Edit Distance Dynamic Programming Algorithm

Lecture 18 of CS502 Fundamentals of Algorithms details the complete tabular dynamic programming algorithm for computing the Edit Distance between two strings X of length m and Y of length n. A two-dimensional table E[0..m, 0..n] is allocated, where cell E[i, j] represents the minimum edit distance between prefix X[1..i] and prefix Y[1..j]. Boundary conditions are initialized such that E[i, 0] = i (requiring i deletions) and E[0, j] = j (requiring j insertions). For general cells, if characters match (X[i] == Y[j]), the cost is E[i-1, j-1]; otherwise, the cell takes the minimum of three possible transitions plus unit cost: deletion E[i-1, j] + 1, insertion E[i, j-1] + 1, or substitution E[i-1, j-1] + 1. The table is filled row-by-row or column-by-column in strictly O(m * n) time and space. Backtracking from cell E[m, n] back to E[0, 0] reconstructs the exact sequence of optimal edit operations and string alignments. The lesson explores intractable computational problems by examining formal polynomial reductions to known NP-complete cores. Practical code illustrations demonstrate iterative and recursive implementations, memoized table lookups, and pointer-based data structures. Geometric computational techniques including sweep-line approaches are investigated to demonstrate spatial dimension reduction in algorithms. Scholars analyze matrix multiplication parenthesization and edit distance dynamic tables to internalize two-dimensional state memoization. Network flow principles and bipartite matching formulations are examined as extensions of core graph optimization methodologies. The relationship between decision problems, certificate verification, and non-deterministic polynomial time is reinforced through formal definitions. In this lecture, students examine formal computational models, asymptotic behavior, and mathematical proofs of algorithmic correctness. Through detailed theoretical breakdowns, the lesson explores running time recurrence equations, recursion trees, and asymptotic bounds. Special emphasis is placed on worst-case execution bounds, operational space requirements, and optimal substructure formulations. Mastering these foundational algorithmic paradigms empowers software engineers to design highly scalable, computationally efficient software applications. Analytical exercises and algorithmic traces help students prepare thoroughly for Virtual University midterm and final term examinations.