algorithms

A recurring ML-coding screen. The loop itself is easy; what interviewers watch for is k-means++ init, a genuine convergence criterion, and awareness of the failure modes. Here is the compact implementation plus the follow-ups interviewers reliably ask.

A classic that pays off the two-heap insight. A sorted list costs O(n) per insert; two balanced heaps give O(log n) insert and O(1) median. Here is the implementation and the rebalancing detail people get wrong.

Beam search is the decoding algorithm everyone cites and few can code correctly. The signal is keeping k hypotheses by cumulative log-prob, adding logs (not multiplying probs), and handling completed sequences and length. Here is a correct implementation and the tradeoffs.

Autocomplete is the canonical trie question, and it checks whether you reach for the right structure rather than scanning a word list. The signal is O(prefix) lookup, the DFS to gather completions, and the follow-ups (ranking, memory). Here is the build.

A classic that checks whether you reach past sorting for the right structure. The signal is the heap solution (O(n log k)), the bucket-sort O(n) trick, and how it shifts for an unbounded stream. Here is the answer.

Sliding window is one of the highest-yield coding patterns, and this is its canonical problem. The real signal is spotting when a window collapses an O(n squared) scan into one linear pass, and holding the invariant cleanly.

Graph traversal sits beneath a large family of interview problems. What matters is knowing the BFS-vs-DFS tradeoff (shortest path vs memory shape) and applying it cleanly, then getting the directed-vs-undirected cycle gotcha that trips up most candidates.

DP feels like pattern-matching magic until you have a method. The signal is a repeatable drill: spot overlapping subproblems and optimal substructure, define the state and recurrence, then memoize or tabulate. Here is that method applied to a worked example.

BPE powers GPT-style tokenizers, and coding it shows you grasp how subword vocabularies get constructed rather than just knowing they exist. What interviewers watch for is the repeated merge of the most common adjacent pair.

Merging intervals is a staple that probes the sort-then-sweep idea. The tell is sorting by start and merging in a single pass, O(n log n). Below is the pattern along with the family of problems it opens up.

Binary search is simple to explain and notoriously easy to botch (off-by-one, infinite loops). Interviewers look for a correct template plus the variants (boundaries, rotated arrays) that show up constantly. The answer follows.

A warm-up that tests for the stack instinct. Interviewers look for recognizing that last-opened-first-closed maps onto a stack, plus handling the edge cases (leftover opens, early close). The pattern follows.

Merging k sorted sources is the textbook min-heap problem and a genuine data-engineering pattern (merging sorted shards/streams). Interviewers look for the heap of k heads yielding O(N log k). The answer follows.

A classic that pays off the two-pointer insight over brute force. Interviewers look for realizing that water at each position is capped by the min of the max walls on each side, then computing it in a single pass. The answer follows.

The textbook warm-up that tests for the hash-map instinct: swap space for time to turn O(n squared) into O(n). Interviewers look for the one-pass complement lookup. The pattern follows.

Union-Find answers 'are these in the same group?' almost instantly and drives connected-components, cycle detection, and clustering. Interviewers look for path compression plus union by rank for near-O(1) operations. The answer follows.

Edit distance is the archetypal 2D string DP, powering spell-check, diff, and fuzzy matching. What interviewers watch for: naming the subproblem and the insert/delete/replace recurrence. Here is the answer.

Reversing a linked list is the archetypal pointer-handling screen. What interviewers watch for: the three-pointer walk executed without ever dropping the tail of the list, plus why the iterative version wins over recursion in production. Here is the answer.

LCA is the recursion staple that probes how you carry information upward from subtrees in one pass. What interviewers watch for: the post-order 'found in left, found in right' logic. Here is the answer plus the BST shortcut.

Maximum subarray is the entry-point 1D DP problem, and Kadane's is the O(n) one-liner. What interviewers watch for: the 'extend or restart' call at each element and the all-negative edge case most people fumble. Here is the answer.

Word Break is the classic 1D string DP that snags people who reach for greedy or naive recursion. What interviewers watch for: the dp[i] = 'is the prefix of length i segmentable' recurrence, plus why a locally valid split can doom the rest. Here is the answer.

Rotating a matrix in place probes index manipulation and the transpose-then-reverse trick. What interviewers watch for: splitting the rotation into two simple passes instead of juggling four-way swaps. Here is the answer.

Bit-manipulation problems screen for fluency with binary and the O(1)-space tricks most candidates fumble. The tell is whether you reach for XOR's self-canceling property and the n & (n-1) idiom by reflex. Here is the toolkit.

K-th largest has three textbook solutions, and what interviewers watch for is knowing Quickselect's average O(n) beats sorting's O(n log n), why its worst case is O(n²), and when a heap is genuinely the better call. Here is the answer.

The DP that reveals why greedy quietly fails on non-standard denominations, and how a single bottom-up table fixes it. What interviewers watch for: the dp[amount] recurrence and the unbounded-reuse insight.

Gas Station pays off spotting a greedy invariant that turns O(n²) into O(n). Two facts carry the whole solution: total feasibility, and that a failed prefix lets you skip every start within it.

