Week 8: Graph Algorithms — Routes, Paths & Adjacency Matrices

Acceptance Standard: This chapter serves as a stand-alone textbook for any beginner studying IIT Madras BSCS1001. No video lecture or external slide deck is required to achieve complete conceptual mastery, trace every loop execution, and solve all eight recurring exam question patterns.

1. Orientation & Mental Model

The Real-World Scenario: The Railway Transit Grid

(Chennai) ----[Train 101]----> (Bangalore) ----[Train 105]----> (Delhi)
    \                              |                              ^
     \                             | [Train 102]                 /
      \                            v                            /
    [Train 103]                (Mumbai) -----[Train 104]-------/
        \                          ^
         \                        /
          +----------------------+

1. Direct Transit (0 Changes / 1 Hop): Is there a single train that leaves Chennai and terminates in Delhi without requiring you to switch trains?
2. One Change (1 Stopover / 2 Hops): Can you reach Delhi by switching trains exactly once at an intermediate junction (e.g., Chennai $\to$ Bangalore $\to$ Delhi, or Chennai $\to$ Mumbai $\to$ Delhi)?
3. Multi-Hop Reachability: If you have infinite time, can you reach Delhi at all through any valid sequence of trains?

• Vertices (Nodes): The stations (Chennai, Bangalore, Mumbai, Delhi).
• Directed Edges (Arcs): The train services flowing strictly from origin to destination.
• Paths / Walks: The sequences of edges traversed from start to finish.

The Computational Breakthrough: Algebraic Graph Theory

• Store direct connections in a square grid of 0s and 1s called the Adjacency Matrix $A$.
• To find all 2-step routes, you do not need recursive path searching; you simply compute the matrix product $A \times A = A^2$.
• The entry $(A^k)[i][j]$ does not merely tell you if you can travel from station $i$ to station $j$ in $k$ steps—it counts the exact number of distinct $k$-step routes available!

2. Terminology, Notation & Etymology

3. Foundational Mechanics: Representing Graphs as Matrices

• 0: Chennai
• 1: Bangalore
• 2: Mumbai
• 3: Delhi

1. Train 101: Chennai $\to$ Bangalore ($0 \to 1$)
2. Train 102: Bangalore $\to$ Mumbai ($1 \to 2$)
3. Train 103: Chennai $\to$ Mumbai ($0 \to 2$)
4. Train 104: Mumbai $\to$ Delhi ($2 \to 3$)
5. Train 105: Bangalore $\to$ Delhi ($1 \to 3$)

Step 1: Constructing the Direct Adjacency Matrix $A$

        To:   0 (Chn)  1 (Blr)  2 (Mum)  3 (Del)
From:
0 (Chn)          0        1        1        0
1 (Blr)          0        0        1        1
2 (Mum)          0        0        0        1
3 (Del)          0        0        0        0

The Adjacency Matrix Construction Algorithm (Pseudocode)

pseudo
Procedure BuildAdjacencyMatrix(trains, stationList)
    n = length(stationList)
    
    // Initialize an n x n matrix with zeros
    A = Create2DArray(n, n, 0)
    
    // Populate indices map: stationName -> index
    stnIndex = Dictionary()
    for i from 0 to n - 1 do
        stnIndex[stationList[i]] = i
    end for
    
    // Insert direct directed edges
    for each train in trains do
        u = stnIndex[train.start]
        v = stnIndex[train.end]
        A[u][v] = 1
    end for
    
    return A
End Procedure

4. Algorithmic Depth & Trace: From Direct Edges to Multi-Hop Paths

Layer 1: The One-Hop (2-Edge) Transfer Algorithm

1. There must be a direct train from $i$ to $k$: A[i][k] == 1
2. There must be a direct train from $k$ to $j$: A[k][j] == 1

pseudo
Procedure ComputeTwoHopRoutes(A, n)
    // Initialize 2-hop matrix
    TwoHop = Create2DArray(n, n, 0)
    
