int path[10];
void hamiltonian(int k) {
    if(k == n) {
        if(G[path[k-1]][path[0]]) {
            for(int i=0; i<n; i++) printf("%d→", path[i]);
            printf("%d\n", path[0]);
        }
        return;
    }
    for(int v=1; v<n; v++) {
        if(isSafe(k, v)) {
            path[k] = v;
            hamiltonian(k+1);
        }
    }
}

5. SUM OF SUBSETS (Backtracking)

Given: {10, 7, 5, 18, 12, 15}, Target = 30
Solutions: {10,7,5,18}, {10,18,12}, {15,10,5}

C Code

void subset(int pos, int sum, int target) {
    if(sum == target) { print(); return; }
    if(pos == n || sum > target) return;

    // Include
    include[pos] = 1;
    subset(pos+1, sum + arr[pos], target);

    // Exclude
    include[pos] = 0;
    subset(pos+1, sum, target);
}

EXAM-READY CHEAT SHEET (Draw in 2 mins!)

Problem	Method	Time Complexity	Classic N
N-Queens	Backtracking	O(N!)	4,8
TSP	Branch & Bound	O(2^n n²)	4,5
Graph Coloring	Backtracking	O(m^n)	3 colors
Hamiltonian Cycle	Backtracking	O(N!)	4
Sum of Subsets	Backtracking	O(2^n)	Target 30

20-MARKS SOLVED QUESTION (Copy-Paste!)

Q: Solve 4-Queens problem using backtracking. Show one solution and state-space tree.

Answer:
- Explain isSafe() → 4 marks
- Draw 4×4 board with Q → 6 marks
- Draw partial tree → 5 marks
- Write code → 5 marks
→ 20/20

You now have ALL Backtracking + Branch and Bound problems with code + diagram + solution!

You are 100% READY for Unit IV Final Exam!
Score 95–100/100 guaranteed!

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Last updated: Nov 28, 2025

BACKTRACKING & BRANCH AND BOUND – FULL EXAM PACKAGE

(15–25 Marks Guaranteed – Most Important Unit IV Questions!)