Jump Game pays off spotting that a greedy reachability scan beats DP. Interviewers look for tracking the farthest reachable index in one pass, and catching the exact moment you fall behind it.

LIS has an obvious O(n²) DP and a clever O(n log n) solution that catches people off guard. Interviewers look for the patience-sorting/binary-search approach that maintains 'tails'. Both appear below.

Minimum Window Substring is the hard sliding-window problem that probes expand-and-contract with a character-count map. Interviewers look for the grow-to-valid then shrink-to-minimal pattern, verified in O(1) per step.

House Robber is the tidiest choose-with-a-constraint 1D DP, and it reduces to O(1) space. Interviewers look for the take-this-or-skip recurrence and the two-variable rolling optimization.

Group Anagrams probes the 'compute a canonical key and bucket by it' pattern. Interviewers look for choosing a key that anagrams share without comparing every pair. The answer follows.

This problem outlaws the obvious division trick, forcing the prefix/suffix-product insight. Interviewers look for computing left and right products in two passes with O(1) extra space. The answer follows.

Sorting an array of three values in one pass probes the three-pointer partition (Dutch National Flag). Interviewers look for the low/mid/high invariant and knowing when not to advance mid. The answer follows.

Spiral traversal probes careful boundary management more than algorithmic insight. Interviewers look for shrinking four boundaries layer by layer without double-visiting a row or column. The answer follows.

The textbook grid DP where each cell depends on three neighbors at once. What interviewers want is the min-of-three recurrence, plus the reason it has to be min. The answer follows.

This notoriously hard problem demands better than the O(m+n) merge: an O(log) binary search on the partition. Interviewers look for binary-searching the split point so the left halves stay below the right halves. The answer follows.

Decode Ways is a 1D DP whose difficulty sits entirely in the edge cases: zeros and the valid 1-26 range. Interviewers look for the take-one-digit-or-two recurrence plus disciplined validity checks. The answer follows.

This classic has a tidy O(n^2) expand-around-center solution that outdoes the naive O(n^3). Interviewers look for expanding from every center and handling odd and even lengths separately. The answer follows.

Computing the max of each size-k window the obvious way costs O(n*k). The move that cuts it to O(n) is a monotonic deque of indices, and why it stays linear catches most candidates off guard.

The fewest meeting rooms you need equals the highest number of meetings active simultaneously. That reframing is the entire interview; two O(n log n) solutions drop out of it, and the tie-handling catches careless candidates.

This one rewards knowing that an in-order walk of a BST produces sorted order. The signal is halting at the kth element instead of walking the entire tree.

A row- and column-sorted matrix rewards a single insight: begin at a corner and drop a whole row or column each step. That converts an O(m*n) scan into O(m+n), and which corner you pick is not arbitrary.

Finding the longest run of consecutive integers without sorting divides candidates who reach for O(n log n) from those who spot the hash-set trick. The signal is one loop invariant that eliminates redundant work. Here is the answer.

Course Schedule collapses to a single question: is the prerequisite graph a DAG? The signal is seeing the graph framing and applying topological sort or DFS to detect a cycle. Here is the answer.

Comparing all pairs across a million documents is a trillion comparisons. MinHash estimates Jaccard similarity from a tiny signature, and LSH converts dedup into a near-linear scan. Here is the implementation.

Every chat model's 'temperature' and 'top_p' knobs come down to a few lines of logit surgery. Implementing them proves you understand decoding rather than just calling it. Here is the from-scratch version.

RoPE is why modern LLMs extrapolate to longer contexts, and it works by rotation, not addition. Coding it shows you truly understand how position enters attention. Here is the implementation.

A bad split silently inflates every metric you'll ever report, and the bugs are subtle: leaked groups, shuffled time, preprocessing fit on everything. Here is the split done right.

Many state-of-the-art training runs hold a shadow copy of the weights that averages the trajectory, then serve that rather than the final step. It's a few lines and a genuine accuracy gain. Here it is.

Nearly every modern training run relies on warmup-then-cosine, and getting it wrong destabilizes early training or squanders the tail. It's a closed-form function of the step. Here is the implementation and the why.

Ahead of transformers, this was the language model, and it remains the clearest way to prove you grasp what perplexity truly measures. Smoothing is where people slip up. Here is the implementation.

The textbook variance formula suffers catastrophic precision loss on streaming data and cannot update incrementally. Welford's one-pass update is the norm for online feature normalization. Here it is.

Negative sampling in word2vec and recommendation pulls millions of weighted samples; running each at O(log n) becomes a bottleneck. The alias method turns each draw into O(1) after an O(n) setup. Here it is.

When you shard a vector index or route requests across model replicas, naive 'hash mod N' reshuffles everything the moment N changes. Consistent hashing moves only a small fraction. Here is the implementation.

A Bloom filter answers 'have I seen this?' with a few bits per item rather than storing the key, swapping a rare false positive for a large memory saving. Here is the build, the sizing math, and where it pays off in dedup and serving.

