L29 · Graphs — BFS, DFS (Bangla)

1. Graphs Are Everywhere

Social network, road map, web pages, dependency — vertex (জিনিস) + edge (সম্পর্ক) থাকলেই graph।

Figure 29.1 — 5-vertex undirected graph।

2. Two Representations

Adjacency Matrix

int adj[V][V];
adj[u][v] = 1; // u→v

O(V²) space; O(1) edge-check; dense graph-এ ভালো।

Adjacency List

Node *adj[V];
// linked list of neighbors

O(V+E) space; O(deg v) neighbors; sparse graph-এ preferred।

3. BFS — Breadth-First Search

bfs.c

#include <stdio.h>
#include <string.h>

#define V 6

int adj[V][V];      // adjacency matrix

void add_edge(int u, int v) { adj[u][v] = adj[v][u] = 1; }

void bfs(int start) {
    int visited[V] = {0};
    int q[V], head = 0, tail = 0;

    q[tail++] = start;
    visited[start] = 1;

    while (head < tail) {
        int u = q[head++];
        printf("%d ", u);
        for (int v = 0; v < V; v++)
            if (adj[u][v] && !visited[v]) {
                visited[v] = 1;
                q[tail++] = v;
            }
    }
    putchar('\n');
}

int main(void) {
    add_edge(0,1); add_edge(0,2);
    add_edge(1,3); add_edge(2,4);
    add_edge(3,5); add_edge(4,5);

    printf("BFS from 0: ");
    bfs(0);
    return 0;
}

BFS unweighted graph-এ shortest path দেয় (edges গোনার দিক থেকে)। Queue-ই মূল hardware।

4. DFS — Depth-First Search

dfs.c

#include <stdio.h>

#define V 6

int adj[V][V];
int visited[V];

void add_edge(int u, int v) { adj[u][v] = adj[v][u] = 1; }

void dfs(int u) {
    visited[u] = 1;
    printf("%d ", u);
    for (int v = 0; v < V; v++)
        if (adj[u][v] && !visited[v]) dfs(v);
}

int main(void) {
    add_edge(0,1); add_edge(0,2);
    add_edge(1,3); add_edge(2,4);
    add_edge(3,5); add_edge(4,5);

    printf("DFS from 0: ");
    dfs(0);
    putchar('\n');
    return 0;
}

Recursion stack = DFS stack। Iterative version-এ নিজে stack maintain করা যায়।

5. Key Algorithms to Know

Connected components — প্রতিটি unvisited vertex থেকে DFS।
Cycle detection — DFS recursion stack।
Topological sort — DFS post-order (DAG-এ)।
Unweighted shortest path — BFS।
Weighted (non-negative) — Dijkstra।
General weighted — Bellman-Ford।
MST — Kruskal / Prim।

6. Practice Problems

Build an adjacency-list graph.

Adjacency list দিয়ে graph।

✨ Show Answer

typedef struct N { int v; struct N *next; } N;
N *adj[V];
void add(int u, int v) {
    N *n = malloc(sizeof *n); n->v = v; n->next = adj[u]; adj[u] = n;
}

BFS from a source.
Source থেকে BFS।

✨ Show Answer
Section 3-এর bfs.c-ই উত্তর।
DFS recursive & iterative.
DFS — recursive ও iterative।

✨ Show Answer
Recursive Section 4। Iterative: stack ব্যবহার করুন — push(start), loop: pop → visit → push all unvisited neighbors।
Count connected components.
Connected component গুনুন।

✨ Show Answer
প্রতিটি unvisited vertex থেকে DFS চালান, counter বাড়ান — total component count।
Detect a cycle in an undirected graph.
Undirected graph-এ cycle detect।

✨ Show Answer
DFS-এ parent track করুন। Neighbor visited + parent না হলে cycle।
Detect a cycle in a directed graph.
Directed graph-এ cycle।

✨ Show Answer
Three-color DFS: WHITE (unvisited), GRAY (in-progress), BLACK (done)। GRAY-এ ফেরত এলে cycle।
Topological sort of a DAG.
DAG-এর topological sort।

✨ Show Answer
DFS post-order-এ stack-এ push; শেষে stack reverse-ই topological order।
Unweighted shortest path (BFS).
Unweighted shortest path।

✨ Show Answer
BFS-এ dist[v] = dist[u] + 1 set করুন; target-এ পৌঁছালে return।
Maze / grid shortest path (2D BFS).
Grid-এ shortest path।

✨ Show Answer
Queue-এ (r, c) push। চারটি dr/dc neighbors দেখুন — bounds + visited + cell-ok চেক।
Island count in a 2D grid.
2D grid-এ island সংখ্যা।

✨ Show Answer
প্রতিটি '1' cell থেকে DFS/BFS; counter বাড়ান; visited চিহ্নিত করুন।
Flood fill.
Flood fill।

✨ Show Answer
DFS recursion: current cell-এর color পরিবর্তন করুন এবং চার দিকে recurse (source color match-এ)।
Bipartite check via 2-coloring BFS.
BFS দিয়ে bipartite check।

