Algorithms on IT Comparison

Memoization vs Tabulation: Top-Down vs Bottom-Up Dynamic Programming

Sun, 02 Aug 2026 23:46:25 +0900

Overview

Both techniques avoid redundant recomputation in dynamic programming problems by caching subproblem results, but they build that cache in opposite directions. Memoization starts from the original problem and recurses downward, caching results lazily as calls return, while tabulation starts from the base cases and iterates upward, filling a table until the final answer is reached.

Comparison Diagram

Comparison Table

Aspect	Memoization (Top-Down)	Tabulation (Bottom-Up)
Entry point	Starts from the original problem call, e.g. solve(n)	Starts from the smallest base cases, e.g. dp[0], dp[1]
Control flow	Recursive function calls, following the natural recurrence	Iterative loop over subproblem indices in dependency order
Subproblems computed	Only those actually reachable from the initial call	All subproblems up to the target, whether needed or not
Storage structure	Hash map or sparse cache keyed by call arguments	Array or table indexed by subproblem parameters
Order dependency	Implicit, resolved automatically by call stack ordering	Explicit, programmer must sort iteration to respect dependencies
Call overhead	Function call and stack frame cost per subproblem	No function call overhead, just array reads and writes
Stack risk	Can hit recursion depth limits or stack overflow on deep inputs	No recursion, so no stack depth concerns
Code shape	Mirrors the recursive definition, often easier to derive first	Requires reformulating the recurrence as an explicit loop

Key Differences

Memoization computes subproblems on demand, while tabulation computes every subproblem exhaustively in order.
Memoization relies on the call stack to sequence work; tabulation relies on a manually ordered loop.
Tabulation typically enables space optimization (e.g. keeping only the last few rows) since access order is known upfront.
Memoization can be stack-limited on deep recursion, whereas tabulation avoids recursion entirely.
Memoization is usually a smaller code change from a naive recursive solution; tabulation requires restructuring it as iteration.

When to Use Each

Memoization (Top-Down)

Recursion vs Iteration: Two Ways to Repeat Work

Sun, 02 Aug 2026 23:45:29 +0900

Overview

Both techniques repeat a computation until some condition is met, but they do it through fundamentally different mechanisms. Recursion repeats by having a function call itself on the call stack, while iteration repeats by looping over the same stack frame. The choice affects memory usage, readability, and how deep a computation can safely go.

Comparison Diagram

Comparison Table

Aspect	Recursion	Iteration
Core mechanism	Function calls itself with a smaller subproblem	Explicit loop (for/while) repeats a block of code
Termination condition	A base case stops further calls	A loop condition evaluates to false
State storage	Held implicitly in parameters and locals of each stack frame	Held explicitly in variables updated each pass
Memory usage	O(n) call stack space, one new frame per call	O(1) auxiliary space, the same frame is reused
Failure mode	Stack overflow on deep or unbounded recursion	Infinite loop if the condition never turns false
Performance overhead	Function-call overhead per level unless tail-call optimized	No call overhead, just a branch/jump back
Best-fit problems	Tree/graph traversal, divide-and-conquer, backtracking	Linear, bounded repetition like counting or array scans

Key Differences

Recursion breaks a problem into self-similar subcalls; iteration repeats a block via an explicit loop.
Recursion grows the call stack by one frame per call; iteration reuses a single frame.
Deep recursion risks stack overflow, while iteration’s main failure mode is an infinite loop.
Some languages apply tail-call optimization to turn tail recursion into iteration under the hood.
Recursion maps naturally onto trees and graphs; iteration suits flat, bounded repetition.

When to Use Each

Recursion

BFS vs DFS: Traversal Order and Data Structure

Sun, 02 Aug 2026 23:44:09 +0900

Overview

Both are algorithms for visiting every node in a tree or graph, but they differ in which node they explore next. BFS spreads outward level by level using a queue, while DFS plunges down one path as far as possible before backtracking, using a stack.

Comparison Diagram

Comparison Table

Aspect	BFS	DFS
Underlying structure	Queue (FIFO)	Stack (FILO), often via recursion
Traversal pattern	Explores all neighbors at current depth before going deeper	Follows one branch to its end before backtracking
Order nodes are visited	Level by level (breadth-first)	Branch by branch (depth-first)
Memory usage	O(width) — can be large for wide/bushy graphs	O(depth) — can be large for deep graphs
Shortest path guarantee	Yes, on unweighted graphs (first visit = shortest path)	No, may find a longer path first
Implementation style	Iterative with explicit queue	Recursive, or iterative with explicit stack
Risk of infinite loop	Low with visited-set on cyclic graphs	Higher on cyclic graphs without visited-set, or infinite depth
Typical use cases	Shortest path, level-order processing, web crawling by hops	Topological sort, cycle detection, maze/backtracking problems

Key Differences

BFS uses a queue and expands outward level by level, while DFS uses a stack (or recursion) and dives deep before backtracking.
BFS guarantees the shortest path on unweighted graphs; DFS does not.
BFS memory cost scales with graph width, while DFS memory cost scales with graph depth.
DFS naturally supports backtracking algorithms like maze solving and topological sort.
Both require a visited set to avoid infinite loops on cyclic graphs.

When to Use Each

BFS

Two Pointers vs Sliding Window: Choosing the Right Array Scanning Technique

Sun, 02 Aug 2026 23:43:16 +0900

Overview

Two Pointers and Sliding Window are both O(n) techniques for scanning arrays or strings, but they solve different shapes of problems: Two Pointers tracks two independent indices that move toward, away from, or alongside each other, while Sliding Window maintains a contiguous subrange that expands and contracts as it scans. Picking the wrong one usually means either overcomplicating a pair-search problem or missing the running aggregate a window naturally provides.

Comparison Diagram

Comparison Table

Aspect	Two Pointers	Sliding Window
Core mechanism	Two independent indices scan or converge across the data	Two indices (left/right) define a contiguous range that grows and shrinks
Pointer movement	Move toward each other, away, or in lockstep at a fixed offset	Right pointer expands the range forward, left pointer contracts it
Input requirement	Usually needs sorted data or a paired structure	Works on any unsorted array or string
State tracked	Just the two positions and the values being compared	A running aggregate of window contents (sum, count, frequency map)
Problem signature	Pair-sum, palindrome check, merging two sorted arrays	Longest/shortest substring or max/min sum under a constraint
Relationship between pointers	No notion of a range between them, only the two positions matter	The range between the pointers IS the answer candidate
Time and space complexity	O(n) time, O(1) space	O(n) time, O(1) to O(k) space for the aggregate
Failure mode	Breaks if data isn’t sorted or orderable for the comparison	Breaks if the target condition isn’t monotonic, so the window can’t shrink safely

Key Differences

Two Pointers tracks two independent indices; Sliding Window tracks a contiguous range between them.
Two Pointers typically requires sorted input; Sliding Window works fine on unsorted sequences.
Sliding Window maintains a running aggregate as it moves; Two Pointers usually just compares individual values.
Sliding Window breaks down when the target condition isn’t monotonic, since the window can’t be safely shrunk.

When to Use Each

Two Pointers