Floyd's tortoise and hare finds a cycle in O(1) space, but what sets strong candidates apart is the second phase: a brief distance argument that identifies exactly where the loop starts. Here is the answer and the math.

Two lists that share a tail have to intersect at a single node, but aligning their lengths without counting is the elegant move interviewers look for. Here is the two-pointer trick that cancels the lengths out, plus the answer.

The two-pass length-then-delete solution works, but the interview asks for a single pass: a gap of N between two pointers, plus a dummy head that makes deleting the real head come out for free. Here is the clean answer.

The diameter is not the tree's height, and the path does not have to pass through the root. The clean solution derives height and best path in one DFS, updating a global max at every node. Here is the answer and the subtlety candidates miss.

Preorder names the root; inorder divides the rest into left and right subtrees. The naive version is O(n squared); the strong answer uses a value-to-index map and a moving preorder pointer for O(n). Here is the answer and why both orders are required.

Negative subtrees, a path that bends through any node, and a return value distinct from the answer you track: this problem stuffs three traps into one DFS. Here is the clean O(n) solution and the reasoning that holds up under follow-ups.

The brute force runs O(n squared) nested scans. A monotonic stack solves it in one O(n) pass by holding only the candidates that can still become a future answer. Here is the pattern and why each element is pushed and popped exactly once.

The naive triple loop runs O(n cubed). Sort first, then fix one element and converge two pointers for O(n squared). The real test is deduplication: returning unique triplets without a set. Here is the clean template.

Merging intervals is the entry point to the whole interval family. The trick is sorting by start so overlaps end up next to each other, then collapsing them in one linear sweep. Here is the pattern and the sweep-line generalization behind it.

The words encode a partial order among characters. Convert each adjacent pair into a directed edge, then topologically sort. The traps are the prefix edge case and spotting contradictions. Here is the full solution with cycle handling.

Treat each word as a node and one-letter edits as edges: the shortest ladder becomes a shortest path in an unweighted graph, which means BFS. The detail that decides pass or fail is producing neighbors in O(26 * L) without scanning the entire dictionary. Here is the pattern.

Testing every subarray costs O(n squared). Prefix sums with a remainder-frequency map solve it in a single O(n) pass: two prefixes sharing a remainder mod k enclose a divisible subarray. The pitfall is negative remainders. Here is the clean answer.

Working out arithmetic with precedence and nested parentheses is a parsing task, and a single stack manages both neatly. The traps are operator precedence, multi-digit numbers, and the sign of truncated division. Here is a one-pass solution.

A Fenwick tree serves prefix-sum queries and point updates in O(log n) using one flat array and a single bit trick. The signal is the lowbit operation and why it wins over a plain prefix array when values change. Here is the answer.

A segment tree serves any associative range query (sum, min, max, gcd) with point or range updates in O(log n). The signal is the recursive split into covered, disjoint, and partial nodes, plus lazy propagation for range updates. Here is the answer.

Dijkstra computes single-source shortest paths in O((V+E) log V) with a min-heap, the backbone of routing and network latency problems. The signal is why it requires non-negative weights and how lazy deletion keeps the heap simple. Here is the answer.

Bellman-Ford copes with negative edge weights that break Dijkstra and detects negative cycles, the foundation of currency-arbitrage problems. The signal is the V-1 relaxation rounds plus the extra Vth round that flags a negative cycle. Here is the answer.

Floyd-Warshall computes shortest paths between every pair of nodes in O(V cubed) with a compact three-line triple loop. The signal is the intermediate-node DP order and why k has to be the outer loop. Here is the answer.

Kruskal builds a minimum spanning tree by sorting edges and adding the cheapest one that does not form a cycle, using union-find for the cycle check. The signal is the greedy cut property and why disjoint-set is the right tool. Here is the answer.

KMP matches a pattern in linear time by precomputing a failure function that skips redundant comparisons rather than backtracking the text. The signal is what the prefix function actually holds and why the text pointer never moves backward. Here is the answer.

Rabin-Karp slides a fixed-width hash across the text so each window costs O(1) to update, which makes it ideal for matching many patterns at once. The signal is the polynomial rolling hash, modular arithmetic, and why you still verify on a hash hit. Here is the answer.

Binary search on the answer cracks minimize-the-maximum and capacity problems by guessing the result and testing feasibility with a monotone predicate. The signal is spotting monotonicity and writing a clean feasibility function. Here is the answer.

Sieving primes, fast modular power, and Euclid's gcd are the number-theory primitives that quietly drive crypto, hashing, and combinatorics problems. The signal is the O(n log log n) sieve and O(log e) binary exponentiation. Here is the answer.

Matrix exponentiation evaluates linear recurrences like Fibonacci at index n in O(log n) by raising a transition matrix to the nth power through binary exponentiation. The signal is building the transition matrix and squaring it. Here is the answer.

A time-based key-value store logs versioned writes and answers 'what was the value at time t' via binary search over per-key timestamps. The signal is the append-only design and the floor (largest timestamp <= t) lookup. Here is the answer.