✨ Show Answer
Source-কে color 0 দিন; BFS-এ প্রতিটি neighbor-কে opposite color। Conflict হলে not bipartite।
Dijkstra with priority queue.
Dijkstra — priority queue।

✨ Show Answer
Min-heap-এ (dist, vertex)। Pop; relax edges; updated হলে push। O((V+E) log V)।
Bellman-Ford with negative edges.
Bellman-Ford।

✨ Show Answer
V−1 বার সব edge relax। V-th iteration-এ update হলে negative cycle। O(VE)।
Floyd-Warshall all-pairs shortest paths.
Floyd-Warshall।

✨ Show Answer
Triple loop: for k for i for j: d[i][j] = min(d[i][j], d[i][k] + d[k][j]); — O(V³)।
Kruskal MST with union-find.
Kruskal MST।

✨ Show Answer
Edges sort করুন weight-এ; union-find ব্যবহার করে একটি একটি edge যোগ করুন — cycle না হলে।
Prim MST with priority queue.
Prim MST।

✨ Show Answer
Min-heap-এ (cost, vertex)। Extract-min; unvisited হলে include; তার edges push।
Number of paths from A to B in a DAG.
DAG-এ A থেকে B পর্যন্ত path সংখ্যা।

✨ Show Answer
Topological order-এ paths[B] += paths[u], যেখানে u → B। DP।
Longest path in a DAG.
DAG-এ longest path।

✨ Show Answer
Topological order-এ DP: dist[v] = max(dist[v], dist[u] + w(u,v))।
Strongly connected components (Kosaraju).
SCC — Kosaraju।

✨ Show Answer
(1) DFS finish order stack-এ রাখুন। (2) Graph reverse। (3) Reversed graph-এ stack order থেকে DFS — প্রতিটি DFS-tree একটি SCC।
Articulation points and bridges.
Articulation points ও bridges।

✨ Show Answer
DFS-এ disc[] ও low[] maintain করুন। Node u-এর কোনো child v-এর low[v] >= disc[u] মানে u articulation point; low[v] > disc[u] মানে edge u-v bridge।
Word ladder (BFS over edit graph).
Word ladder — BFS।

✨ Show Answer
Vertex = word; edge = এক-অক্ষর পার্থক্য। BFS দিয়ে shortest transformation।
Knight's shortest path on a chessboard.
দাবার ঘোড়ার shortest path।

✨ Show Answer
8-move BFS on 8×8 grid; visited[8][8]। Queue-এ (r, c, dist)।
A* search intuition — what heuristic makes it consistent?
A*-এর heuristic কখন consistent?

✨ Show Answer
Consistent (monotone): সকল edge (u, v)-র জন্য h(u) ≤ c(u,v) + h(v)। গ্রিডে Manhattan বা Euclidean distance consistent। এই condition-এ একবার close-list-এ গেলে আর re-open লাগে না।
Why is almost every interesting CS problem really a graph problem in disguise?
CS-এর প্রায় সব সমস্যা কেন graph-এর সমস্যা?

✨ Show Answer
যেকোনো "state → state" transition mental model-ই একটি graph — state হলো vertex, legal move হলো edge। Dijkstra, BFS, DFS, topo sort — এই সামান্য primitive দিয়েই অজস্র সমস্যা সমাধান সম্ভব।

Glossary (শব্দকোষ)

Term	Meaning	বাংলায়
Graph	A collection of vertices connected by edges.	Vertex ও edge দিয়ে গঠিত structure।
Vertex (Node)	A point in the graph.	Graph-এর একটি বিন্দু।
Edge	A connection between two vertices.	দুই vertex-এর মধ্যে সংযোগ।
Directed Graph	Edges have direction (one-way).	Edge-এর নির্দিষ্ট দিক আছে।
Undirected Graph	Edges have no direction.	Edge-এর দিক নেই।
Weighted Graph	Edges carry numeric weights.	Edge-এ সংখ্যাগত ওজন।
Adjacency Matrix	2D array where `A[i][j]` means edge i→j.	2D array — `A[i][j]` মানে i→j edge।
Adjacency List	Each vertex stores its list of neighbors.	প্রতি vertex তার neighbor-list রাখে।
BFS	Breadth-First Search — explore neighbors level by level using a queue.	Queue দিয়ে level ধরে ধরে অনুসন্ধান।
DFS	Depth-First Search — go as deep as possible using recursion or a stack.	Recursion/stack দিয়ে যত গভীরে সম্ভব যাওয়া।
Connected Component	A maximal set of mutually reachable vertices.	পরস্পর-পৌঁছনো-যোগ্য vertex-এর সর্বোচ্চ set।
Cycle	A path that returns to its starting vertex.	শুরুতে ফিরে আসা path।
Topological Sort	Linear ordering of a DAG respecting edges.	DAG-এর edge মেনে linear ক্রম।
Shortest Path	Path with minimum total edge weight or count.	সর্বনিম্ন ওজন বা edge-সংখ্যার path।

Graphs — Representation, BFS, DFS