    for i from 0 to n - 1 do           // Origin station
        for j from 0 to n - 1 do       // Destination station
            for k from 0 to n - 1 do   // Candidate transfer junction
                if A[i][k] == 1 and A[k][j] == 1 then
                    TwoHop[i][j] = 1   // Route confirmed through junction k
                end if
            end for
        end for
    end for
    
    return TwoHop
End Procedure

Layer 2: The Algebraic Derivation of Matrix Powers

The Fundamental Theorem of Graph Powers

• $(A^2)[0][0] = (0\cdot 0) + (1\cdot 0) + (1\cdot 0) + (0\cdot 0) = 0$
• $(A^2)[0][1] = (0\cdot 1) + (1\cdot 0) + (1\cdot 0) + (0\cdot 0) = 0$
• $(A^2)[0][2] = (0\cdot 1) + (1\cdot 1) + (1\cdot 0) + (0\cdot 0) = 1$ (via Bangalore: Chennai $\to$ Bangalore $\to$ Mumbai)
• $(A^2)[0][3] = (0\cdot 0) + (1\cdot 1) + (1\cdot 1) + (0\cdot 0) = 2$ (Two distinct paths: Chennai $\to$ Bangalore $\to$ Delhi AND Chennai $\to$ Mumbai $\to$ Delhi)

• $(A^2)[1][3] = (0\cdot 0) + (0\cdot 1) + (1\cdot 1) + (1\cdot 0) = 1$ (via Mumbai: Bangalore $\to$ Mumbai $\to$ Delhi)

Layer 3: Computing 3-Hop Paths ($A^3$)

5. Failure Modes & Common Exam Pitfalls

Common Student Misconception:
"If A^2[i][j] > 0, then the shortest path from i to j is 2."
                |
                +---> WRONG!
                      A direct edge A[i][j] = 1 might also exist!
                      Shortest path is min { k : A^k[i][j] > 0 }.

Pitfall 1: Confusing Reachability ($T$) with Route Count ($A^k$)

• Matrix power $A^k$: An arithmetic integer matrix where entries represent the number of walks.
• Boolean Reachability $R_k$: A binary matrix where $R_k[i][j] = 1$ if $(A^k)[i][j] \ge 1$, else $0$.
• Transitive Closure $T$: The cumulative logical OR across all possible path lengths: $T = A \lor A^2 \lor A^3 \lor \dots \lor A^{n-1}$

Pitfall 2: The Two-Way Edge / Bounce-Back Walk Trap

Pitfall 3: Off-By-One Transfer Counting

• Problem statement: "Find all pairs of stations reachable with at most 2 train changes."
• Correct computation: 2 changes = 3 hops. The answer is the logical OR of $A \lor A^2 \lor A^3$.
• Common blunder: Computing only $A^2$ (which corresponds to 1 change / 2 hops).

6. Question-Pattern Anatomy (8 Finite Exam Shapes)

Shape 1: Direct Verification of Adjacency

• Cue: "Given the edge list, find the entry $A[i][j]$ or total number of non-zero entries."
• Attack Sequence:
  1. Count total directed edges $m$. The number of 1s in $A$ is exactly $m$.
  2. For $A[i][j]$, verify if directed edge $i \to j$ exists.
• Planted Trap: Mistaking an undirected edge for 1 entry instead of 2 symmetric entries $A[i][j] = A[j][i] = 1$.

Shape 2: In-Degree and Out-Degree from Matrix Rows/Cols

• Cue: "Which station has no outgoing trains? / Which vertex is a sink?"
• Attack Sequence:
  • $\text{Out-degree}(i) = \text{Row sum of row } i$. If $\sum_j A[i][j] = 0$, vertex $i$ is a sink.
  • $\text{In-degree}(j) = \text{Column sum of col } j$. If $\sum_i A[i][j] = 0$, vertex $j$ is a source.
• Planted Trap: Summing the column when asked for out-degree. Row = Outgoing, Column = Incoming.

