Spiral Matrix: Layer-by-Layer Traversal

Matrix problems require careful index management and often reduce to graph traversal or dynamic programming on a 2D grid. For this problem, we treat the matrix as a structured 2D array where each cell can interact with its neighbors in specific ways. Common patterns include DFS or BFS traversal treating cells as graph nodes connected to their four or eight neighbors. Spiral traversal uses boundary tracking with top, bottom, left, and right pointers that contract inward after each layer is processed. Matrix rotation and transposition involve systematic swapping of elements: rotating 90 degrees clockwise equals transposing followed by reversing each row. For DP on matrices, the state is typically dp[i][j] representing the answer for the subproblem ending at cell (i,j), with transitions from adjacent cells. Prefix sum matrices enable O(1) range sum queries after O(mn) preprocessing. When working with matrices in interviews, always clarify whether the matrix is square or rectangular, whether indices are zero-based, and what values cells can contain. Watch for off-by-one errors when computing boundaries. Use direction arrays like dx=[0,0,1,-1] and dy=[1,-1,0,0] for clean traversal code. Common problems include spiral order, rotation, search in sorted matrices, island counting, and path finding.