Shape 3: Number of Paths of Length 2 Between Two Nodes

• Cue: "How many 2-hop paths exist from vertex $u$ to vertex $v$?"
• Attack Sequence: Compute the dot product of row $u$ and column $v$: $(A^2)[u][v] = \sum_{k=0}^{n-1} A[u][k] \cdot A[k][v]$ Count how many indices $k$ satisfy $A[u][k] == 1$ AND $A[k][v] == 1$.
• Planted Trap: Counting junctions where an edge exists to $k$, but no edge exists from $k$ to $v$.

Shape 4: Shortest Route Identification

• Cue: "What is the minimum number of hops required to travel from $u$ to $v$?"
• Attack Sequence:
  1. Check if $A[u][v] == 1$ $\to$ 1 hop.
  2. Check if $(A^2)[u][v] > 0$ $\to$ 2 hops.
  3. Check if $(A^3)[u][v] > 0$ $\to$ 3 hops.
• Planted Trap: Choosing an answer based on geographic distance instead of topological edge count.

Shape 5: Detecting Cycles via Diagonal Entries

• Cue: "Does the graph contain a directed triangle (cycle of length 3)?"
• Attack Sequence: Compute the diagonal of $A^3$: $\text{Cycle of length 3 through vertex } i \iff (A^3)[i][i] > 0$ The total number of directed triangles in the graph is $\frac{1}{3} \sum_{i=0}^{n-1} (A^3)[i][i]$.
• Planted Trap: Forgetting to divide by 3 when counting undirected triangles.

Shape 6: Reachability Within $K$ Steps

• Cue: "Find the reachability matrix for all journeys with at most 2 hops."
• Attack Sequence: Compute $R_{\le 2} = A \lor A^2$ (or $I \lor A \lor A^2$ if zero-hop self-reachability is included).
• Planted Trap: Computing $A \times A^2 = A^3$ instead of matrix addition/boolean OR.

Shape 7: Loop Termination in Reachability Exploration

• Cue: "In an $n$-vertex graph, what is the maximum power of $A$ needed to check all simple paths?"
• Attack Sequence: A simple path in an $n$-vertex graph visits each vertex at most once, meaning it has at most $n-1$ edges. Therefore, the search stops at $A^{n-1}$.
• Planted Trap: Answering $n$ or $n^2$ instead of $n-1$.

Shape 8: Pseudocode Tracing of 3-Nested Matrix Loops

• Cue: "What is the final value of count in the given 3-nested loop?"
• Attack Sequence: Identify if the inner loop computes matrix multiplication ($A \times B$) or transitive closure (Warshall-like update). Trace the outer loop index order: $(i, j, k)$ vs $(k, i, j)$.
• Planted Trap: Swapping the roles of $k$ as intermediate junction with $j$ as destination.

7. Interactive Verification & Attempt-First Self-Checks

pseudo
Procedure PathCheck(A, n, startNode, endNode)
    visited = CreateArray(n, False)
    frontier = [startNode]
    visited[startNode] = True
    hops = 0
    
    while length(frontier) > 0 do
        nextFrontier = []
        for each u in frontier do
            if u == endNode then
                return hops
            end if
            for v from 0 to n - 1 do
                if A[u][v] == 1 and not visited[v] then
                    visited[v] = True
                    Append(nextFrontier, v)
                end if
            end for
        end for
        frontier = nextFrontier
        hops = hops + 1
    end while
    return -1
End Procedure

8. Next Action & Spacing Rail

• Next Invariant Chapter: Proceed to Week 9: Recursion & Call Stacks (Layered) to master how recursive stack frames traverse branched decision trees.
• Direct Application in Week 9 DFS: Notice how PathCheck uses an iterative level frontier (BFS). In Week 9, we convert this into a recursive call stack to perform Depth-First Search (DFS).
• Cross-Course Connection: In Mathematics I (BSMA1001), see how eigenvalues of adjacency matrix $A$ define graph connectivity and spectral clustering.